Finite Group Databasebeta

← #29708574

Subgroups — #29708574

Order 15616.

970 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839809C211yesmaximal, characteristic
2#21454406C211yesmaximal, characteristic
3#21454380C211yesmaximal, characteristic
4C2×#16839769C211yesmaximal, characteristic
5#419×D61C211yesmaximal, characteristic
6C2×#101×D61C211yesmaximal, characteristic
7#21447187C211yesmaximal, characteristic, complemented
8#21447186C211yesmaximal, characteristic, complemented
9#21447185C211yesmaximal, characteristic, complemented
10#21447184C211yesmaximal, characteristic, complemented
11#21447183C211yesmaximal, characteristic, complemented
12#21447182C211yesmaximal, characteristic, complemented
13#21447181C211yesmaximal, characteristic, complemented
14#21445556C211yesmaximal, characteristic
15C61×#2064C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C2×M16×D61C2211yescharacteristic
19C2×M16×D61C2211yescharacteristic
20C2×M16×D61C2211yescharacteristic
21C2×#14519735C2211yescharacteristic
22#117×D61C2211yescharacteristic
23#16839809—24no—
24C2×#14519713C2211yescharacteristic
25C2×#14519712C2211yescharacteristic
26C2×#14519712C2211yescharacteristic
27#16839805C2211yescharacteristic
28#16839804C2211yescharacteristic
29#16839803—21no—
30#16839802C2211yescharacteristic
31#16839801—21no—
32#16839799—21no—
33#16839797—21no—
34#16839790C2211yescharacteristic
35#16839789C2211yescharacteristic
36#16839788C2211yescharacteristic
37#16839787—22no—
38#16839784—22no—
39#16839772C2211yescharacteristic
40#16839771C2211yescharacteristic
41#16839770C2211yescharacteristic
42#16839769—22no—
43#16839769—22no—
44#16839769—21no—
45#16839769—21no—
46C2×#14519702C2211yescharacteristic
47C2×#14519701C2211yescharacteristic
48C2×#14519700C2211yescharacteristic
49#103×D61—21no—
50#103×D61—21no—
51C2×#14519696C2211yescharacteristic
52C2×#14519695C2211yescharacteristic
53C2×#14519695C2211yescharacteristic
54#101×D61—24no—
55#16839531C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58C2×#108×C61C2211yescharacteristic
59C61×#428C2211yescharacteristic
60C61×#424C2211yescharacteristic
61C61×#419C2211yescharacteristic
62C2×#103×C61C2211yescharacteristic
63C2×#101×C61C2211yescharacteristic
64C61×#343C2211yescharacteristic
65C22×Dic122—21no—
66C22×Dic122C2311yescharacteristic
67C2×Q8×D61—22no—
68C2×Q8×D61—21no—
69C2×Q8×D61—21no—
70C2×Q8×D61—21no—
71C2×Q8×D61—21no—
72C2×Q8×D61—22no—
73C2×Q8×D61—22no—
74C2×Q8×D61Q814yes—
75C22×C4×D61—21no—
76C22×C4×D61C2311yescharacteristic
77C2×#11758194C2311yescharacteristic
78C2×#11758194C2311yescharacteristic
79C2×#11758194C2311yescharacteristic
80C2×#11758189—21no—
81C2×#11758189—22no—
82C2×#11758189—22no—
83C2×#11758189—21no—
84C2×#11758189—21no—
85C2×#11758189D412yes—
86C2×#11758189D412yes—
87M16×D61—22no—
88M16×D61—22no—
89M16×D61—22no—
90M16×D61—22no—
91M16×D61—22no—
92M16×D61D414yes—
93C22×#92418C2311yescharacteristic
94C2×C8×D61—21no—
95C2×C8×D61—21no—
96C2×C8×D61—21no—
97#14519736C2311yescharacteristic
98#14519735—21no—
99#14519735—21no—
100#14519735—21no—
