Finite Group Databasebeta

← #33089539

Subgroups — #33089539

Order 18688.

882 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988634C211yesmaximal, characteristic
2#22618518C211yesmaximal, characteristic
3C2×#17988626C211yesmaximal, characteristic
4#428×D73C211yesmaximal, characteristic
5#22618347C211yesmaximal, characteristic
6#22610717C211yesmaximal, characteristic, complemented
7#22610716C211yesmaximal, characteristic, complemented
8#22610715C211yesmaximal, characteristic, complemented
9#22610713C211yesmaximal, characteristic, complemented
10#22610710C211yesmaximal, characteristic, complemented
11#22610709C211yesmaximal, characteristic, complemented
12#22610708C211yesmaximal, characteristic, complemented
13#22609668C211yesmaximal, characteristic
14#343×D73C211yesmaximal, characteristic
15C73×#1951C211yesmaximal, characteristic, complemented
16C22×#11924011C2211yescharacteristic
17C2×M16×D73C2211yescharacteristic
18C2×M16×D73C2211yescharacteristic
19C2×M16×D73C2211yescharacteristic
20C2×#14556549C2211yescharacteristic
21#117×D73C2211yescharacteristic
22C2×C4⋊C4×D73C2211yescharacteristic
23#17988634C418yescomplemented
24C2×#14556527C2211yescharacteristic
25C2×#14556526C2211yescharacteristic
26C2×#14556526C2211yescharacteristic
27#17988630C2211yescharacteristic
28#17988630C2211yescharacteristic
29#17988629C2211yescharacteristic
30#17988629C2211yescharacteristic
31#17988628—21no—
32#17988628—21no—
33#17988627C2211yescharacteristic
34#17988627C2211yescharacteristic
35#17988626—21no—
36#17988626—21no—
37#17988626—21no—
38#17988626—21no—
39#17988626—21no—
40#17988626—21no—
41#17988624—21no—
42#17988622—21no—
43C2×#14556525C2211yescharacteristic
44C2×#14556524C2211yescharacteristic
45#107×D73—21no—
46C2×#14556522C2211yescharacteristic
47#106×D73—21no—
48#17988545C2211yescharacteristic
49#17988544—21no—
50#17988543C2211yescharacteristic
51#17988542C2211yescharacteristic
52#17988541—21no—
53#17988539—21no—
54C4×#11924011—21no—
55#17988356C2211yescharacteristic
56#17988356C2211yescharacteristic
57#17988355C2211yescharacteristic
58#17988355C2211yescharacteristic
59#17988355C2211yescharacteristic
60#17988355C2211yescharacteristic
61C2×#108×C73C2211yescharacteristic
62#428×C73C2211yescharacteristic
63#428×C73C2211yescharacteristic
64C2×#106×C73C2211yescharacteristic
65#405×C73C2211yescharacteristic
66#343×C73C2211yescharacteristic
67#343×C73C2211yescharacteristic
68C22×C4×D73—21no—
69C22×C4×D73C2311yescharacteristic
70C2×#11924012C2311yescharacteristic
71C2×#11924012C2311yescharacteristic
72C2×#11924012C2311yescharacteristic
73C2×#11924011—21no—
74C2×#11924011—21no—
75C2×#11924011—21no—
76C2×#11924011C2×C414yes—
77C2×#11924011D411yescharacteristic
78C2×#11924011D411yescharacteristic
79C2×#11924011D411yescharacteristic
80C2×#11924011D411yescharacteristic
81C2×#11924011Q814yes—
82M16×D73—22no—
83M16×D73—22no—
84M16×D73—22no—
85M16×D73C2×C414yes—
86M16×D73C2×C414yes—
87M16×D73Q814yes—
88C22×#10596831C2311yescharacteristic
89C2×C8×D73—21no—
90C2×C8×D73—21no—
91C2×C8×D73—21no—
92#14556550C2311yescharacteristic
93#14556549—21no—
