Finite Group Databasebeta

← #29723971

Subgroups — #29723971

Order 15616.

958 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839922C211yesmaximal, characteristic
2C2×#16839854C211yesmaximal, characteristic, complemented
3#21449318C211yesmaximal, characteristic, complemented
4#21448656C211yesmaximal, characteristic
5#21447341C211yesmaximal, characteristic, complemented
6#21447250C211yesmaximal, characteristic, complemented
7#21447133C211yesmaximal, characteristic, complemented
8#21446500C211yesmaximal, characteristic, complemented
9#21446428C211yesmaximal, characteristic, complemented
10#21446172C211yesmaximal, characteristic, complemented
11#21446157C211yesmaximal, characteristic, complemented
12#21446084C211yesmaximal, characteristic, complemented
13#21445833C211yesmaximal, characteristic, complemented
14#21445760C211yesmaximal, characteristic
15C61×#2035C211yesmaximal, characteristic, complemented
16C22×#11758222C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C22×#11758201C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22#16839922—24no—
23#16839854—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×C22⋊#92418C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839701—21no—
35#16839687—21no—
36#16839669C2211yescharacteristic
37#16839663C2211yescharacteristic
38#16839663C2211yescharacteristic
39#16839662—21no—
40#16839662—21no—
41#16839658C2211yescharacteristic
42#16839650—21no—
43#16839643—21no—
44#16839627C2211yescharacteristic
45#16839622C2211yescharacteristic
46#16839621—21no—
47C4×C22⋊Dic61C2211yescharacteristic
48#16839609—21no—
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C22⋊Q8×C61C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C2×#11758222—22no—
62C2×#11758222—22no—
63C2×#11758222—22no—
64C2×#11758222—22no—
65C2×#11758222—41no—
66C2×#11758222—41no—
67C22×Dic122C2311yescharacteristic
68C22×D244C2311yescharacteristic
69C22×C22⋊D61—21no—
70C22×C4×D61—21no—
71C2×#11758202—22no—
72C2×#11758202—22no—
73C2×#11758202D414yes—
74C2×#11758201—22no—
75C2×#11758201—22no—
76C2×#11758201—41no—
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79Q8×Dic61—22no—
80Q8×Dic61—41no—
81#14519742—22no—
82#14519742—22no—
83#14519742—41no—
84#14519742—41no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61D411yescharacteristic
93C2×C22⋊Dic61D411yescharacteristic
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103#14519710—22no—
104#14519706—41no—
105#14519702—22no—
106#14519702—22no—
107#14519700—22no—
108#14519699—22no—
109#14519699—41no—
110#14519697—22no—
111C22⋊#92418—22no—
112C8×Dic61—22no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684D411yescharacteristic
120#14519684D411yescharacteristic
121C22×Q8×C61C2311yescharacteristic
122C23×C4×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22⋊Q8×C61—22no—
126C22⋊Q8×C61—41no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C22⋊C4×C61—21no—
132C2×C22⋊C4×C61—21no—
133#106×C61—41no—
134#103×C61—22no—
135#103×C61—41no—
136C22⋊C8×C61—41no—
137#95×C61—21no—
138C24×C61C2×D411yescharacteristic
139C23×D61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61C2×D412yes—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C2×C22⋊D61—24no—
159C2×C22⋊D61—42no—
160C2×C22⋊D61—42no—
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168#11758222—44no—
