Finite Group Databasebeta

← #33149593

Subgroups — #33149593

Order 18688.

818 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22613633C211yesmaximal, characteristic
2#22612934C211yesmaximal, characteristic
3#22611490C211yesmaximal, characteristic, complemented
4#22611429C211yesmaximal, characteristic, complemented
5#22611311C211yesmaximal, characteristic, complemented
6#22611073C211yesmaximal, characteristic, complemented
7#22610882C211yesmaximal, characteristic, complemented
8#22610510C211yesmaximal, characteristic, complemented
9#22610354C211yesmaximal, characteristic
10#22610236C211yesmaximal, characteristic
11#22610202C211yesmaximal, characteristic, complemented
12#22610126C211yesmaximal, characteristic
13#22609937C211yesmaximal, characteristic, complemented
14C2×#17988350C211yesmaximal, characteristic
15C73×#1850C211yesmaximal, characteristic
16C2×#14556591C2211yescharacteristic
17C2×#14556570C2211yescharacteristic
18C2×#14556568—21no—
19C2×C4×Dic146C2211yescharacteristic
20C22×C4⋊Dic73C2211yescharacteristic
21C22×C4⋊Dic73C2211yescharacteristic
22C2×#14556523C2211yescharacteristic
23C2×#14556523C2211yescharacteristic
24C2×#14556516C2211yescharacteristic
25C2×#14556516C2211yescharacteristic
26C2×#14556516C2211yescharacteristic
27C2×#14556516C2211yescharacteristic
28#17988519—21no—
29#17988508—21no—
30#17988493C2211yescharacteristic
31#17988485C2211yescharacteristic
32#17988485C2211yescharacteristic
33#17988485C2211yescharacteristic
34#17988484—21no—
35#17988474—21no—
36#17988462—21no—
37#17988459—21no—
38#17988457C2211yescharacteristic
39#17988447C2211yescharacteristic
40C4×C4⋊Dic73C2211yescharacteristic
41C22⋊C4×Dic73—21no—
42#17988350C2211yescharacteristic
43#17988350C2211yescharacteristic
44#17988350C2211yescharacteristic
45#17988350C2211yescharacteristic
46#17988350C2211yescharacteristic
47#17988350C2211yescharacteristic
48#17988350C2211yescharacteristic
49#17988350C2211yescharacteristic
50#17988350C2211yescharacteristic
51#17988350C2211yescharacteristic
52C23×C8×C73C2211yescharacteristic
53C2×C4⋊C8×C73C2211yescharacteristic
54C2×C4⋊C8×C73C2211yescharacteristic
55C2×C22⋊C8×C73C2211yescharacteristic
56C4×C22⋊C4×C73C2211yescharacteristic
57#336×C73C2211yescharacteristic
58#336×C73C2211yescharacteristic
59C22×Dic146—21no—
60C22×Dic146C2311yescharacteristic
61C23×Dic73—21no—
62C23×Dic73—21no—
63#14556591—22no—
64#14556591—41no—
65#14556570—22no—
66#14556570Q814yescomplemented
67#14556568—44no—
68C4×Dic146—22no—
69C4×Dic146—41no—
70C2×C4⋊Dic73—22no—
71C2×C4⋊Dic73—22no—
72C2×C4⋊Dic73—21no—
73C2×C4⋊Dic73—21no—
74C2×C4⋊Dic73—21no—
75C2×C4⋊Dic73—21no—
76C2×C4⋊Dic73—21no—
77C2×C4⋊Dic73—41no—
78C2×C4⋊Dic73—41no—
79C2×C4⋊Dic73C2311yescharacteristic
80C2×C4⋊Dic73C2311yescharacteristic
81C2×C4⋊Dic73C2311yescharacteristic
82C2×C4⋊Dic73C2311yescharacteristic
83C2×C4⋊Dic73C2311yescharacteristic
84C2×C4⋊Dic73C2311yescharacteristic