94#14556549—21no—
95#14556549—21no—
96#14556549—21no—
97#14556549D411yescharacteristic
98#14556549D411yescharacteristic
99#14556548—21no—
100#14556547—21no—
101C2×C4⋊Dic73C2311yescharacteristic
102C2×C4⋊Dic73C2311yescharacteristic
103C2×#11924007—21no—
104C4⋊C4×D73—21no—
105C4⋊C4×D73—21no—
106C4⋊C4×D73—21no—
107C4⋊C4×D73—21no—
108C4⋊C4×D73D411yescharacteristic
109C4⋊C4×D73D411yescharacteristic
110C2×#11924005—21no—
111C22⋊C4×D73—21no—
112C2×C4×Dic73—21no—
113C42×D73—21no—
114#14556527C2×C414yescomplemented
115#14556526C2×C414yescomplemented
116#14556526C2×C414yescomplemented
117#14556525—21no—
118#14556525—21no—
119#14556525—21no—
120#14556525—21no—
121#14556524—21no—
122#14556524—21no—
123#14556524—21no—
124#14556524—21no—
125#14556524—21no—
126#14556524—21no—
127#14556523—21no—
128#14556523—21no—
129#14556522—21no—
130#14556522—21no—
131#14556522—21no—
132#14556522—21no—
133#14556522—21no—
134#14556522—21no—
135#14556502—21no—
136#14556501—21no—
137#14556501—21no—
138C4×#10596831—21no—
139C8×Dic73—21no—
140C8×Dic73—21no—
141C2×M16×C73C2311yescharacteristic
142C2×M16×C73C2311yescharacteristic
143C2×M16×C73C2311yescharacteristic
144C22×C8×C73C2311yescharacteristic
145#117×C73C2311yescharacteristic
146C2×C4⋊C4×C73C2311yescharacteristic
147C2×C4⋊C4×C73C2311yescharacteristic
148#108×C73C2×C414yescomplemented
149#107×C73—21no—
150#107×C73—21no—
151#106×C73—21no—
152#106×C73—21no—
153#106×C73—21no—
154#106×C73—21no—
155#97×C73—21no—
156C4×C8×C73—21no—
157C23×D73#4111yescharacteristic
158C22×Dic73—21no—
159C22×Dic73—21no—
160C22×Dic73#4111yescharacteristic
161C2×C4×D73—21no—
162C2×C4×D73—21no—
163C2×C4×D73—41no—
164C2×C4×D73—41no—
165C2×C4×D73—82no—
166C2×C4×D73C22×C412yes—
167C2×C4×D73C2×D411yescharacteristic
168C2×C4×D73C2×D411yescharacteristic
169C2×C4×D73C2×D411yescharacteristic
170C2×C4×D73C2×D411yescharacteristic
171C2×C4×D73C2×Q812yes—
172C2×C4×D73C2×Q812yes—
173C2×C4×D73#4112yes—
174#11924012—21no—
175#11924012—21no—
176#11924012—21no—
177#11924012C22×C412yes—
178#11924012C22×C412yes—
179#11924012C2×Q812yes—
180#11924011—24no—
181#11924011—24no—
182#11924011—24no—
183#11924011C4⋊C414yes—
184#11924011C4⋊C414yes—
185#11924011C4⋊C414yes—
186#11924011C4⋊C414yes—
187C2×#10596831—21no—
188C2×#10596831—21no—
189C2×#10596831—21no—
190C2×#10596831C22×C412yes—
191C2×#10596831C2×D411yescharacteristic
192C2×#10596831C2×D411yescharacteristic
193C2×#10596831C2×Q812yes—
194C8×D73—24no—
195C8×D73—24no—
196C8×D73—24no—
197C4⋊Dic73—21no—
198C4⋊Dic73—21no—
199C4⋊Dic73—21no—
200C4⋊Dic73—21no—
201C4⋊Dic73C2×D411yescharacteristic
202C4⋊Dic73C2×D411yescharacteristic
203#11924007—21no—
204#11924007—21no—
205#11924007—41no—
206#11924007—41no—
207C22⋊Dic73—21no—
208#11924005—21no—
209#11924005—21no—
210#11924005—41no—
211#11924005—41no—
212C4×Dic73—21no—
213C4×Dic73—21no—
214C4×Dic73—21no—
215C4×Dic73—41no—
216C4×Dic73—41no—
217C22×C4×C73—21no—
218C22×C4×C73—21no—
219C22×C4×C73C2411yescharacteristic
220M16×C73—21no—
221M16×C73—21no—
222M16×C73—21no—
223M16×C73C22×C412yes—
224M16×C73C22×C412yes—
225M16×C73C2×Q812yes—
226C2×C8×C73—21no—