169#11758222—44no—
170#11758222—44no—
171#11758222—44no—
172C2×D244—21no—
173C2×D244—21no—
174C2×D244—21no—
175C2×D244—21no—
176C2×D244—21no—
177C2×D244#4112yes—
178C2×C4×D61—24no—
179C2×C4×D61—42no—
180C2×C4×D61—42no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122C2×D412yes—
185C2×Dic122#4112yes—
186C2×Dic122#4112yes—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—42no—
200C22⋊C4×C61—42no—
201C22⋊C4×C61—81no—
202C22⋊Dic61—22no—
203C22⋊Dic61—41no—
204C22⋊Dic61—42no—
205#11758202—28no—
206#11758201—44no—
207#11758198—42no—
208#11758198—42no—
209#11758198—81no—
210#11758198—81no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—42no—
216#11758196—42no—
217#11758196—42no—
218#11758196—42no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61—42no—
224C4×Dic61—42no—
225C4×Dic61—42no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418C2×D412yes—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61—21no—
236C23×C61—21no—
237C23×C61—21no—
238C23×C61—21no—
239C23×C61C22×D411yescharacteristic
240C22×D61—22no—
241C22×D61—41no—
242C22×D61—41no—
243C22×D61—41no—
244C22×D61—41no—
245C22×D61—41no—
246C22×D61—41no—
247Q8×C61—22no—
248Q8×C61—22no—
249Q8×C61—41no—
250Q8×C61—41no—
251C2×C4×C61—21no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61H(5,𝔽2)11yescharacteristic
275C2×C4×C61H(5,𝔽2)11yescharacteristic
276C22⋊D61—48no—
277C2×Dic61—22no—
278C2×Dic61—21no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—21no—
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—21no—
285C2×Dic61—21no—
286C2×Dic61—21no—
287C2×Dic61—21no—
288C2×Dic61—21no—
289C2×Dic61—21no—
290C2×Dic61—21no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—41no—
296C2×Dic61—41no—
297C2×Dic61#13512yes—
298D244—22no—
299D244—22no—
300D244—41no—
301D244—41no—
302C4×D61—48no—
303Dic122—22no—
304Dic122—22no—
305Dic122—22no—
306Dic122#12314yes—
307C8×C61—41no—
308C8×C61—41no—
309#92418—22no—
310#92418—41no—
311#24908—611nomaximal, Sylow
312C22×C61—41no—
313C22×C61—41no—
314C22×C61—41no—
315C22×C61—41no—
316C22×C61—41no—
317C22×C61—41no—
318C22×C61—41no—
319C22×C61#52511yescharacteristic
320C22×C61D4211yescharacteristic
321C22×C61#54611yescharacteristic
322C22×C61C2×#13511yescharacteristic
323C22×C61#57511yescharacteristic
324C22×C61#57711yescharacteristic
325C22×C61#57811yescharacteristic
326C2×D61—42no—
327C2×D61—42no—
328C2×D61—42no—
329C2×D61—81no—
330C2×D61—81no—
331C2×D61—81no—
332C2×D61—81no—
333C4×C61—41no—
334C4×C61—42no—
335C4×C61—81no—
336C4×C61—81no—
337C4×C61—81no—
338C4×C61C2×#12311yescharacteristic
339C4×C61#53811yescharacteristic
340C4×C61#54611yescharacteristic
341C4×C61#55111yescharacteristic
342Dic61—24no—
343Dic61—22no—
344Dic61—41no—
345Dic61—41no—
346Dic61—41no—
347Dic61—81no—
348Dic61—81no—
349C2×#452—611no—
350C2×C4×Q16—611no—
351#2421—611no—
352#2341—611no—
353#2092—611no—
354#2076—611no—
355#2055—611no—
356#2035D6111yescharacteristic, complemented
357#1934—611no—
358#1923—611no—
359#1880—611no—
360#1878—611no—
361#1865—611no—
362#1805—611no—
363#1788—611no—
364C2×C61—81no—
365C2×C61#241811yescharacteristic
366C2×C61#310211yescharacteristic
367C2×C61#312011yescharacteristic
368C2×C61#327111yescharacteristic
369C2×C61#328511yescharacteristic
370C2×C61#330411yescharacteristic
371C2×C61#332711yescharacteristic