85C2×C4⋊Dic73C2311yescharacteristic
86C2×#11924007—21no—
87C2×#11924007—21no—
88C2×#11924007—21no—
89C2×#11924007—21no—
90C2×#11924007—21no—
91C2×C22⋊Dic73—21no—
92C2×C22⋊Dic73—21no—
93C2×C4×Dic73—21no—
94C2×C4×Dic73—21no—
95#14556523—22no—
96#14556523—41no—
97#14556516—22no—
98#14556516—41no—
99#14556516—41no—
100#14556516—41no—
101#14556498—21no—
102#14556498—21no—
103#14556498—21no—
104#14556498—21no—
105#14556498—21no—
106#14556498—21no—
107#14556498—21no—
108C23×C4×C73C2311yescharacteristic
109C22×C8×C73—22no—
110C22×C8×C73—41no—
111C22×C8×C73—41no—
112C22×C8×C73C2311yescharacteristic
113C22×C8×C73C2311yescharacteristic
114C22×C8×C73C2311yescharacteristic
115C22×C8×C73C2311yescharacteristic
116C2×C22⋊C4×C73D411yescharacteristic
117C2×C22⋊C4×C73D411yescharacteristic
118C2×C42×C73C2311yescharacteristic
119C2×C42×C73C2311yescharacteristic
120C4⋊C8×C73—41no—
121C4⋊C8×C73—41no—
122C22⋊C8×C73—22no—
123#95×C73D411yescharacteristic
124#95×C73D411yescharacteristic
125C2×Dic146—21no—
126C2×Dic146—21no—
127C2×Dic146—21no—
128C2×Dic146—21no—
129C2×Dic146—21no—
130C2×Dic146—42no—
131C2×Dic146—42no—
132C2×Dic146—81no—
133C2×Dic146C2×Q812yes—
134C22×Dic73—21no—
135C22×Dic73—21no—
136C22×Dic73—21no—
137C22×Dic73—21no—
138C22×Dic73—21no—
139C22×Dic73—21no—
140C22×Dic73—21no—
141C22×Dic73—21no—
142C22×Dic73—42no—
143C22×Dic73—42no—
144C22×Dic73—42no—
145C22×Dic73—42no—
146C22×Dic73—81no—
147C22×Dic73—81no—
148C4⋊Dic73—21no—
149C4⋊Dic73—21no—
150C4⋊Dic73—21no—
151C4⋊Dic73—21no—
152C4⋊Dic73—21no—
153C4⋊Dic73—21no—
154C4⋊Dic73—21no—
155C4⋊Dic73—21no—
156C4⋊Dic73—21no—
157C4⋊Dic73—21no—
158C4⋊Dic73—21no—
159C4⋊Dic73—42no—
160C4⋊Dic73—42no—
161C4⋊Dic73—42no—
162C4⋊Dic73—42no—
163C4⋊Dic73—81no—
164C4⋊Dic73C2×Q812yes—
165C4⋊Dic73#4112yes—
166C4⋊Dic73#4112yes—
167#11924007—24no—
168#11924007—42no—
169#11924007—42no—
170#11924007—42no—
171#11924007—81no—
172C22⋊Dic73—24no—
173C22⋊Dic73—42no—
174C4×Dic73—42no—
175C4×Dic73—81no—
176C24×C73C2×D411yescharacteristic
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×D411yescharacteristic
185C22×C4×C73C2×D411yescharacteristic
186C22×C4×C73C2×Q812yes—
187C22×C4×C73#4112yes—
188C22×C4×C73C2411yescharacteristic
189C2×C8×C73—21no—
190C2×C8×C73—21no—
191C2×C8×C73—21no—
192C2×C8×C73—21no—
193C2×C8×C73—21no—
194C2×C8×C73—21no—
195C2×C8×C73—21no—
196C2×C8×C73—21no—
197C2×C8×C73—21no—
198C2×C8×C73—21no—
199C2×C8×C73—21no—
200C2×C8×C73—21no—
201C2×C8×C73—44no—
202C22⋊C4×C73—22no—
203C22⋊C4×C73SD1614yes—
204C42×C73—21no—
205C42×C73—21no—
206C42×C73—21no—
207C42×C73—21no—
208C23×C73—21no—
209C23×C73—21no—
210C23×C73—21no—
211C23×C73—21no—
212C23×C73C22⋊Q812yes—
213C23×C73C2×SD1612yes—
214C23×C73#13712yes—
215C23×C73C22×D411yescharacteristic
216C2×C4×C73—21no—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
222C2×C4×C73—22no—
223C2×C4×C73—22no—
224C2×C4×C73—22no—
225C2×C4×C73—22no—