227C2×C8×C73—21no—
228C2×C8×C73—21no—
229C2×C8×C73C22×C412yes—
230C2×C8×C73C2×D411yescharacteristic
231C2×C8×C73C2×D411yescharacteristic
232C2×C8×C73C2×Q812yes—
233C4⋊C4×C73—21no—
234C4⋊C4×C73—21no—
235C4⋊C4×C73—21no—
236C4⋊C4×C73—21no—
237C4⋊C4×C73C2×D411yescharacteristic
238C4⋊C4×C73C2×D411yescharacteristic
239C22⋊C4×C73—21no—
240C42×C73—21no—
241C23×C73C2×#4111yescharacteristic
242C22×D73—41no—
243C22×D73—41no—
244C22×D73C4×Q812yes—
245C22×D73#12111yescharacteristic
246C22×D73#12111yescharacteristic
247C22×D73#12512yes—
248C2×C4×C73—21no—
249C2×C4×C73—21no—
250C2×C4×C73—41no—
251C2×C4×C73—41no—
252C2×C4×C73—81no—
253C2×C4×C73—81no—
254C2×C4×C73C23×C411yescharacteristic
255C2×C4×C73C22×D411yescharacteristic
256C2×C4×C73C22×D411yescharacteristic
257C2×C4×C73C22×Q811yescharacteristic
258C2×C4×C73C22×Q811yescharacteristic
259C2×C4×C73C2×#4111yescharacteristic
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—81no—
265C2×Dic73—81no—
266C2×Dic73C4×Q812yes—
267C2×Dic73#12111yescharacteristic
268C2×Dic73#12111yescharacteristic
269C2×Dic73#12512yes—
270C4×D73—81no—
271C4×D73—81no—
272C4×D73—82no—
273C4×D73—82no—
274C4×D73C2×C4⋊C412yes—
275C4×D73C2×C4⋊C412yes—
276C4×D73C4×D412yes—
277C4×D73C4×D412yes—
278C4×D73C22⋊Q812yes—
279C4×D73C22⋊Q812yes—
280C4×D73C4⋊Q812yes—
281C4×D73C4⋊Q812yes—
282C8×C73—22no—
283C8×C73—22no—
284C8×C73—22no—
285C8×C73C2×C4⋊C412yes—
286C8×C73C2×C4⋊C412yes—
287#10596831—22no—
288#10596831—22no—
289#10596831—22no—
290#10596831C2×C4⋊C412yes—
291#10596831C2×C4⋊C412yes—
292C22×C73—21no—
293C22×C73—21no—
294C22×C73C2×C4×Q811yescharacteristic
295C22×C73C2×#12111yescharacteristic
296C22×C73C2×#12511yescharacteristic
297C2×D73—21no—
298C2×D73—21no—
299C2×D73—21no—
300C2×D73—21no—
301C2×D73—42no—
302C2×D73—42no—
303C2×D73#38912yes—
304C2×D73#38912yes—
305C4×C73—41no—
306C4×C73—41no—
307C4×C73—81no—
308C4×C73—81no—
309C4×C73C22×C4⋊C411yescharacteristic
310C4×C73C2×C4×D411yescharacteristic
311C4×C73C2×C22⋊Q811yescharacteristic
312C4×C73C2×C4⋊Q811yescharacteristic
313Dic73—41no—
314Dic73—41no—
315Dic73—81no—
316Dic73—81no—
317Dic73#38912yes—
318Dic73#38912yes—
319#21182—731nomaximal, Sylow
320C2×C73—21no—
321C2×C73—21no—
322C2×C73C2×#38911yescharacteristic
323C2×C73#306811yescharacteristic
324C2×C73#306911yescharacteristic
325D73—22no—
326D73—22no—
327C2×#430—731no—
328#2911—731no—
329C2×#428—731no—
330C2×#428—731no—
331#2873—731no—
332#1952—731no—
333#1952—731no—
334#1952—731no—
335#1952—731no—
336#1951—731no—
337#1951—731no—
338#1951—731no—
339#1951D7311yescharacteristic, complemented
340#1742—731no—
341C2×#343—731no—
342C73#2118211yescharacteristic, complemented, Sylow
343C22×M16—731no—
344C22×M16—731no—
345C22×M16—731no—
346C22×M16—731no—
347C2×#117—731no—
348C2×#117—731no—
349C22×C4⋊C4—731no—
350#430—738no—
351C2×#108—731no—
352C2×#108—731no—
353C2×#108—731no—
354C2×#108C2×D7311yescharacteristic
355#428—731no—
356#428—731no—
357#428—731no—
358#428—731no—
359#428—731no—
360#428—731no—
361#428—1461no—
362#428—1461no—
363#428—1461no—
364#428—1461no—
365#428—1461no—
366#428—1461no—
367#428—1461no—