101#14519735—21no—
102#14519735D411yescharacteristic
103#14519735D411yescharacteristic
104#14519734—21no—
105#14519733—21no—
106C2×C4⋊Dic61C2311yescharacteristic
107C4⋊C4×D61D411yescharacteristic
108C4⋊C4×D61D411yescharacteristic
109C2×#11758198—21no—
110C22⋊C4×D61—21no—
111C2×C4×Dic61—21no—
112C42×D61—21no—
113#14519713—22no—
114#14519712—22no—
115#14519712—22no—
116#14519711—21no—
117#14519710—21no—
118#14519710—21no—
119#14519709—21no—
120#14519708—21no—
121#14519708—21no—
122C4⋊#92418—22no—
123#14519706—22no—
124#14519706—22no—
125#14519702—22no—
126#14519702—21no—
127#14519702—21no—
128#14519701—21no—
129#14519701—21no—
130#14519701—22no—
131#14519700—21no—
132#14519700—21no—
133#14519700—22no—
134#14519696—22no—
135#14519695—22no—
136#14519695—22no—
137C22×Q8×C61C2311yescharacteristic
138C2×M16×C61C2311yescharacteristic
139C2×M16×C61C2311yescharacteristic
140C2×M16×C61C2311yescharacteristic
141C22×C8×C61C2311yescharacteristic
142#117×C61C2311yescharacteristic
143C2×C4⋊C4×C61C2311yescharacteristic
144#108×C61—22no—
145#107×C61—21no—
146#106×C61—21no—
147C4⋊C8×C61—22no—
148#103×C61—22no—
149#103×C61—21no—
150#103×C61—21no—
151#101×C61—22no—
152C23×D61#4111yescharacteristic
153C2×Q8×C61—21no—
154C2×Q8×C61—21no—
155C2×Q8×C61—21no—
156C2×Q8×C61—21no—
157C2×Q8×C61—21no—
158C2×Q8×C61C2×Q812yes—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—21no—
163C22×Dic61—21no—
164C22×Dic61#4111yescharacteristic
165Q8×D61—24no—
166Q8×D61—24no—
167Q8×D61—24no—
168Q8×D61—24no—
169Q8×D61—42no—
170Q8×D61—42no—
171Q8×D61—42no—
172Q8×D61—42no—
173C2×C4×D61—24no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—41no—
177C2×C4×D61—41no—
178C2×C4×D61—42no—
179C2×C4×D61C2×D412yes—
180C2×C4×D61C2×D411yescharacteristic
181C2×C4×D61C2×D411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×C4×D61C2×D411yescharacteristic
184C2×C4×D61C2×Q812yes—
185C2×C4×D61#4112yes—
186C2×C4×D61#4112yes—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122—21no—
190C2×Dic122—21no—
191C2×Dic122—21no—
192C2×Dic122—24no—
193C2×Dic122—41no—
194C2×Dic122—41no—
195C2×Dic122—42no—
196C2×Dic122C2×Q812yes—
197M16×C61—21no—
198M16×C61—21no—
199M16×C61—21no—
200M16×C61—21no—
201M16×C61—21no—
202M16×C61C2×D412yes—
203C2×C8×C61—21no—
204C2×C8×C61—21no—
205C2×C8×C61—21no—
206C2×C8×C61—21no—
207C2×C8×C61—21no—
208C2×C8×C61C2×D412yes—
209C4⋊C4×C61—21no—
210C4⋊C4×C61—21no—
211C4⋊C4×C61C2×D411yescharacteristic
212C4⋊C4×C61C2×D411yescharacteristic
213C22⋊C4×C61—21no—
214C42×C61—21no—
215C22⋊Dic61—21no—
216#11758198—21no—
217#11758198—21no—
218#11758198—41no—
219#11758198—41no—
220C4⋊Dic61—21no—
221C4⋊Dic61—21no—
222C4⋊Dic61C2×D411yescharacteristic
223C4⋊Dic61C2×D411yescharacteristic
224#11758196—21no—
225#11758196—21no—
226C4×Dic61—21no—
227C4×Dic61—21no—
228C4×Dic61—21no—
229C4×Dic61—41no—
230C4×Dic61—41no—
231#11758194—21no—
232#11758194—21no—
233#11758194—21no—
234#11758194—21no—
235#11758194—21no—
236#11758194C2×D412yes—
237C2×#92418—21no—
238C2×#92418—21no—
239C2×#92418—21no—
240C2×#92418—21no—
241C2×#92418—21no—