226C2×C4×C73—21no—
227C2×C4×C73—21no—
228C2×C4×C73—21no—
229C2×C4×C73—21no—
230C2×C4×C73—21no—
231C2×C4×C73—21no—
232C2×C4×C73C22⋊Q812yes—
233C2×C4×C73C2×SD1612yes—
234C2×C4×C73C2×SD1612yes—
235C2×C4×C73C22×Q811yescharacteristic
236C2×C4×C73C2×#4111yescharacteristic
237C2×C4×C73C2×#4111yescharacteristic
238C2×C4×C73C2×#4111yescharacteristic
239C2×C4×C73H(5,𝔽2)11yescharacteristic
240C2×C4×C73H(5,𝔽2)11yescharacteristic
241C2×Dic73—22no—
242C2×Dic73—22no—
243C2×Dic73—22no—
244C2×Dic73—22no—
245C2×Dic73—41no—
246C2×Dic73—41no—
247C2×Dic73—41no—
248C2×Dic73—41no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—41no—
252C2×Dic73—41no—
253C2×Dic73—41no—
254C2×Dic73—41no—
255C2×Dic73—41no—
256C2×Dic73—41no—
257C2×Dic73—41no—
258C2×Dic73—41no—
259C2×Dic73—41no—
260C2×Dic73—41no—
261C2×Dic73—41no—
262C2×Dic73—41no—
263C2×Dic73—41no—
264C2×Dic73—41no—
265C2×Dic73—84no—
266C2×Dic73—84no—
267Dic146—22no—
268Dic146—22no—
269Dic146—41no—
270Dic146—41no—
271Dic146—84no—
272C8×C73—41no—
273C8×C73—41no—
274C8×C73—41no—
275C8×C73—41no—
276C22×C73—22no—
277C22×C73—22no—
278C22×C73—22no—
279C22×C73—22no—
280C22×C73—22no—
281C22×C73—22no—
282C22×C73#47514yes—
283C22×C73C2×C22⋊Q811yescharacteristic
284C22×C73#54611yescharacteristic
285C22×C73#54611yescharacteristic
286C22×C73C22×SD1611yescharacteristic
287C22×C73C2×#13711yescharacteristic
288C22×C73#57611yescharacteristic
289C22×C73#57811yescharacteristic
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—22no—
293C4×C73—41no—
294C4×C73—41no—
295C4×C73#52911yescharacteristic
296C4×C73#53811yescharacteristic
297C4×C73#55411yescharacteristic
298C4×C73#55411yescharacteristic
299Dic73—42no—
300Dic73—42no—
301Dic73—42no—
302Dic73—42no—
303Dic73—81no—
304Dic73—81no—
305Dic73—81no—
306Dic73—81no—
307#26412—731nomaximal, Sylow
308C2×C73—24no—
309C2×C73#245011yescharacteristic
310C2×C73C2×#47511yescharacteristic
311C2×C73#308311yescharacteristic
312C2×C73#329511yescharacteristic
313C2×C73#329811yescharacteristic
314C2×C73#331211yescharacteristic
315C2×C73#331411yescharacteristic
316#2440—731no—
317#2360—731no—
318#2099—731no—
319#2088—731no—
320#2066—731no—
321#2024—731no—
322#1988—731no—
323#1919—731no—
324#1892—731no—
325#1872—731no—
326#1865—731no—
327#1853—731no—
328#1850D7311yescharacteristic
329#1803—731no—
330C2×#340—731no—
331C73#2641211yescharacteristic, complemented, Sylow
332C23×C8C2×D7311yescharacteristic
333C2×C4⋊Q8—731no—
334C2×C22⋊Q8—731no—
335C2×C22⋊Q8—1461no—
336C2×C4×Q8—731no—
337C22×C4⋊C4—731no—
338C22×C4⋊C4—731no—
339C2×#106—731no—
340C2×#106—731no—
341C2×C4⋊C8C2×D7311yescharacteristic
342C2×C4⋊C8C2×D7311yescharacteristic
343C2×#103—731no—
344C2×#103—731no—
345C2×#103—731no—
346C2×#103—731no—
347C2×C22⋊C8C2×D7311yescharacteristic
348#398—1461no—
349#395—1461no—
350#391—731no—
351#389—731no—
352#389—731no—
353#389—731no—
354#389—1461no—
355#387—1461no—
356#385—1461no—
