Finite Group Databasebeta

← #29723941

Subgroups — #29723941

Order 15616.

940 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839917C211yesmaximal, characteristic
2C2×#16839854C211yesmaximal, characteristic, complemented
3#21449340C211yesmaximal, characteristic
4#21448651C211yesmaximal, characteristic
5#21447354C211yesmaximal, characteristic, complemented
6#21447250C211yesmaximal, characteristic
7#21447140C211yesmaximal, characteristic, complemented
8#21446493C211yesmaximal, characteristic, complemented
9#21446428C211yesmaximal, characteristic
10#21446158C211yesmaximal, characteristic, complemented
11#21446157C211yesmaximal, characteristic, complemented
12#21446070C211yesmaximal, characteristic, complemented
13#21445823C211yesmaximal, characteristic, complemented
14#21445755C211yesmaximal, characteristic
15C61×#2036C211yesmaximal, characteristic
16C23×Dic122C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C22×#11758202C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22#16839917—24no—
23#16839854—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×C22⋊#92418C2211yescharacteristic
33C2×C8×Dic61C2211yescharacteristic
34#16839698—21no—
35#16839692—21no—
36#16839669C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839667C2211yescharacteristic
39#16839666—21no—
40#16839666—21no—
41#16839658C2211yescharacteristic
42#16839650—21no—
43#16839643—21no—
44#16839631C2211yescharacteristic
45#16839622C2211yescharacteristic
46#16839620—21no—
47C22⋊C4×Dic61—21no—
48C4×C22⋊Dic61C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C22⋊Q8×C61C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×C22⋊C8×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122—22no—
63C22×Dic122—22no—
64C22×Dic122—22no—
65C22×Dic122—22no—
66C22×Dic122—41no—
67C22×Dic122—41no—
68C22×Dic122C2311yescharacteristic
69C22×Dic122C2311yescharacteristic
70C23×Dic61—21no—
71C2×#11758202—22no—
72C2×#11758202—22no—
73C2×#11758202—22no—
74C2×#11758202—22no—
75C2×#11758202—41no—
76C2×#11758202D414yes—
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×#11758196—21no—
92C2×#11758196—21no—
93C2×C22⋊Dic61—21no—
94C2×C22⋊Dic61D411yescharacteristic
95C2×C22⋊Dic61D411yescharacteristic
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61C2311yescharacteristic
100C2×C4×Dic61C2311yescharacteristic
101#14519710—22no—
102#14519706—41no—
103#14519702—22no—
104#14519702—22no—
105#14519702—22no—
106#14519702—41no—
107#14519700—22no—
108#14519700—22no—
109C22⋊#92418—22no—
110C8×Dic61—22no—
111#14519684—21no—
112#14519684—21no—
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#107×C61—41no—
134#103×C61—22no—
135#103×C61—41no—
136C22⋊C8×C61—41no—
137#95×C61—21no—
138C24×C61C2×D411yescharacteristic
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61C2×D412yes—
145C22×C4×C61—21no—
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×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—24no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—24no—
176C2×Dic122—21no—
177C2×Dic122—44no—
178C2×Dic122—44no—
179C2×Dic122—44no—
180C2×Dic122—42no—
181C2×Dic122—42no—
182C2×Dic122—44no—
183C2×Dic122C2×D412yes—
184C2×Dic122#4112yes—
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—42no—
202C22⋊Dic61—41no—
203C22⋊Dic61—81no—
204C22⋊Dic61Q1614yescomplemented
205#11758202—28no—
206#11758202—44no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—42no—
212#11758196—42no—
213#11758196—42no—
214#11758196—42no—
215#11758196—81no—
216#11758196—81no—
217C4×Dic61—21no—
218C4×Dic61—21no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—42no—
222C4×Dic61—42no—
223C4×Dic61—42no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418C2×D412yes—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61—21no—
236C23×C61C2×Q1612yes—
237C23×C61C22×D411yescharacteristic
