Finite Group Databasebeta

← #34028010

Subgroups — #34028010

Order 18688.

860 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989639C211yesmaximal, characteristic, complemented
2C2×#17989592C211yesmaximal, characteristic
3#22626102C211yesmaximal, characteristic, complemented
4#22626030C211yesmaximal, characteristic, complemented
5#22626030C211yesmaximal, characteristic, complemented
6#22626029C211yesmaximal, characteristic, complemented
7#22625991C211yesmaximal, characteristic, complemented
8#22625990C211yesmaximal, characteristic, complemented
9#22625879C211yesmaximal, characteristic
10#22625877C211yesmaximal, characteristic
11#22625723C211yesmaximal, characteristic, complemented
12#22625672C211yesmaximal, characteristic
13#22625672C211yesmaximal, characteristic
14C4×#14556648C211yesmaximal, characteristic
15#22612465C211yesmaximal, characteristic
16C22×#11924038C2211yescharacteristic
17C22×#11924038C2211yescharacteristic
18#17989639—24no—
19#17989592—24no—
20C2×#14556658C2211yescharacteristic
21C2×#14556658C2211yescharacteristic
22C2×#14556658C2211yescharacteristic
23C2×#14556658C2211yescharacteristic
24C2×#14556658C2211yescharacteristic
25C2×#14556658C2211yescharacteristic
26C2×C4⋊#10596833C2211yescharacteristic
27C2×C4⋊#10596833C2211yescharacteristic
28C2×#14556648C2211yescharacteristic
29C2×#14556648C2211yescharacteristic
30C2×#14556646C2211yescharacteristic
31C2×#14556646C2211yescharacteristic
32C2×C4×#10596833C2211yescharacteristic
33C2×C4×#10596833C2211yescharacteristic
34C2×C4×#10596833C2211yescharacteristic
35C2×C4×#10596833C2211yescharacteristic
36#17989453C2211yescharacteristic
37#17989453C2211yescharacteristic
38#17989453C2211yescharacteristic
39#17989453C2211yescharacteristic
40#17989453C2211yescharacteristic
41#17989453C2211yescharacteristic
42#17989453C2211yescharacteristic
43#17989453C2211yescharacteristic
44#17989453C2211yescharacteristic
45#17989453C2211yescharacteristic
46C2×#14556569C411yescharacteristic
47C2×#14556556C2211yescharacteristic
48C2×C42×D73C2211yescharacteristic
49#17988491C411yescharacteristic
50#17988484C411yescharacteristic
51#17988471C411yescharacteristic
52#17988461C411yescharacteristic
53#17988458C411yescharacteristic
54#17988446C411yescharacteristic
55C4⋊C4×Dic73C2211yescharacteristic
56#17988438C2211yescharacteristic
57#17988438C2211yescharacteristic
58#17988434C2211yescharacteristic
59#17988434C2211yescharacteristic
60#387×C73C411yescharacteristic
61C2×#11924038—22no—
62C2×#11924038—22no—
63C2×#11924038—22no—
64C2×#11924038—22no—
65C2×#11924038—41no—
66C2×#11924038—41no—
67C22×#10596833C2311yescharacteristic
68C22×#10596833C2311yescharacteristic
69C22×#10596833C2311yescharacteristic
70C22×#10596833C2311yescharacteristic
71C22×#10596833C2311yescharacteristic
72C22×#10596833C2311yescharacteristic
73C22×#10596833C2311yescharacteristic
74C22×#10596833C2311yescharacteristic
75#14556658—22no—
76#14556658—22no—
77#14556658—22no—
78#14556658—22no—
79#14556658—41no—
80#14556658D414yes—
81C4⋊#10596833—22no—
82C4⋊#10596833—41no—
83#14556648—22no—
84#14556648Q814yescomplemented
85#14556646—22no—
86#14556646—22no—
87C4×#10596833—22no—
88C4×#10596833—22no—
89C4×#10596833—41no—
90C4×#10596833—41no—
91C22×Dic146C2×C411yescharacteristic
92C22×C4×D73C2×C411yescharacteristic
