Finite Group Databasebeta

← #29708557

Subgroups — #29708557

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#108×D61C211yesmaximal, characteristic, complemented
2#21454406C211yesmaximal, characteristic, complemented
3#424×D61C211yesmaximal, characteristic, complemented
4#21454333C211yesmaximal, characteristic
5C2×#16839755C211yesmaximal, characteristic
6C2×#16839743C211yesmaximal, characteristic
7#21447187C211yesmaximal, characteristic, complemented
8#21447185C211yesmaximal, characteristic, complemented
9#21447183C211yesmaximal, characteristic, complemented
10#21447181C211yesmaximal, characteristic, complemented
11#21447179C211yesmaximal, characteristic, complemented
12#21447177C211yesmaximal, characteristic, complemented
13#21447175C211yesmaximal, characteristic, complemented
14#21445556C211yesmaximal, characteristic, complemented
15C61×#2063C211yesmaximal, characteristic, complemented
16C22×#11758224C2211yescharacteristic
17C2×M16×D61C2211yescharacteristic
18C2×M16×D61C2211yescharacteristic
19C2×M16×D61C2211yescharacteristic
20C22×C8×D61C2211yescharacteristic
21C2×#14519735C2211yescharacteristic
22#117×D61C2211yescharacteristic
23C2×#14519713C2211yescharacteristic
24C2×#14519712C2211yescharacteristic
25C2×#14519712C2211yescharacteristic
26#108×D61—24no—
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—
38C4⋊C8×D61—22no—
39#16839772C2211yescharacteristic
40#16839770C2211yescharacteristic
41#16839767C2211yescharacteristic
42#16839760—21no—
43#16839760—21no—
44#16839755—22no—
45#16839755—22no—
46#16839755—21no—
47#16839755—21no—
48C2×#14519702C2211yescharacteristic
49C2×#14519700C2211yescharacteristic
50C2×#14519698C2211yescharacteristic
51#16839743—24no—
52C2×#14519695C2211yescharacteristic
53C2×#14519695C2211yescharacteristic
54C2×#14519694C2211yescharacteristic
55#16839531C2211yescharacteristic
56#16839530C2211yescharacteristic
57#16839530C2211yescharacteristic
58C2×#108×C61C2211yescharacteristic
59C61×#428C2211yescharacteristic
60C61×#424C2211yescharacteristic
61C61×#418C2211yescharacteristic
62C2×#102×C61C2211yescharacteristic
63C2×#100×C61C2211yescharacteristic
64C61×#343C2211yescharacteristic
65C2×#11758224—21no—
66C2×#11758224—21no—
67C2×#11758224—21no—
68C2×#11758224—21no—
69C2×#11758224—22no—
70C2×#11758224—22no—
71C2×#11758224—22no—
72C2×#11758224Q814yes—
73C22×Dic122C2311yescharacteristic
74C22×C22⋊D61—21no—
75C23×Dic61—21no—
76C22×C4×D61C2311yescharacteristic
77C2×#11758194C2311yescharacteristic
78C2×#11758194C2311yescharacteristic
79C2×#11758194C2311yescharacteristic
80C2×#11758189—21no—
81C2×#11758189—21no—
82C2×#11758189—21no—
83M16×D61—22no—
84M16×D61—22no—
85M16×D61—22no—
86M16×D61—22no—
87M16×D61—22no—
88M16×D61D414yes—
89C22×#92418C2311yescharacteristic