242C2×#92418C2×D412yes—
243#11758189—24no—
244#11758189—24no—
245#11758189—24no—
246#11758189—42no—
247#11758189—42no—
248C8×D61—24no—
249C8×D61—42no—
250C8×D61—42no—
251C23×C61C2×#4111yescharacteristic
252C22×D61—41no—
253C22×D61—41no—
254C22×D61#12111yescharacteristic
255C22×D61#12111yescharacteristic
256C22×D61C22⋊Q812yes—
257C22×D61#12612yes—
258Q8×C61—22no—
259Q8×C61—22no—
260Q8×C61—41no—
261Q8×C61—41no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—22no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—81no—
269C2×C4×C61C22×D411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×Q811yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—22no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—81no—
282C2×Dic61#12111yescharacteristic
283C2×Dic61#12111yescharacteristic
284C2×Dic61C22⋊Q812yes—
285C2×Dic61#12612yes—
286C4×D61—44no—
287C4×D61—44no—
288C4×D61—81no—
289C4×D61—81no—
290C4×D61#12112yes—
291C4×D61#12112yes—
292C4×D61#12112yes—
293C4×D61#12112yes—
294C4×D61C22⋊Q812yes—
295C4×D61C22⋊Q812yes—
296C4×D61C22⋊Q812yes—
297C4×D61C22⋊Q812yes—
298Dic122—22no—
299Dic122—22no—
300Dic122—41no—
301Dic122—41no—
302Dic122—44no—
303Dic122—44no—
304C8×C61—22no—
305C8×C61—22no—
306C8×C61—41no—
307C8×C61—41no—
308#92418—22no—
309#92418—22no—
310#92418—41no—
311#92418—41no—
312#23603—611nomaximal, Sylow
313C22×C61—21no—
314C22×C61—21no—
315C22×C61C2×#12111yescharacteristic
316C22×C61C2×C22⋊Q811yescharacteristic
317C22×C61C2×#12611yescharacteristic
318C2×D61—21no—
319C2×D61—21no—
320C2×D61—21no—
321C2×D61—21no—
322C2×D61—42no—
323C2×D61—42no—
324C2×D61#39612yes—
325C2×D61#39612yes—
326C4×C61—42no—
327C4×C61—41no—
328C4×C61—41no—
329C4×C61—81no—
330C4×C61C2×#12111yescharacteristic
331C4×C61C2×#12111yescharacteristic
332C4×C61C2×C22⋊Q811yescharacteristic
333C4×C61C2×C22⋊Q811yescharacteristic
334Dic61—42no—
335Dic61—41no—
336Dic61—41no—
337Dic61—81no—
338Dic61#39612yes—
339Dic61#39612yes—
340C2×#430—611no—
341#2911—611no—
342#2905—611no—
343C2×#419—611no—
344C2×#419—611no—
345C22×#101—611no—
346#2064—611no—
347#2064—611no—
348#2064—611no—
349#2064—611no—
350#2064—611no—
351#2064—611no—
352#2064—611no—
353#2064D6111yescharacteristic, complemented
354#1742—611no—
355C2×C61—21no—
356C2×C61—21no—
357C2×C61C2×#39611yescharacteristic
358C2×C61#306911yescharacteristic
359C2×C61#306911yescharacteristic
360D61—22no—
361D61—22no—
362C23×Q8—611no—
363C22×M16—611no—
364C22×M16—611no—
365C22×M16—611no—
366C22×M16—611no—
367C2×#117—611no—
368C2×#117—611no—
369#430—1224no—
370C2×#108—611no—
371C2×#108—611no—
372C2×#108—611no—
373C2×#108C2×D6111yescharacteristic
374#428—611no—
375#428—611no—
376#428—611no—
377#428—1221no—
378#428—1221no—
379#428C2×D6111yescharacteristic
380#427—1221no—
381#427—1221no—
382#424—611no—
383#424—611no—
384#424—611no—
385#424—1222no—
386#424C2×D6111yescharacteristic
387#423—1222no—
388#419—611no—
389#419—611no—
390#419—611no—
391#419—1221no—
392#419—1221no—
393#419—1222no—
394#419—1222no—
395#419C2×D6111yescharacteristic
396C2×#103—611no—
397C2×#103—611no—
398C2×#103—611no—
