Finite Group Databasebeta

← #29721574

Subgroups — #29721574

Order 15616.

946 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839982C211yesmaximal, characteristic
2C2×#16839970C211yesmaximal, characteristic, complemented
3C2×SD16×Dic61C211yesmaximal, characteristic
4#21450812C211yesmaximal, characteristic
5Q8×#11758196C211yesmaximal, characteristic, complemented
6#21447199C211yesmaximal, characteristic, complemented
7#21447198C211yesmaximal, characteristic, complemented
8#21447058C211yesmaximal, characteristic
9#21447055C211yesmaximal, characteristic
10#21446619C211yesmaximal, characteristic, complemented
11#21446534C211yesmaximal, characteristic, complemented
12#21446162C211yesmaximal, characteristic, complemented
13#21446160C211yesmaximal, characteristic, complemented
14#21445817C211yesmaximal, characteristic
15C2×C61×#464C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C2×#14519775C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22#16839982—24no—
23#16839970—24no—
24SD16×Dic61—24no—
25C2×#14519709C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519698C2211yescharacteristic
35C2×#14519698C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839690—21no—
38#16839690—21no—
39#16839669C2211yescharacteristic
40#16839668C2211yescharacteristic
41#16839660—21no—
42#16839660C2211yescharacteristic
43#16839658—21no—
44#16839658C2211yescharacteristic
45#16839631—21no—
46#16839631C2211yescharacteristic
47#16839620—21no—
48#16839620—21no—
49C4×#11758196—21no—
50C4×#11758196C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55C22×Q16×C61C2211yescharacteristic
56C22×SD16×C61C2211yescharacteristic
57C2×#124×C61C2211yescharacteristic
58C2×C4×Q8×C61C2211yescharacteristic
59C61×#464—24no—
60C2×C4⋊C8×C61C2211yescharacteristic
61C2×#103×C61C2211yescharacteristic
62C2×#102×C61C2211yescharacteristic
63C22×Dic122C2311yescharacteristic
64C23×Dic61—21no—
65C2×#11758202—22no—
66C2×#11758202—22no—
67C2×#11758202—22no—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519775—22no—
71#14519775—22no—
72#14519764—22no—
73#14519764—41no—
74Q8×Dic61—22no—
75Q8×Dic61—22no—
76Q8×Dic61—41no—
77Q8×Dic61D414yes—
78D4×Dic61—22no—
79D4×Dic61—22no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196D411yescharacteristic
92C2×#11758196D411yescharacteristic
93C2×C22⋊Dic61—21no—
94C2×C22⋊Dic61—21no—
95C2×C22⋊Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100#14519709—22no—
101#14519706—22no—
102#14519706—22no—
103#14519702—22no—
104#14519702—22no—
105#14519701—22no—
106#14519701—22no—
107#14519701—41no—
108#14519700—41no—
109#14519698—22no—
110#14519698—22no—
111C8×Dic61—22no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684D411yescharacteristic
119#14519684D411yescharacteristic
120C22×Q8×C61C2311yescharacteristic
121C22×Q8×C61C2311yescharacteristic
122C22×D4×C61C2311yescharacteristic
123C2×Q16×C61—22no—
124C2×Q16×C61—22no—
125C2×Q16×C61—22no—
126C2×SD16×C61—22no—
127C2×SD16×C61—22no—
128C2×SD16×C61D414yes—
129C22×C8×C61C2311yescharacteristic
130C22×C8×C61C2311yescharacteristic
131#124×C61—22no—
132#124×C61—41no—
133C4×Q8×C61—22no—
134C4×Q8×C61—41no—
135C2×C4⋊C4×C61—21no—
136C2×C4⋊C4×C61C2311yescharacteristic
137C2×C22⋊C4×C61—21no—
138C2×C22⋊C4×C61—21no—
139C2×C42×C61—21no—
140C2×C42×C61C2311yescharacteristic
141C4⋊C8×C61—22no—
142#103×C61—22no—