90C2×C8×D61—22no—
91C2×C8×D61—21no—
92C2×C8×D61—22no—
93C2×C8×D61—21no—
94C2×C8×D61—21no—
95C2×C8×D61D412yes—
96C2×C8×D61D412yes—
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#14519700—21no—
129#14519700—21no—
130#14519700—22no—
131#14519698—21no—
132#14519698—21no—
133#14519698—22no—
134#14519695—22no—
135#14519695—22no—
136#14519694—22no—
137C22×D4×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#102×C61—22no—
149#102×C61—21no—
150#102×C61—21no—
151#100×C61—22no—
152C24×C61—21no—
153C23×D61#4111yescharacteristic
154C2×D4×C61—21no—
155C2×D4×C61—21no—
156C2×D4×C61—21no—
157C2×D4×C61—21no—
158C2×D4×C61—21no—
159C2×D4×C61C2×Q812yes—
160C22×C4×C61—21no—
161C22×C4×C61C2411yescharacteristic
162C2×C22⋊D61—24no—
163C2×C22⋊D61—41no—
164C2×C22⋊D61—41no—
165C2×C22⋊D61—42no—
166C22×Dic61—21no—
167C22×Dic61—24no—
168C22×Dic61—21no—
169C22×Dic61—41no—
170C22×Dic61—41no—
171C22×Dic61—42no—
172C22×Dic61#4111yescharacteristic
173#11758224—24no—
174#11758224—24no—
175#11758224—24no—
176#11758224—24no—
177#11758224—42no—
178#11758224—42no—
179#11758224—42no—
180#11758224—42no—
181C2×C4×D61—21no—
182C2×C4×D61—21no—
183C2×C4×D61C2×D412yes—
184C2×C4×D61C2×D411yescharacteristic
185C2×C4×D61C2×D411yescharacteristic
186C2×C4×D61C2×D411yescharacteristic
187C2×C4×D61C2×D411yescharacteristic
188C2×C4×D61C2×Q812yes—
189C2×C4×D61#4112yes—
190C2×C4×D61#4112yes—
191C2×Dic122—21no—
192C2×Dic122—21no—
193C2×Dic122—21no—
194C2×Dic122—21no—
195C2×Dic122—21no—
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—42no—
245#11758189—42no—
246C8×D61—24no—
247C8×D61—24no—
248C8×D61—24no—
249C8×D61—42no—
250C8×D61—42no—
251C23×C61—22no—
252C23×C61—41no—
253C23×C61—41no—
254C23×C61—81no—
255C23×C61—81no—
256C23×C61C2×#4111yescharacteristic
257C22×D61—41no—
258C22×D61—41no—
259C22×D61#12111yescharacteristic
260C22×D61#12111yescharacteristic
261C22×D61C22⋊Q812yes—
262C22×D61#12612yes—
263D4×C61—22no—
264D4×C61—22no—
265D4×C61—41no—
266D4×C61—41no—
267C2×C4×C61—21no—
268C2×C4×C61—21no—
269C2×C4×C61—41no—
270C2×C4×C61—81no—
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×Q811yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C22⋊D61—44no—
278C22⋊D61—44no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—22no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—44no—
285C2×Dic61—44no—
286C2×Dic61—41no—
287C2×Dic61—81no—
288C2×Dic61#12111yescharacteristic
289C2×Dic61#12111yescharacteristic
290C2×Dic61C22⋊Q812yes—
291C2×Dic61#12612yes—
292C4×D61—42no—
293C4×D61—42no—
294C4×D61#12112yes—
295C4×D61#12112yes—
296C4×D61#12112yes—
297C4×D61#12112yes—
298C4×D61C22⋊Q812yes—
299C4×D61C22⋊Q812yes—
300C4×D61C22⋊Q812yes—
301C4×D61C22⋊Q812yes—
302Dic122—22no—
303Dic122—22no—
304Dic122—41no—
305Dic122—41no—
306C8×C61—22no—