372D61—82no—
373C22×#41—611no—
374C22×Q16—611no—
375C22×SD16—611no—
376C2×C22⋊Q8C2×D6111yescharacteristic
377C2×C4×Q8—611no—
378C2×C4×Q8—611no—
379C2×C4×Q8—611no—
380#452—1224no—
381C4×Q16—1224no—
382C2×#107—611no—
383C2×#106C2×D6111yescharacteristic
384C2×C4⋊C8—611no—
385C2×#103—611no—
386C2×#103—611no—
387C2×#103—611no—
388C2×#103C2×D6111yescharacteristic
389C2×#103C2×D6111yescharacteristic
390C2×#102—611no—
391C2×#102—611no—
392C2×#102—611no—
393C2×C22⋊C8—611no—
394C2×C22⋊C8C2×D6111yescharacteristic
395C2×C4×C8—611no—
396#400—1221no—
397#397—1221no—
398#391—611no—
399#390—611no—
400#390—611no—
401#390—1221no—
402#390—1221no—
403#389—611no—
404#388—1221no—
405#387—1221no—
406#382—611no—
407#380—611no—
408#380—1221no—
409#380C2×D6111yescharacteristic
410C4×C22⋊C4—611no—
411C4×C22⋊C4—1221no—
412#340—611no—
413#340—611no—
414#340—611no—
415#340C2×D6111yescharacteristic
416#336—611no—
417#336—611no—
418C61#2490811yescharacteristic, complemented, Sylow
419C2×#41—1222no—
420C2×#41—1222no—
421C2×#41—1222no—
422C2×#41—1222no—
423C2×#41—2441no—
424C2×#41—2441no—
425C22×Q8—611no—
426C22×Q8C22×D6111yescharacteristic
427C22×D4—611no—
428C22×D4—1221no—
429C23×C4—1221no—
430C23×C4C22×D6111yescharacteristic
431C2×Q16—614no—
432C2×Q16—1222no—
433C2×Q16—1222no—
434C2×SD16—1222no—
435C2×SD16—1222no—
436C2×SD16—2441no—
437C22×C8—611no—
438C22×C8—611no—
439C22×C8C22×D6111yescharacteristic
440C22×C8C22×D6111yescharacteristic
441C22⋊Q8—22no—
442C22⋊Q8—41no—
443C4×Q8—1222no—
444C4×Q8—1222no—
445C4×Q8—1222no—
446C4×Q8—2441no—
447C4×Q8—2441no—
448C4×Q8—2441no—
449C2×C4⋊C4—21no—
450C2×C4⋊C4—611no—
451C2×C4⋊C4—611no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4C22×D6111yescharacteristic
457C2×C4⋊C4C22×D6111yescharacteristic
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C22⋊C4—21no—
460C2×C22⋊C4—21no—
461C2×C22⋊C4—611no—
462C2×C22⋊C4—611no—
463C2×C22⋊C4—1221no—
464C2×C22⋊C4—1221no—
465C2×C22⋊C4—1221no—
466C2×C22⋊C4—1221no—
467C2×C22⋊C4—1221no—
468C2×C42—611no—
469C2×C42—611no—
470C2×C42—1221no—
471C2×C42—1221no—
472C2×C42—1221no—
473#107—1222no—
474#106—41no—
475C4⋊C8—2441no—
476#103—22no—
477#103—41no—
478#103—1222no—
479#103—1222no—
480#103—1222no—
481#102—1222no—
482#102—1222no—
483#102—2441no—
484C22⋊C8—41no—
485C22⋊C8—1222no—
486C4×C8—1222no—
487#95—21no—
488#95—611no—
489#95—611no—
490#95—1221no—
491#95—1221no—
492#95—1221no—
493#95—1221no—
494#95—1221no—
495#95—1221no—
496C24—1221no—
497C24D4×D6111yescharacteristic
498#41—2444no—
499#41—2444no—
500#41—2444no—
501#41—2444no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—21no—
507C2×Q8—612no—
508C2×Q8—612no—
509C2×Q8—612no—
510C2×Q8—1221no—
511C2×Q8—1221no—
512C2×Q8—1221no—
513C2×Q8D4×D6112yes—
514C2×D4—612no—
515C2×D4—1224no—
516C2×D4—1221no—
517C2×D4—1221no—
518C2×D4—1221no—
519C2×D4—1221no—
520C2×D4—1221no—
521C2×D4—2442no—
522C2×D4—2442no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—21no—
529C22×C4—21no—
530C22×C4—21no—
531C22×C4—21no—
532C22×C4—21no—
533C22×C4—611no—
534C22×C4—611no—
535C22×C4—611no—
536C22×C4—611no—
537C22×C4—1224no—
538C22×C4—1221no—
539C22×C4—1221no—
540C22×C4—1221no—
541C22×C4—2442no—
542C22×C4—2442no—
543C22×C4D4×D6111yescharacteristic
544C22×C4C23×D6111yescharacteristic
545Q16—1228no—
546SD16—2444no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—612no—