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—41no—
241Q8×C61—41no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
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×C61C22×D411yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×Dic61—22no—
268C2×Dic61—21no—
269C2×Dic61—22no—
270C2×Dic61—21no—
271C2×Dic61—22no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—22no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—48no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61C2×Q1612yes—
291C2×Dic61C2×Q1612yes—
292Dic122—22no—
293Dic122—22no—
294Dic122—22no—
295Dic122—22no—
296Dic122—22no—
297Dic122—48no—
298Dic122—41no—
299Dic122—41no—
300Dic122#12314yes—
301C8×C61—41no—
302C8×C61—41no—
303#92418—22no—
304#92418—41no—
305#24906—611nomaximal, Sylow
306C22×C61—22no—
307C22×C61—22no—
308C22×C61—22no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61—41no—
313C22×C61#52511yescharacteristic
314C22×C61D4211yescharacteristic
315C22×C61#54611yescharacteristic
316C22×C61C22×Q1611yescharacteristic
317C22×C61#57511yescharacteristic
318C22×C61#57711yescharacteristic
319C22×C61#57811yescharacteristic
320C4×C61—41no—
321C4×C61—42no—
322C4×C61—81no—
323C4×C61—81no—
324C4×C61—81no—
325C4×C61C2×#12311yescharacteristic
326C4×C61#53811yescharacteristic
327C4×C61#54611yescharacteristic
328C4×C61#55111yescharacteristic
329Dic61—24no—
330Dic61—22no—
331Dic61—22no—
332Dic61—42no—
333Dic61—41no—
334Dic61—41no—
335Dic61—81no—
336Dic61—81no—
337C2×#451—611no—
338C2×C4×Q16—611no—
339#2423—611no—
340Q8×C22⋊C4—611no—
341#2095—611no—
342#2076—611no—
343#2056—611no—
344#2036D6111yescharacteristic
345#1933—611no—
346#1923—611no—
347#1878—611no—
348#1878—611no—
349#1863—611no—
350#1803—611no—
351#1787—611no—
352C2×C61—42no—
353C2×C61#241811yescharacteristic
354C2×C61#310411yescharacteristic
355C2×C61#312111yescharacteristic
356C2×C61#327111yescharacteristic
357C2×C61D4×Q1611yescharacteristic
358C2×C61#329911yescharacteristic
359C2×C61#332711yescharacteristic
360C23×Q8—611no—
361C22×Q16—611no—
362C22×Q16—611no—
363C2×C22⋊Q8C2×D6111yescharacteristic
364C2×C4×Q8—611no—
365C2×C4×Q8—611no—
366C2×C4×Q8—611no—
367#451—1224no—
368C4×Q16—1224no—
369C2×#107—611no—
370C2×#107C2×D6111yescharacteristic
371C2×C4⋊C8—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#103C2×D6111yescharacteristic
379C2×#103C2×D6111yescharacteristic
380C2×C22⋊C8—611no—
381C2×C22⋊C8C2×D6111yescharacteristic
382C2×C4×C8—611no—
383#399—1221no—
384#398—1221no—
385#391—611no—
386#391—611no—
387#391—611no—
388#391—1221no—
389#391—1221no—
390#389—611no—
391#388—1221no—
392#387—1221no—
393#384—611no—
394#380—611no—
395#380—1221no—
396#380C2×D6111yescharacteristic
397C4×C22⋊C4—611no—
398C4×C22⋊C4—1221no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340C2×D6111yescharacteristic
403#336—611no—
404#336—611no—
405C61#2490611yescharacteristic, complemented, Sylow
406C22×Q8—611no—
407C22×Q8—611no—
408C22×Q8—1222no—
409C22×Q8—1222no—
410C22×Q8—1222no—
411C22×Q8—1222no—
412C22×Q8—1221no—
413C22×Q8—2441no—
414C22×Q8—2441no—
415C22×Q8C22×D6111yescharacteristic
416C23×C4—1221no—
417C23×C4C22×D6111yescharacteristic
418C2×Q16—614no—
419C2×Q16—1222no—
420C2×Q16—1222no—
421C2×Q16—1222no—
422C2×Q16—1222no—
423C2×Q16—2441no—
424C22×C8—611no—
425C22×C8—611no—
426C22×C8C22×D6111yescharacteristic
427C22×C8C22×D6111yescharacteristic
428C22⋊Q8—22no—
429C22⋊Q8—41no—
430C4×Q8—1222no—
431C4×Q8—1222no—
432C4×Q8—1222no—
433C4×Q8—2441no—
434C4×Q8—2441no—
435C4×Q8—2441no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—611no—
451C2×C22⋊C4—611no—
452C2×C22⋊C4—1221no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42—1221no—
456C2×C42—1221no—
457C2×C42—1221no—
458#107—41no—
459#107—1222no—
460C4⋊C8—2441no—
461#103—22no—
462#103—41no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#103—1222no—