143#102×C61—22no—
144C24×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×Q8×C61—21no—
154C2×Q8×C61C2×D412yes—
155C2×Q8×C61C2×D412yes—
156C2×Q8×C61#4112yes—
157C2×D4×C61—21no—
158C2×D4×C61—21no—
159C2×D4×C61—21no—
160C2×D4×C61—21no—
161C2×D4×C61—21no—
162C2×D4×C61C2×D412yes—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2411yescharacteristic
169C22×Dic61—21no—
170C22×Dic61—21no—
171C22×Dic61—21no—
172C22×Dic61—21no—
173C22×Dic61—42no—
174C22×Dic61—42no—
175C22×Dic61—42no—
176C22×Dic61C2×D411yescharacteristic
177C22×Dic61C2×D411yescharacteristic
178C22×Dic61C2×D411yescharacteristic
179C22×Dic61C2×D411yescharacteristic
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186Q16×C61—44no—
187SD16×C61—28no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61C2×D412yes—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—42no—
197C22⋊C4×C61—42no—
198C22⋊C4×C61—42no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—42no—
202C22⋊Dic61—42no—
203C22⋊Dic61—42no—
204C22⋊Dic61—42no—
205#11758202—44no—
206C4⋊Dic61—21no—
207C4⋊Dic61—24no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—42no—
213C4⋊Dic61C2×D412yes—
214#11758196—22no—
215#11758196—22no—
216#11758196—42no—
217#11758196—42no—
218#11758196—42no—
219#11758196—81no—
220#11758196—81no—
221#11758196—81no—
222C4×Dic61—21no—
223C4×Dic61—21no—
224C4×Dic61—24no—
225C4×Dic61—42no—
226C4×Dic61C2×D412yes—
227C4×Dic61#4112yes—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—21no—
233C2×#92418—21no—
234C23×C61—22no—
235C23×C61—41no—
236C23×C61—41no—
237C23×C61—41no—
238C23×C61—41no—
239C23×C61—41no—
240C23×C61—41no—
241C23×C61C22×D411yescharacteristic
242Q8×C61—22no—
243Q8×C61—22no—
244Q8×C61—22no—
245Q8×C61—41no—
246Q8×C61—41no—
247Q8×C61—41no—
248Q8×C61—41no—
249Q8×C61#12114yes—
250D4×C61—22no—
251D4×C61—22no—
252D4×C61—41no—
253D4×C61—41no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
256C2×C4×C61—22no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
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×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61H(5,𝔽2)11yescharacteristic
274C2×C4×C612−1+411yescharacteristic
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—22no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—84no—
296C2×Dic61#12112yes—
297C2×Dic61#12112yes—
298C2×Dic61#13512yes—
299C2×Dic61#13712yes—
300Dic122—41no—
301Dic122—41no—
302Dic122—41no—
303Dic122—41no—
304C8×C61—22no—
305C8×C61—41no—
306#92418—41no—
307#92418—41no—
308#24729—611nomaximal, Sylow
309C22×C61—42no—
310C22×C61—42no—
311C22×C61—42no—
312C22×C61—81no—
313C22×C61—81no—
314C22×C61—81no—
315C22×C61—81no—
316C22×C61C2×#12111yescharacteristic
317C22×C61D4211yescharacteristic
318C22×C61#54811yescharacteristic
319C22×C61C2×#13511yescharacteristic
320C22×C61C2×#13711yescharacteristic
321C22×C61#57711yescharacteristic
322C22×C61#57811yescharacteristic
323C4×C61—24no—
324C4×C61—41no—
325C4×C61—41no—
326C4×C61—81no—
327C4×C61—81no—
328C4×C61C2×#12111yescharacteristic
329C4×C61#53711yescharacteristic
330C4×C61#54711yescharacteristic
331C4×C61#55311yescharacteristic
332Dic61—22no—
333Dic61—41no—
334Dic61—41no—
335Dic61—42no—
336Dic61—81no—
337Dic61—81no—
338Dic61—81no—
339Dic61#46414yescomplemented
340C2×#464—611no—
341C2×#464D6111yescharacteristic
342C2×#462—611no—
343C2×C4×SD16—611no—
344#2547—611no—