357#385—1461no—
358#384—731no—
359#380—731no—
360C4×C4⋊C4—731no—
361C4×C22⋊C4—1461no—
362C4×C22⋊C4C2×D7311yescharacteristic
363#340—731no—
364#340—731no—
365#340—731no—
366#340—731no—
367#340—731no—
368#340—731no—
369#340—731no—
370#340—731no—
371#340—731no—
372#340—731no—
373#336C2×D7311yescharacteristic
374#336C2×D7311yescharacteristic
375C22×Q8—731no—
376C22×Q8—1461no—
377C23×C4—1461no—
378C23×C4—1461no—
379C23×C4C22×D7311yescharacteristic
380C22×C8—22no—
381C22×C8—41no—
382C22×C8—41no—
383C22×C8C22×D7311yescharacteristic
384C22×C8C22×D7311yescharacteristic
385C22×C8C22×D7311yescharacteristic
386C22×C8C22×D7311yescharacteristic
387C4⋊Q8—2921no—
388C4⋊Q8—1462no—
389C22⋊Q8—2924no—
390C22⋊Q8—734no—
391C22⋊Q8—1462no—
392C4×Q8—2921no—
393C4×Q8—1462no—
394C2×C4⋊C4—2921no—
395C2×C4⋊C4—2921no—
396C2×C4⋊C4—731no—
397C2×C4⋊C4—731no—
398C2×C4⋊C4—731no—
399C2×C4⋊C4—731no—
400C2×C4⋊C4—731no—
401C2×C4⋊C4—731no—
402C2×C4⋊C4—731no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4—1461no—
406C2×C4⋊C4—1461no—
407C2×C4⋊C4—1461no—
408C2×C4⋊C4—1461no—
409C2×C4⋊C4—1461no—
410C2×C4⋊C4—1462no—
411C2×C4⋊C4—1462no—
412C2×C4⋊C4—1461no—
413C2×C4⋊C4—1461no—
414C2×C4⋊C4—1461no—
415C2×C22⋊C4—1461no—
416C2×C22⋊C4—1461no—
417C2×C22⋊C4D29211yescharacteristic
418C2×C22⋊C4D29211yescharacteristic
419C2×C42—1461no—
420C2×C42—1461no—
421C2×C42C22×D7311yescharacteristic
422C2×C42C22×D7311yescharacteristic
423#106—2921no—
424#106—1462no—
425C4⋊C8—41no—
426C4⋊C8—41no—
427#103—2921no—
428#103—2921no—
429#103—2921no—
430#103—1462no—
431C22⋊C8—22no—
432#95—1461no—
433#95—1461no—
434#95—1461no—
435#95—1461no—
436#95—1461no—
437#95—1461no—
438#95—1461no—
439#95D29211yescharacteristic
440#95D29211yescharacteristic
441C24C2×D29211yescharacteristic
442C2×Q8—2922no—
443C2×Q8—2922no—
444C2×Q8—5841no—
445C2×Q8—732no—
446C2×Q8—1461no—
447C2×Q8—1461no—
448C2×Q8—1461no—
449C2×Q8—1461no—
450C2×Q8—1461no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—2922no—
456C22×C4—2922no—
457C22×C4—2922no—
458C22×C4—2922no—
459C22×C4—5841no—
460C22×C4—5841no—
461C22×C4—1461no—
462C22×C4—1461no—
463C22×C4—1461no—
464C22×C4—1461no—
465C22×C4—1461no—
466C22×C4—1461no—
467C22×C4—1461no—
468C22×C4—1461no—
469C22×C4C2×D29211yescharacteristic
470C22×C4C2×D29211yescharacteristic
471C22×C4C2×D29211yescharacteristic
472C22×C4C2×D29211yescharacteristic
473C22×C4C2×D29211yescharacteristic
474C22×C4Q8×D7312yes—
475C22×C4#1192402712yes—
476C22×C4C23×D7311yescharacteristic
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—44no—
490C4⋊C4—2922no—
491C4⋊C4—2922no—
492C4⋊C4—2922no—
493C4⋊C4—2922no—
494C4⋊C4—2922no—
495C4⋊C4—2922no—
496C4⋊C4—2922no—
497C4⋊C4—5841no—
498C4⋊C4—5841no—
499C4⋊C4—732no—
500C4⋊C4—732no—
501C4⋊C4—732no—
502C4⋊C4—1461no—
503C4⋊C4—1461no—
504C4⋊C4—1461no—
505C4⋊C4—1461no—
506C4⋊C4—1461no—
507C4⋊C4—1461no—
508C4⋊C4—1461no—
509C4⋊C4—1461no—
510C4⋊C4—1464no—
511C4⋊C4—1461no—