368#428—1461no—
369#428C2×D7311yescharacteristic
370#428C2×D7311yescharacteristic
371#427—1461no—
372#427—1461no—
373C2×#107—731no—
374C2×#107—731no—
375C2×#107—1461no—
376C2×#106—731no—
377C2×#106—1461no—
378C2×#106C2×D7311yescharacteristic
379#405—731no—
380#405—731no—
381#405—731no—
382#405—1461no—
383#405—1461no—
384#405C2×D7311yescharacteristic
385C4×M16—1461no—
386C4×M16—1461no—
387#343—731no—
388#343—731no—
389#343—731no—
390#343—731no—
391#343—731no—
392#343—731no—
393#343C2×D7311yescharacteristic
394#343C2×D7311yescharacteristic
395C23×C4—731no—
396C23×C4—1461no—
397C2×M16—734no—
398C2×M16—731no—
399C2×M16—731no—
400C2×M16—731no—
401C2×M16—731no—
402C2×M16—734no—
403C2×M16—731no—
404C2×M16—731no—
405C2×M16—731no—
406C2×M16—734no—
407C2×M16—734no—
408C2×M16—734no—
409C2×M16—1461no—
410C2×M16—1461no—
411C2×M16—1461no—
412C2×M16—1462no—
413C2×M16—1462no—
414C2×M16—1462no—
415C2×M16C22×D7311yescharacteristic
416C2×M16C22×D7311yescharacteristic
417C2×M16C22×D7311yescharacteristic
418C22×C8—731no—
419C22×C8—1461no—
420C22×C8—1461no—
421C22×C8—1461no—
422C22×C8C22×D7311yescharacteristic
423#117—731no—
424#117—731no—
425#117—731no—
426#117—1461no—
427#117—1461no—
428#117—1461no—
429#117—1461no—
430#117—1461no—
431#117—1461no—
432#117C22×D7311yescharacteristic
433C2×C4⋊C4—731no—
434C2×C4⋊C4—731no—
435C2×C4⋊C4—731no—
436C2×C4⋊C4—731no—
437C2×C4⋊C4—1461no—
438C2×C4⋊C4—1461no—
439C2×C4⋊C4—1461no—
440C2×C4⋊C4—1461no—
441C2×C4⋊C4—1461no—
442C2×C4⋊C4C22×D7311yescharacteristic
443C2×C4⋊C4C22×D7311yescharacteristic
444C2×C22⋊C4—1461no—
445C2×C22⋊C4—1461no—
446C2×C42—1461no—
447C2×C42—1461no—
448#108—734no—
449#108—734no—
450#108—734no—
451#108C4×D7314yescomplemented
452#107—21no—
453#107—21no—
454#107—1461no—
455#107—1461no—
456#107—1461no—
457#107—1461no—
458#107—1461no—
459#107—1461no—
460#107—1461no—
461#107—1461no—
462#107—1461no—
463#107—1461no—
464#106—21no—
465#106—21no—
466#106—21no—
467#106—21no—
468#106—1461no—
469#106—1461no—
470#106—1461no—
471#106—1461no—
472#106—1461no—
473#106—1461no—
474#106—1461no—
475#106—1461no—
476#97—21no—
477#97—1461no—
478#97—1461no—
479#97—1461no—
480C4×C8—21no—
481C4×C8—1461no—
482C4×C8—1461no—
483C4×C8—1461no—
484C24—731no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—2921no—
488C22×C4—2921no—
489C22×C4—5842no—
490C22×C4—731no—
491C22×C4—732no—
492C22×C4—731no—
493C22×C4—731no—
494C22×C4—731no—
495C22×C4—731no—
496C22×C4—732no—
497C22×C4—732no—
498C22×C4—732no—
499C22×C4—1461no—
500C22×C4—1461no—
501C22×C4—1461no—
502C22×C4—1461no—
503C22×C4C23×D7311yescharacteristic
504M16—21no—
505M16—21no—
506M16—21no—
507M16—734no—
508M16—734no—
509M16—734no—
510M16—734no—
511M16—732no—
512M16—732no—
513M16—732no—
514M16—1464no—
515M16—1464no—
516M16—1464no—
517M16—1461no—
518M16—1461no—
519M16—1461no—
520M16C2×C4×D7312yes—
521M16C2×C4×D7312yes—
522M16Q8×D7312yes—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—732no—
527C2×C8—731no—
528C2×C8—731no—
529C2×C8—732no—
530C2×C8—1464no—
531C2×C8—1464no—
532C2×C8—1464no—
533C2×C8—1461no—
534C2×C8—1461no—
535C2×C8—1461no—