345Q8×C4⋊C4—611no—
346#2066—611no—
347#2066—611no—
348#2042—611no—
349#2042—611no—
350#1956—611no—
351#1939—611no—
352#1878—611no—
353#1878—611no—
354#1802—611no—
355C2×C61—82no—
356C2×C61#241611yescharacteristic
357C2×C61C2×#46411yescharacteristic
358C2×C61#304611yescharacteristic
359C2×C61#327611yescharacteristic
360C2×C61#328411yescharacteristic
361C2×C61#336411yescharacteristic
362C2×C61#336911yescharacteristic
363C22×Q16—611no—
364C22×Q16C2×D6111yescharacteristic
365C22×SD16C2×D6111yescharacteristic
366C2×C4⋊Q8—611no—
367C2×#124—611no—
368C2×#124C2×D6111yescharacteristic
369C2×C4×Q8—611no—
370C2×C4×Q8—611no—
371C2×C4×Q8C2×D6111yescharacteristic
372C2×C4×D4—611no—
373#464—24no—
374#464—1224no—
375#462—1224no—
376C4×SD16—1224no—
377C2×#106—611no—
378C2×C4⋊C8—611no—
379C2×C4⋊C8—611no—
380C2×C4⋊C8C2×D6111yescharacteristic
381C2×#103—611no—
382C2×#103—611no—
383C2×#103—611no—
384C2×#103—611no—
385C2×#103—611no—
386C2×#103—611no—
387C2×#103C2×D6111yescharacteristic
388C2×#102—611no—
389C2×#102—611no—
390C2×#102C2×D6111yescharacteristic
391C2×C4×C8—611no—
392#397—1221no—
393#397—1221no—
394#391—611no—
395#391—611no—
396#389—611no—
397#389—611no—
398#389—1221no—
399#389—1221no—
400#384—611no—
401#384—1221no—
402#380—1221no—
403#380—1221no—
404C4×C4⋊C4—611no—
405C4×C4⋊C4—1221no—
406#340—611no—
407#340—611no—
408#336—611no—
409#336—611no—
410C61#2472911yescharacteristic, complemented, Sylow
411C22×Q8—611no—
412C22×Q8C22×D6111yescharacteristic
413C22×Q8C22×D6111yescharacteristic
414C22×D4C22×D6111yescharacteristic
415C23×C4—1221no—
416C2×Q16—22no—
417C2×Q16—22no—
418C2×Q16—22no—
419C2×Q16—1222no—
420C2×Q16—1222no—
421C2×Q16—1222no—
422C2×SD16—22no—
423C2×SD16—22no—
424C2×SD16C22⋊D6114yes—
425C22×C8—611no—
426C22×C8—611no—
427C22×C8C22×D6111yescharacteristic
428C22×C8C22×D6111yescharacteristic
429C4⋊Q8—1222no—
430C4⋊Q8—1222no—
431#124—22no—
432#124—41no—
433#124—1222no—
434#124—2441no—
435C4×Q8—22no—
436C4×Q8—41no—
437C4×Q8—614no—
438C4×Q8—1222no—
439C4×Q8—1222no—
440C4×Q8—2441no—
441C4×D4—1222no—
442C4×D4—1222no—
443C2×C4⋊C4—21no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—611no—
448C2×C4⋊C4—611no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4—1221no—
457C2×C4⋊C4C22×D6111yescharacteristic
458C2×C22⋊C4—21no—
459C2×C22⋊C4—21no—
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C42—21no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—1221no—
467C2×C42—1221no—
468C2×C42C22×D6111yescharacteristic
469#106—1222no—
470C4⋊C8—22no—
471C4⋊C8—1222no—
472C4⋊C8—1222no—
473#103—22no—
474#103—1222no—
475#103—1222no—
476#103—1222no—
477#103—1222no—
478#103—2441no—
479#103—2441no—
480#102—22no—
481#102—1222no—
482#102—1222no—
483C4×C8—1222no—
484#95—611no—
485#95—611no—
486#95—1221no—
487#95—1221no—
488#95—1221no—
489#95—1221no—
490#95—1221no—
491#95—1221no—
492C24—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—21no—
501C2×Q8—21no—
502C2×Q8—1221no—
503C2×Q8—1221no—
504C2×Q8—1221no—
505C2×Q8—1221no—
506C2×Q8—1221no—
507C2×Q8—1221no—
508C2×Q8D4×D6112yes—
509C2×Q8#1175822412yes—
510C2×Q8C2×C22⋊D6112yes—
511C2×D4—21no—
512C2×D4—21no—
513C2×D4—21no—
514C2×D4—21no—
515C2×D4—21no—
516C2×D4C2×C22⋊D6112yes—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—1221no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—2442no—
529C22×C4—2442no—