307C8×C61—22no—
308C8×C61—41no—
309C8×C61—41no—
310#92418—22no—
311#92418—22no—
312#92418—41no—
313#92418—41no—
314#23601—611nomaximal, Sylow
315C22×C61—21no—
316C22×C61—21no—
317C22×C61—42no—
318C22×C61—82no—
319C22×C61—82no—
320C22×C61—81no—
321C22×C61—81no—
322C22×C61C2×#12111yescharacteristic
323C22×C61C2×C22⋊Q811yescharacteristic
324C22×C61C2×#12611yescharacteristic
325C2×D61—21no—
326C2×D61—21no—
327C2×D61—21no—
328C2×D61—21no—
329C2×D61—42no—
330C2×D61—42no—
331C2×D61#39612yes—
332C2×D61#39612yes—
333C4×C61—41no—
334C4×C61—41no—
335C4×C61—81no—
336C4×C61C2×#12111yescharacteristic
337C4×C61C2×#12111yescharacteristic
338C4×C61C2×C22⋊Q811yescharacteristic
339C4×C61C2×C22⋊Q811yescharacteristic
340Dic61—42no—
341Dic61—41no—
342Dic61—41no—
343Dic61—81no—
344Dic61#39612yes—
345Dic61#39612yes—
346C22×#108—611no—
347#2911—611no—
348C2×#424—611no—
349#2896—611no—
350C2×#416—611no—
351C2×#413—611no—
352#2064—611no—
353#2064—611no—
354#2064—611no—
355#2064—611no—
356#2063—611no—
357#2063—611no—
358#2063—611no—
359#2063D6111yescharacteristic, complemented
360#1742—611no—
361C2×C61—21no—
362C2×C61—21no—
363C2×C61—82no—
364C2×C61C2×#39611yescharacteristic
365C2×C61#306711yescharacteristic
366C2×C61#306711yescharacteristic
367D61—22no—
368D61—22no—
369C22×#41—611no—
370C22×M16—611no—
371C22×M16—611no—
372C22×M16—611no—
373C23×C8—611no—
374C2×#117—611no—
375C2×#117—611no—
376C2×#108—611no—
377C2×#108—611no—
378C2×#108—611no—
379C2×#108—1224no—
380C2×#108C2×D6111yescharacteristic
381#428—611no—
382#428—611no—
383#428—611no—
384#428—1221no—
385#428—1221no—
386#428C2×D6111yescharacteristic
387#427—1221no—
388#427—1221no—
389#424—611no—
390#424—611no—
391#424—611no—
392#424—1222no—
393#424C2×D6111yescharacteristic
394C2×C4⋊C8—1222no—
395#419—611no—
396#419—611no—
397#418—611no—
398#418C2×D6111yescharacteristic
399#417—1221no—
400#417—1221no—
401#416—1221no—
402#416—1221no—
403#416—1222no—
404#416—1222no—
405C2×#103—611no—
406C2×#103—611no—
407C2×#102—611no—
408C2×#102C2×D6111yescharacteristic
409#413—1224no—
410C2×#101—611no—
411C2×#101—611no—
412C2×#100—611no—
413C2×#100C2×D6111yescharacteristic
414#343—611no—
415#343—611no—
416#343—611no—
417#343C2×D6111yescharacteristic
418C61#2360111yescharacteristic, complemented, Sylow
419C2×#41—614no—
420C2×#41—1222no—
421C2×#41—1222no—
422C2×#41—1222no—
423C2×#41—1221no—
424C2×#41—1221no—
425C2×#41—1221no—
426C2×#41—1221no—
427C22×Q8—611no—
428C22×D4—1221no—
429C22×D4C22×D6111yescharacteristic
430C23×C4—611no—
431C23×C4—1221no—
432C2×M16—611no—
433C2×M16—614no—
434C2×M16—611no—
435C2×M16—611no—
436C2×M16—1221no—
437C2×M16—1221no—
438C2×M16—1221no—
439C2×M16—1222no—
440C2×M16—1222no—
441C2×M16—1222no—
442C2×M16—1222no—
443C2×M16—1222no—
444C2×M16C22×D6111yescharacteristic