536C2×C8C2×C4×D7312yes—
537C2×C8D4×D7311yescharacteristic
538C2×C8D4×D7311yescharacteristic
539C2×C8Q8×D7312yes—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—2921no—
545C4⋊C4—2921no—
546C4⋊C4—731no—
547C4⋊C4—731no—
548C4⋊C4—1461no—
549C4⋊C4—1461no—
550C4⋊C4—1461no—
551C4⋊C4—1461no—
552C4⋊C4—1461no—
553C4⋊C4—1461no—
554C4⋊C4D4×D7311yescharacteristic
555C4⋊C4D4×D7311yescharacteristic
556C22⋊C4—21no—
557C22⋊C4—2921no—
558C22⋊C4—2921no—
559C22⋊C4—1461no—
560C22⋊C4—1461no—
561C22⋊C4—1461no—
562C42—21no—
563C42—2921no—
564C42—2921no—
565C42—1461no—
566C42—1461no—
567C42—1461no—
568C23—2921no—
569C23—2921no—
570C23—732no—
571C23—732no—
572C23—731no—
573C23—731no—
574C23#41×D7311yescharacteristic
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—5841no—
584C2×C4—5841no—
585C2×C4—5842no—
586C2×C4—5842no—
587C2×C4—5841no—
588C2×C4—5841no—
589C2×C4—732no—
590C2×C4—731no—
591C2×C4—731no—
592C2×C4—732no—
593C2×C4—732no—
594C2×C4—732no—
595C2×C4—732no—
596C2×C4—732no—
597C2×C4—732no—
598C2×C4—732no—
599C2×C4—732no—
600C2×C4—732no—
601C2×C4—1461no—
602C2×C4—1461no—
603C2×C4C22×C4×D7311yescharacteristic
604C2×C4C2×D4×D7311yescharacteristic
605C2×C4C2×D4×D7311yescharacteristic
606C2×C4C2×Q8×D7311yescharacteristic
607C2×C4C2×Q8×D7311yescharacteristic
608C2×C4#41×D7311yescharacteristic
609C8—22no—
610C8—22no—
611C8—22no—
612C8—732no—
613C8—732no—
614C8—1462no—
615C8—1462no—
616C8—1462no—
617C8C4⋊C4×D7312yes—
618C8C4⋊C4×D7312yes—
619C22—21no—
620C22—21no—
621C22—2922no—
622C22—2922no—
623C22—732no—
624C22—732no—
625C22—1461no—
626C22—1461no—
627C22—1461no—
628C22—1461no—
629C22C4×Q8×D7311yescharacteristic
630C22#121×D7311yescharacteristic
631C22#125×D7311yescharacteristic
632C4—41no—
633C4—41no—
634C4—81no—
635C4—81no—
636C4—2921no—
637C4—2921no—
638C4—5841no—
639C4—5841no—
640C4—732no—
641C4—732no—
642C4C2×C4⋊C4×D7311yescharacteristic
643C4C4×D4×D7311yescharacteristic
644C4C22⋊Q8×D7311yescharacteristic
645C4C4⋊Q8×D7311yescharacteristic
646C2—21no—
647C2—21no—
648C2—1462no—
649C2—1462no—
650C2#389×D7311yescharacteristic
651C2#2261965411yescharacteristic
652C2#2261966311yescharacteristic
subgroupG/H
centerC22#121×D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1951C2
socleC22×C73C2×#121
prime radicalC23×D73#41
cosocleC22×C4×C73C24
hypercenter#1951D73
hypocenterC73#21182
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1951

Complements

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

NH ≅ G/Ntype
6. #22610717C2semidirect
7. #22610716C2semidirect
8. #22610715C2semidirect
9. #22610713C2semidirect
10. #22610710C2semidirect
11. #22610709C2semidirect
12. #22610708C2semidirect
15. C73×#1951C2semidirect
23. #17988634C4semidirect
114. #14556527C2×C4semidirect
115. #14556526C2×C4semidirect
116. #14556526C2×C4semidirect
148. #108×C73C2×C4semidirect
339. #1951D73semidirect
342. C73#21182semidirect
451. #108C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886342
#226185182
C2×#179886262
#428×D732
#226183472
#226107172
#226107162
#226107152
#226107132
#226107102
#226107092
#226107082
#226096682
#343×D732
C73×#19512
#2118273