512C4⋊C4—1461no—
513C4⋊C4—1461no—
514C22⋊C4—22no—
515C22⋊C4—2922no—
516C22⋊C4—1464no—
517C22⋊C4#1192401714yes—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—2922no—
523C42—5841no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23#1455657112yes—
529C23C2×#1192401712yes—
530C23#1455662412yes—
531C23C22×D29211yescharacteristic
532Q8—2921no—
533Q8—2921no—
534Q8—5844no—
535Q8—1462no—
536Q8—1462no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—2921no—
554C2×C4—2921no—
555C2×C4—2921no—
556C2×C4—2921no—
557C2×C4—2921no—
558C2×C4—2921no—
559C2×C4—2921no—
560C2×C4—2921no—
561C2×C4—2921no—
562C2×C4—2921no—
563C2×C4—2921no—
564C2×C4—2921no—
565C2×C4—2921no—
566C2×C4—2921no—
567C2×C4—2921no—
568C2×C4—2921no—
569C2×C4—2921no—
570C2×C4—2921no—
571C2×C4—2921no—
572C2×C4—2921no—
573C2×C4—5844no—
574C2×C4—5844no—
575C2×C4—1462no—
576C2×C4—1462no—
577C2×C4—1462no—
578C2×C4—1462no—
579C2×C4#1455657112yes—
580C2×C4C2×#1192401712yes—
581C2×C4C2×#1192401712yes—
582C2×C4C2×Q8×D7311yescharacteristic
583C2×C4#41×D7311yescharacteristic
584C2×C4#41×D7311yescharacteristic
585C2×C4C2×#1192402711yescharacteristic
586C2×C4#1455663611yescharacteristic
587C2×C4#1455663611yescharacteristic
588C8—41no—
589C8—41no—
590C8—41no—
591C8—41no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1798887314yes—
599C22C2×#1455657111yescharacteristic
600C22#1798923211yescharacteristic
601C22#1798923211yescharacteristic
602C22C22×#1192401711yescharacteristic
603C22C2×#1455662411yescharacteristic
604C22#1798939211yescharacteristic
605C22#1798940411yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—41no—
610C4—41no—
611C4—2922no—
612C4—2922no—
613C4—2922no—
614C4—2922no—
615C4—5841no—
616C4—5841no—
617C4—5841no—
618C4—5841no—
619C4#1798911411yescharacteristic
620C4#1798917211yescharacteristic
621C4#1798929111yescharacteristic
622C4#1798929111yescharacteristic
623C2—24no—
624C2#2261375311yescharacteristic
625C2C2×#1798887311yescharacteristic
626C2#2261978411yescharacteristic
627C2#2262175611yescharacteristic
628C2#2262179711yescharacteristic
629C2#2262198411yescharacteristic
630C2#2262201411yescharacteristic
subgroupG/H
centerC23C22×D292
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#1850C2
socleC23×C73C22×D4
prime radicalC24×C73C2×D4
hypercenter#1850D73
hypocenterC73#26412
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
3. #22611490C2semidirect
4. #22611429C2semidirect
5. #22611311C2semidirect
6. #22611073C2semidirect
7. #22610882C2semidirect
8. #22610510C2semidirect
11. #22610202C2semidirect
13. #22609937C2semidirect
66. #14556570Q8semidirect
331. C73#26412semidirect

Maximal subgroups

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

subgroupindex
#226136332
#226129342
#226114902
#226114292
#226113112
#226110732
#226108822
#226105102
#226103542
#226102362
#226102022
#226101262
#226099372
C2×#179883502
C73×#18502
#2641273