445C2×M16C22×D6111yescharacteristic
446C2×M16C22×D6111yescharacteristic
447C22×C8—611no—
448C22×C8—612no—
449C22×C8—612no—
450C22×C8—1222no—
451C22×C8—1222no—
452C22×C8—1221no—
453C22×C8—1221no—
454C22×C8—1221no—
455C22×C8C22×D6111yescharacteristic
456#117—611no—
457#117—611no—
458#117—611no—
459#117—1221no—
460#117—1221no—
461#117—1221no—
462#117—1221no—
463#117—1221no—
464#117—1221no—
465#117C22×D6111yescharacteristic
466C2×C4⋊C4—611no—
467C2×C4⋊C4—611no—
468C2×C4⋊C4—611no—
469C2×C4⋊C4C22×D6111yescharacteristic
470C2×C22⋊C4—1221no—
471C2×C22⋊C4—1221no—
472C2×C42—1221no—
473C2×C42—1221no—
474#108—22no—
475#108—1222no—
476#108—1222no—
477#108—1222no—
478#107—21no—
479#107—1221no—
480#107—1221no—
481#107—1221no—
482#106—21no—
483#106—1221no—
484#106—1221no—
485#106—1221no—
486C4⋊C8—22no—
487C4⋊C8—1222no—
488C4⋊C8—1222no—
489C4⋊C8—1222no—
490#103—1222no—
491#103—1221no—
492#103—1221no—
493#103—1222no—
494#103—1221no—
495#103—1221no—
496#102—22no—
497#102—21no—
498#102—21no—
499#102—1221no—
500#102—1221no—
501#102—1222no—
502#101—1222no—
503#101—1222no—
504#100—22no—
505#100—1222no—
506C24—21no—
507C24—611no—
508#41—1224no—
509#41—1224no—
510#41—1224no—
511#41—1224no—
512#41—2442no—
513#41—2442no—
514#41—2442no—
515#41—2442no—
516C2×Q8—612no—
517C2×Q8—1221no—
518C2×Q8—1221no—
519C2×Q8—1221no—
520C2×Q8—1221no—
521C2×Q8—1221no—
522C2×D4—21no—
523C2×D4—21no—
524C2×D4—21no—
525C2×D4—21no—
526C2×D4—21no—
527C2×D4—1224no—
528C2×D4—2441no—
529C2×D4—2441no—
530C2×D4—2442no—
531C2×D4Q8×D6112yes—
532C22×C4—21no—
533C22×C4—611no—
534C22×C4—611no—
535C22×C4—611no—
536C22×C4—611no—
537C22×C4—611no—
538C22×C4—612no—
539C22×C4—612no—
540C22×C4—612no—
541C22×C4—612no—
542C22×C4—1221no—
543C22×C4—1221no—
544C22×C4—1224no—
545C22×C4—1221no—
546C22×C4—1221no—
547C22×C4—2441no—
548C22×C4—2441no—
549C22×C4—2442no—
550C22×C4C23×D6111yescharacteristic
551M16—21no—
552M16—21no—
553M16—21no—
554M16—21no—
555M16—21no—
556M16—612no—
557M16—1224no—
558M16—1221no—
559M16—1221no—
560M16—1221no—
561M16—1221no—
562M16—1221no—
563M16—2442no—
564M16—2442no—
565M16D4×D6112yes—
566C2×C8—21no—
567C2×C8—21no—
568C2×C8—21no—
569C2×C8—21no—
570C2×C8—21no—
571C2×C8—612no—
572C2×C8—1224no—
573C2×C8—1224no—
574C2×C8—1224no—
575C2×C8—1221no—
576C2×C8—1221no—
577C2×C8—1221no—
578C2×C8—1221no—
579C2×C8—1221no—
580C2×C8—2442no—
581C2×C8—2442no—
582C2×C8D4×D6112yes—
583C4⋊C4—21no—
584C4⋊C4—21no—
585C4⋊C4—611no—
586C4⋊C4—611no—
587C4⋊C4—1221no—
588C4⋊C4—1221no—
589C4⋊C4—1221no—
590C4⋊C4—1221no—
591C4⋊C4D4×D6111yescharacteristic
592C4⋊C4D4×D6111yescharacteristic
593C22⋊C4—21no—
594C22⋊C4—1221no—
595C22⋊C4—1221no—
596C22⋊C4—1221no—
597C22⋊C4—2441no—
598C22⋊C4—2441no—
599C42—21no—
600C42—1221no—
601C42—1221no—
602C42—1221no—
603C42—2441no—