554C2×C8—1221no—
555C2×C8—1221no—
556C2×C8—1221no—
557C2×C8—1221no—
558C2×C8—1221no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—42no—
566C4⋊C4—1221no—
567C4⋊C4—1221no—
568C4⋊C4—1221no—
569C4⋊C4—1221no—
570C4⋊C4—2442no—
571C4⋊C4—2442no—
572C4⋊C4—2442no—
573C4⋊C4—2442no—
574C22⋊C4—42no—
575C22⋊C4—81no—
576C22⋊C4—1222no—
577C22⋊C4—4881no—
578C22⋊C4—4881no—
579C22⋊C4—2442no—
580C22⋊C4—2442no—
581C22⋊C4—2441no—
582C22⋊C4—2442no—
583C42—1221no—
584C42—1221no—
585C42—1221no—
586C42—1221no—
587C42—2442no—
588C42—2442no—
589C42—2442no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—1222no—
598C23—2441no—
599C23—2441no—
600C23—2441no—
601C23—2441no—
602C23—2441no—
603C23—2441no—
604C23C2×D4×D6111yescharacteristic
605Q8—22no—
606Q8—22no—
607Q8—41no—
608Q8—41no—
609Q8—614no—
610Q8—1222no—
611Q8—1222no—
612Q8—1222no—
613D4—1222no—
614D4—1222no—
615D4—2448no—
616D4—2441no—
617D4—2441no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—612no—
638C2×C4—1222no—
639C2×C4—1221no—
640C2×C4—1221no—
641C2×C4—1222no—
642C2×C4—1222no—
643C2×C4—1221no—
644C2×C4—1221no—
645C2×C4—1221no—
646C2×C4—1221no—
647C2×C4—1221no—
648C2×C4—1221no—
649C2×C4—1221no—
650C2×C4—1221no—
651C2×C4—1221no—
652C2×C4—2448no—
653C2×C4—2441no—
654C2×C4—2441no—
655C2×C4—2441no—
656C2×C4—2441no—
657C2×C4—2441no—
658C2×C4—2441no—
659C2×C4C2×D4×D6111yescharacteristic
660C2×C4#41×D6111yescharacteristic
661C2×C4#41×D6111yescharacteristic
662C2×C4#41×D6111yescharacteristic
663C2×C4#1451982211yescharacteristic
664C2×C4#1451982211yescharacteristic
665C8—41no—
666C8—41no—
667C8—1222no—
668C8—2441no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—4881no—
677C22—4881no—
678C22—4881no—
679C22—4881no—
680C22—2442no—
681C22—2442no—
682C22—2442no—
683C22#1684027011yescharacteristic
684C22#1684039411yescharacteristic
685C22#1684040111yescharacteristic
686C22#135×D6111yescharacteristic
687C22#1684056111yescharacteristic
688C22#1684056811yescharacteristic
689C22#1684057811yescharacteristic
690C4—41no—
691C4—42no—
692C4—81no—
693C4—81no—
694C4—81no—
695C4—1224no—
696C4—1222no—
697C4—4881no—
698C4—4881no—
699C4—2441no—
700C4—2441no—
701C4—2441no—
702C4#123×D6111yescharacteristic
703C4#1684034411yescharacteristic
704C4#1684040111yescharacteristic
705C4#1684044211yescharacteristic
706C2—81no—
707C2—4882no—
708C2#2144927411yescharacteristic
709C2#2145584011yescharacteristic
710C2#2145603611yescharacteristic
711C2#2145740711yescharacteristic
712C2#2145755811yescharacteristic
713C2#2145775311yescharacteristic
714C2#2145808211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2035C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#2035D61
hypocenterC61#24908
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16839854C2semidirect
3. #21449318C2semidirect
5. #21447341C2semidirect
6. #21447250C2semidirect
7. #21447133C2semidirect
8. #21446500C2semidirect
9. #21446428C2semidirect
10. #21446172C2semidirect
11. #21446157C2semidirect
12. #21446084C2semidirect
13. #21445833C2semidirect
15. C61×#2035C2semidirect
356. #2035D61semidirect
418. C61#24908semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168399222
C2×#168398542
#214493182
#214486562
#214473412
#214472502
#214471332
#214465002
#214464282
#214461722
#214461572
#214460842
#214458332
#214457602
C61×#20352
#2490861