399C2×#103—1221no—
400C2×#103—1221no—
401C2×#103C2×D6111yescharacteristic
402C2×#101—611no—
403C2×#101—611no—
404C2×#101—611no—
405C2×#101—1224no—
406C2×#101C2×D6111yescharacteristic
407#343—611no—
408#343—611no—
409#343—611no—
410#343C2×D6111yescharacteristic
411C61#2360311yescharacteristic, complemented, Sylow
412C22×Q8—614no—
413C22×Q8—611no—
414C22×Q8—1221no—
415C22×Q8—1222no—
416C22×Q8—1222no—
417C22×Q8—1222no—
418C22×Q8—1221no—
419C22×Q8—1221no—
420C22×Q8—1221no—
421C22×Q8—1221no—
422C22×Q8C22×D6111yescharacteristic
423C23×C4—611no—
424C23×C4—1221no—
425C2×M16—612no—
426C2×M16—612no—
427C2×M16—611no—
428C2×M16—614no—
429C2×M16—611no—
430C2×M16—611no—
431C2×M16—1221no—
432C2×M16—1222no—
433C2×M16—1222no—
434C2×M16—1221no—
435C2×M16—1221no—
436C2×M16—1222no—
437C2×M16—1222no—
438C2×M16—1222no—
439C2×M16—1222no—
440C2×M16—1222no—
441C2×M16C22×D6111yescharacteristic
442C2×M16C22×D6111yescharacteristic
443C2×M16C22×D6111yescharacteristic
444C22×C8—611no—
445C22×C8—1221no—
446C22×C8—1221no—
447C22×C8—1221no—
448C22×C8C22×D6111yescharacteristic
449#117—611no—
450#117—611no—
451#117—611no—
452#117—1221no—
453#117—1221no—
454#117—1221no—
455#117—1221no—
456#117—1221no—
457#117—1221no—
458#117C22×D6111yescharacteristic
459C2×C4⋊C4—611no—
460C2×C4⋊C4—611no—
461C2×C4⋊C4—611no—
462C2×C4⋊C4C22×D6111yescharacteristic
463C2×C22⋊C4—1221no—
464C2×C22⋊C4—1221no—
465C2×C42—1221no—
466C2×C42—1221no—
467#108—22no—
468#108—1222no—
469#108—1222no—
470#108—1222no—
471#107—21no—
472#107—1221no—
473#107—1221no—
474#107—1221no—
475#106—21no—
476#106—1221no—
477#106—1221no—
478#106—1221no—
479C4⋊C8—22no—
480C4⋊C8—1222no—
481C4⋊C8—1222no—
482C4⋊C8—1222no—
483#103—22no—
484#103—21no—
485#103—21no—
486#103—1222no—
487#103—1221no—
488#103—1221no—
489#103—1221no—
490#103—1221no—
491#103—1222no—
492#103—1222no—
493#103—1221no—
494#103—1221no—
495#101—22no—
496#101—1222no—
497#101—1222no—
498#101—1222no—
499C24—611no—
500C2×Q8—21no—
501C2×Q8—21no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—612no—
506C2×Q8—1224no—
507C2×Q8—1224no—
508C2×Q8—1224no—
509C2×Q8—1224no—
510C2×Q8—1224no—
511C2×Q8—1221no—
512C2×Q8—1221no—
513C2×Q8—1221no—
514C2×Q8—1221no—
515C2×Q8—1221no—
516C2×Q8—2441no—
517C2×Q8—2441no—
518C2×Q8—2442no—
519C2×Q8—2442no—
520C2×Q8—2442no—
521C2×Q8—2442no—
522C2×Q8—2442no—
523C2×Q8Q8×D6112yes—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—611no—
527C22×C4—611no—
528C22×C4—611no—
529C22×C4—611no—
530C22×C4—611no—
531C22×C4—612no—
532C22×C4—612no—
533C22×C4—612no—
534C22×C4—612no—
535C22×C4—1221no—
536C22×C4—1221no—
537C22×C4—1224no—
538C22×C4—1221no—
539C22×C4—1221no—
540C22×C4—2441no—
541C22×C4—2441no—
542C22×C4—2442no—
543C22×C4C23×D6111yescharacteristic
544M16—21no—
545M16—21no—
546M16—21no—
547M16—21no—
548M16—21no—
549M16—612no—
550M16—1224no—
551M16—1224no—
552M16—1224no—
553M16—1221no—
554M16—1221no—
555M16—1221no—
556M16—1221no—
557M16—1221no—
558M16—2442no—
559M16—2442no—
560M16D4×D6112yes—
561C2×C8—21no—
562C2×C8—21no—
563C2×C8—21no—
564C2×C8—21no—
565C2×C8—21no—
566C2×C8—612no—