604C42—2441no—
605C23—22no—
606C23—41no—
607C23—41no—
608C23—81no—
609C23—81no—
610C23—611no—
611C23—611no—
612C23—612no—
613C23—612no—
614C23—2441no—
615C23—2441no—
616C23#41×D6111yescharacteristic
617Q8—1222no—
618Q8—1222no—
619Q8—2441no—
620Q8—2441no—
621D4—22no—
622D4—22no—
623D4—41no—
624D4—41no—
625D4—2444no—
626D4—2444no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—41no—
630C2×C4—81no—
631C2×C4—611no—
632C2×C4—611no—
633C2×C4—612no—
634C2×C4—612no—
635C2×C4—612no—
636C2×C4—612no—
637C2×C4—612no—
638C2×C4—612no—
639C2×C4—612no—
640C2×C4—612no—
641C2×C4—612no—
642C2×C4—612no—
643C2×C4—1221no—
644C2×C4—1221no—
645C2×C4—1222no—
646C2×C4—4881no—
647C2×C4—2442no—
648C2×C4—2442no—
649C2×C4—2444no—
650C2×C4—2444no—
651C2×C4—2441no—
652C2×C4—2441no—
653C2×C4—2441no—
654C2×C4C2×D4×D6111yescharacteristic
655C2×C4C2×D4×D6111yescharacteristic
656C2×C4C2×D4×D6111yescharacteristic
657C2×C4C2×Q8×D6111yescharacteristic
658C2×C4#41×D6111yescharacteristic
659C2×C4#41×D6111yescharacteristic
660C8—22no—
661C8—22no—
662C8—41no—
663C8—41no—
664C8—1222no—
665C8—1222no—
666C8—2441no—
667C8—2441no—
668C22—21no—
669C22—21no—
670C22—42no—
671C22—82no—
672C22—82no—
673C22—81no—
674C22—81no—
675C22—612no—
676C22—612no—
677C22—1221no—
678C22—1221no—
679C22—1221no—
680C22—1221no—
681C22—2442no—
682C22—2442no—
683C22#121×D6111yescharacteristic
684C22C22⋊Q8×D6111yescharacteristic
685C22#126×D6111yescharacteristic
686C4—41no—
687C4—41no—
688C4—81no—
689C4—612no—
690C4—612no—
691C4—4881no—
692C4—2442no—
693C4—2441no—
694C4—2441no—
695C4#121×D6111yescharacteristic
696C4#121×D6111yescharacteristic
697C4C22⋊Q8×D6111yescharacteristic
698C4C22⋊Q8×D6111yescharacteristic
699C2—21no—
700C2—21no—
701C2—82no—
702C2—1222no—
703C2—1222no—
704C2#396×D6111yescharacteristic
705C2#2145553411yescharacteristic
706C2#2145553711yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2063C2
socleC22×C61C2×#121
prime radicalC22×#11758224C22
hypercenter#2063D61
hypocenterC61#23601
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2063

Complements

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

NH ≅ G/Ntype
1. C2×#108×D61C2semidirect
2. #21454406C2semidirect
3. #424×D61C2semidirect
7. #21447187C2semidirect
8. #21447185C2semidirect
9. #21447183C2semidirect
10. #21447181C2semidirect
11. #21447179C2semidirect
12. #21447177C2semidirect
13. #21447175C2semidirect
14. #21445556C2semidirect
15. C61×#2063C2semidirect
359. #2063D61semidirect
418. C61#23601semidirect

Maximal subgroups

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

subgroupindex
C2×#108×D612
#214544062
#424×D612
#214543332
C2×#168397552
C2×#168397432
#214471872
#214471852
#214471832
#214471812
#214471792
#214471772
#214471752
#214455562
C61×#20632
#2360161