467#103—1222no—
468#103—2441no—
469C22⋊C8—41no—
470C22⋊C8—1222no—
471C4×C8—1222no—
472#95—21no—
473#95—611no—
474#95—611no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483C24D4×D6111yescharacteristic
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—612no—
490C2×Q8—612no—
491C2×Q8—612no—
492C2×Q8—612no—
493C2×Q8—1224no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—1221no—
501C2×Q8—1221no—
502C2×Q8—2442no—
503C2×Q8—2442no—
504C2×Q8—2444no—
505C2×Q8—2444no—
506C2×Q8—2444no—
507C2×Q8—2444no—
508C2×Q8D4×D6112yes—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—611no—
523C22×C4—1221no—
524C22×C4—1224no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—2442no—
529C22×C4—2442no—
530C22×C4D4×D6111yescharacteristic
531C22×C4C23×D6111yescharacteristic
532Q16—1228no—
533Q16—2444no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—612no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—42no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—4881no—
558C4⋊C4—4881no—
559C4⋊C4—2442no—
560C4⋊C4—2442no—
561C4⋊C4—2442no—
562C4⋊C4—2442no—
563C22⋊C4—42no—
564C22⋊C4—42no—
565C22⋊C4—614no—
566C22⋊C4—4881no—
567C22⋊C4—2441no—
568C42—1221no—
569C42—1221no—
570C42—1221no—
571C42—1221no—
572C42—2442no—
573C42—2442no—
574C42—2442no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23Q16×D6112yes—
582C23C2×D4×D6111yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—41no—
586Q8—41no—
587Q8—614no—
588Q8—1222no—
589Q8—1222no—
590Q8—1222no—
591Q8—1222no—
592Q8—1222no—
593Q8—2448no—
594Q8—2441no—
595Q8—2441no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—612no—
616C2×C4—612no—
617C2×C4—1222no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1222no—
622C2×C4—1221no—
623C2×C4—1222no—
624C2×C4—1222no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—1221no—
631C2×C4—2448no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4C2×D4×D6111yescharacteristic
641C2×C4#41×D6111yescharacteristic
642C2×C4#41×D6111yescharacteristic
643C2×C4#41×D6111yescharacteristic
644C2×C4#1451982211yescharacteristic
645C2×C4#1451982211yescharacteristic
646C8—41no—
647C8—41no—
648C8—1222no—
649C8—2441no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22#1684027011yescharacteristic
658C22#1684039411yescharacteristic
659C22#1684040111yescharacteristic
660C22C2×Q16×D6111yescharacteristic
661C22#1684055811yescharacteristic
662C22#1684056911yescharacteristic
663C22#1684057811yescharacteristic
664C4—41no—
665C4—42no—
666C4—81no—
667C4—81no—
668C4—81no—
669C4—1222no—
670C4—1224no—
671C4—1222no—
672C4—4881no—
673C4—4881no—
674C4—2442no—
675C4—2441no—
676C4—2441no—
677C4#123×D6111yescharacteristic
678C4#1684034411yescharacteristic
679C4#1684040111yescharacteristic
680C4#1684044211yescharacteristic
681C2—42no—
682C2#2144927411yescharacteristic
683C2#2145585511yescharacteristic
684C2#2145605111yescharacteristic
685C2#2145740211yescharacteristic
686C2#2145756411yescharacteristic
687C2#2145769011yescharacteristic
688C2#2145809311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2036C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2036D61
hypocenterC61#24906
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2036

Complements

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

NH ≅ G/Ntype
2. C2×#16839854C2semidirect
5. #21447354C2semidirect
7. #21447140C2semidirect
8. #21446493C2semidirect
10. #21446158C2semidirect
11. #21446157C2semidirect
12. #21446070C2semidirect
13. #21445823C2semidirect
204. C22⋊Dic61Q16semidirect
405. C61#24906semidirect

Maximal subgroups

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

subgroupindex
C2×#168399172
C2×#168398542
#214493402
#214486512
#214473542
#214472502
#214471402
#214464932
#214464282
#214461582
#214461572
#214460702
#214458232
#214457552
C61×#20362
#2490661