530C22×C4—2442no—
531C22×C4D4×D6111yescharacteristic
532C22×C4D4×D6111yescharacteristic
533C22×C4C23×D6111yescharacteristic
534Q16—44no—
535Q16—2444no—
536SD16—28no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—1221no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8C2×C22⋊D6112yes—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—42no—
552C4⋊C4—612no—
553C4⋊C4—1222no—
554C4⋊C4—1222no—
555C4⋊C4—1224no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4—1221no—
561C4⋊C4—4881no—
562C4⋊C4—4881no—
563C4⋊C4—4881no—
564C4⋊C4—2442no—
565C4⋊C4—2442no—
566C4⋊C4—2442no—
567C4⋊C4—2442no—
568C22⋊C4—42no—
569C22⋊C4—42no—
570C22⋊C4—2442no—
571C22⋊C4—2442no—
572C22⋊C4—2442no—
573C42—21no—
574C42—21no—
575C42—42no—
576C42—612no—
577C42—612no—
578C42—1224no—
579C42—1221no—
580C42—1221no—
581C42—2442no—
582C23—22no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23—41no—
589C23C2×D4×D6111yescharacteristic
590Q8—22no—
591Q8—22no—
592Q8—22no—
593Q8—41no—
594Q8—41no—
595Q8—41no—
596Q8—41no—
597Q8—2441no—
598Q8—2441no—
599Q8—2441no—
600Q8—2441no—
601Q8#1451974914yes—
602D4—22no—
603D4—22no—
604D4—41no—
605D4—41no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—612no—
622C2×C4—612no—
623C2×C4—612no—
624C2×C4—612no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—1222no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—1221no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—1221no—
634C2×C4—4884no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4C2×D4×D6111yescharacteristic
647C2×C4C22×C22⋊D6111yescharacteristic
648C2×C4#41×D6111yescharacteristic
649C2×C4C2×#1175822411yescharacteristic
650C2×C4#1451982211yescharacteristic
651C2×C4#1451982411yescharacteristic
652C8—22no—
653C8—41no—
654C8—2441no—
655C8—2441no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22#121×D6111yescharacteristic
664C22D4×C22⋊D6111yescharacteristic
665C22#1684041811yescharacteristic
666C22#135×D6111yescharacteristic
667C22#137×D6111yescharacteristic
668C22#1684056911yescharacteristic
669C22#1684057811yescharacteristic
670C4—24no—
671C4—41no—
672C4—41no—
673C4—81no—
674C4—81no—
675C4—614no—
676C4—1222no—
677C4—4881no—
678C4—4881no—
679C4—4881no—
680C4—2442no—
681C4—2441no—
682C4—2441no—
683C4C2×#1451974911yescharacteristic
684C4#1684033911yescharacteristic
685C4#1684041111yescharacteristic
686C4#1684045511yescharacteristic
687C2—82no—
688C2#2144925611yescharacteristic
689C2#464×D6111yescharacteristic
690C2#2145537211yescharacteristic
691C2#2145745911yescharacteristic
692C2#2145754311yescharacteristic
693C2#2145849911yescharacteristic
694C2#2145855411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#464C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#464D61
hypocenterC61#24729
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
2. C2×#16839970C2semidirect
5. Q8×#11758196C2semidirect
6. #21447199C2semidirect
7. #21447198C2semidirect
10. #21446619C2semidirect
11. #21446534C2semidirect
12. #21446162C2semidirect
13. #21446160C2semidirect
339. Dic61#464semidirect
410. C61#24729semidirect

Maximal subgroups

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

subgroupindex
C2×#168399822
C2×#168399702
C2×SD16×Dic612
#214508122
Q8×#117581962
#214471992
#214471982
#214470582
#214470552
#214466192
#214465342
#214461622
#214461602
#214458172
C2×C61×#4642
#2472961