567C2×C8—1224no—
568C2×C8—1221no—
569C2×C8—1221no—
570C2×C8—1221no—
571C2×C8—1221no—
572C2×C8—1221no—
573C2×C8—2442no—
574C2×C8—2442no—
575C2×C8D4×D6112yes—
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—611no—
579C4⋊C4—611no—
580C4⋊C4—1221no—
581C4⋊C4—1221no—
582C4⋊C4—1221no—
583C4⋊C4—1221no—
584C4⋊C4D4×D6111yescharacteristic
585C4⋊C4D4×D6111yescharacteristic
586C22⋊C4—21no—
587C22⋊C4—1221no—
588C22⋊C4—1221no—
589C22⋊C4—1221no—
590C22⋊C4—2441no—
591C22⋊C4—2441no—
592C42—21no—
593C42—1221no—
594C42—1221no—
595C42—1221no—
596C42—2441no—
597C42—2441no—
598C23—611no—
599C23—611no—
600C23—612no—
601C23—612no—
602C23—2441no—
603C23—2441no—
604C23#41×D6111yescharacteristic
605Q8—22no—
606Q8—22no—
607Q8—41no—
608Q8—41no—
609Q8—1222no—
610Q8—1222no—
611Q8—2444no—
612Q8—2444no—
613Q8—2441no—
614Q8—2441no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—81no—
622C2×C4—611no—
623C2×C4—611no—
624C2×C4—612no—
625C2×C4—612no—
626C2×C4—612no—
627C2×C4—612no—
628C2×C4—612no—
629C2×C4—612no—
630C2×C4—612no—
631C2×C4—612no—
632C2×C4—612no—
633C2×C4—612no—
634C2×C4—1221no—
635C2×C4—1221no—
636C2×C4—1222no—
637C2×C4—4881no—
638C2×C4—4881no—
639C2×C4—4881no—
640C2×C4—2444no—
641C2×C4—2444no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4C2×D4×D6111yescharacteristic
646C2×C4C2×D4×D6111yescharacteristic
647C2×C4C2×D4×D6111yescharacteristic
648C2×C4C2×Q8×D6111yescharacteristic
649C2×C4#41×D6111yescharacteristic
650C2×C4#41×D6111yescharacteristic
651C8—22no—
652C8—22no—
653C8—41no—
654C8—41no—
655C8—1222no—
656C8—1222no—
657C8—2441no—
658C8—2441no—
659C22—21no—
660C22—21no—
661C22—612no—
662C22—612no—
663C22—1221no—
664C22—1221no—
665C22—1221no—
666C22—1221no—
667C22—2442no—
668C22—2442no—
669C22#121×D6111yescharacteristic
670C22C22⋊Q8×D6111yescharacteristic
671C22#126×D6111yescharacteristic
672C4—42no—
673C4—41no—
674C4—41no—
675C4—81no—
676C4—612no—
677C4—612no—
678C4—4881no—
679C4—2442no—
680C4—2441no—
681C4—2441no—
682C4#121×D6111yescharacteristic
683C4#121×D6111yescharacteristic
684C4C22⋊Q8×D6111yescharacteristic
685C4C22⋊Q8×D6111yescharacteristic
686C2—21no—
687C2—21no—
688C2—1222no—
689C2—1222no—
690C2#396×D6111yescharacteristic
691C2#2145555111yescharacteristic
692C2#2145555411yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2064C2
socleC22×C61C2×#121
prime radicalC23×D61#41
hypercenter#2064D61
hypocenterC61#23603
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2064

Complements

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

NH ≅ G/Ntype
7. #21447187C2semidirect
8. #21447186C2semidirect
9. #21447185C2semidirect
10. #21447184C2semidirect
11. #21447183C2semidirect
12. #21447182C2semidirect
13. #21447181C2semidirect
15. C61×#2064C2semidirect
353. #2064D61semidirect
411. C61#23603semidirect

Maximal subgroups

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

subgroupindex
C2×#168398092
#214544062
#214543802
C2×#168397692
#419×D612
C2×#101×D612
#214471872
#214471862
#214471852
#214471842
#214471832
#214471822
#214471812
#214455562
C61×#20642
#2360361