93C22×C4×D73C2×C411yescharacteristic
94C22×C4×D73C2311yescharacteristic
95#14556569—24no—
96#14556556—22no—
97#14556556—22no—
98C2×C4⋊Dic73C2×C411yescharacteristic
99C2×#11924007C2×C411yescharacteristic
100C2×#11924007C2×C411yescharacteristic
101C2×#11924007C2×C411yescharacteristic
102C2×#11924007C2×C411yescharacteristic
103C2×#11924007C2×C411yescharacteristic
104C2×#11924007C2×C411yescharacteristic
105C2×#11924005C2×C411yescharacteristic
106C2×#11924005C2×C411yescharacteristic
107C2×#11924005C2×C411yescharacteristic
108C2×#11924005C2×C411yescharacteristic
109C2×C4×Dic73C2×C411yescharacteristic
110C2×C4×Dic73C2311yescharacteristic
111C2×C4×Dic73C2311yescharacteristic
112C2×C4×Dic73C2311yescharacteristic
113C2×C4×Dic73C2311yescharacteristic
114C2×C4×Dic73C2311yescharacteristic
115C2×C4×Dic73C2311yescharacteristic
116C42×D73—22no—
117C42×D73—22no—
118#14556498C2×C411yescharacteristic
119#14556498C2×C411yescharacteristic
120#14556498C2×C411yescharacteristic
121#14556498C2×C411yescharacteristic
122#14556498C2×C411yescharacteristic
123#14556498C2×C411yescharacteristic
124C2×C4⋊C4×C73C2×C411yescharacteristic
125C2×C4⋊C4×C73C2×C411yescharacteristic
126C2×C42×C73C2×C411yescharacteristic
127#95×C73C2×C411yescharacteristic
128#95×C73C2×C411yescharacteristic
129#95×C73C2×C411yescharacteristic
130#95×C73C2×C411yescharacteristic
131#11924038—44no—
132#11924038—44no—
133C2×#10596833—21no—
134C2×#10596833—21no—
135C2×#10596833—21no—
136C2×#10596833—21no—
137C2×#10596833—21no—
138C2×#10596833—21no—
139C2×#10596833—21no—
140C2×#10596833—21no—
141C2×#10596833—21no—
142C2×#10596833—21no—
143C2×#10596833—21no—
144C2×#10596833—21no—
145C2×#10596833—21no—
146C2×#10596833—21no—
147C2×#10596833—21no—
148C2×#10596833—21no—
149C2×#10596833—21no—
150C2×#10596833—21no—
151C2×#10596833—21no—
152C2×#10596833—21no—
153C2×#10596833C2×D412yes—
154C2×#10596833C2×D412yes—
155C2×#10596833C2×Q812yes—
156C2×#10596833C2×Q812yes—
157C23×D73C22×C411yescharacteristic
158C2×Dic146—22no—
159C2×Dic146—22no—
160C2×Dic146—22no—
161C22×Dic73C22×C411yescharacteristic
162C22×Dic73C22×C411yescharacteristic
163C22×Dic73C22×C411yescharacteristic
164C22×Dic73C22×C411yescharacteristic
165C22×Dic73C22×C411yescharacteristic
166C22×Dic73C22×C411yescharacteristic
167C22×Dic73C2411yescharacteristic
168C2×C4×D73—21no—
169C2×C4×D73—21no—
170C2×C4×D73—21no—
171C2×C4×D73—21no—
172C2×C4×D73—22no—
173C2×C4×D73—22no—
174C2×C4×D73—22no—
175C2×C4×D73—22no—
176C2×C4×D73—22no—
177C2×C4×D73—41no—
178C2×C4×D73C2×Q812yes—
179C2×C4×D73#4112yes—
180C4⋊Dic73—22no—
181#11924007—22no—
182#11924007—22no—
183#11924007—22no—
184#11924007—22no—
185#11924007—22no—
186#11924007M1614yescomplemented
187#11924005—22no—
188#11924005—22no—
189#11924005—41no—
190#11924005—41no—
191C4×Dic73—22no—
192C4×Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73—21no—
197C4×Dic73—21no—
198C4×Dic73—21no—
199C4×Dic73—21no—
200C4×Dic73—21no—
201C4×Dic73—21no—
202C4×Dic73C2×D412yes—
203C4×Dic73#4112yes—
204C22×C4×C73C22×C411yescharacteristic
205C22×C4×C73C22×C411yescharacteristic
206C22×C4×C73C22×C411yescharacteristic
207C22×C4×C73C22×C411yescharacteristic
208C22×C4×C73C22×C411yescharacteristic
209C22×C4×C73C22×C411yescharacteristic
210C22×C4×C73C22×C411yescharacteristic
211C4⋊C4×C73—22no—
212C4⋊C4×C73—22no—
213C42×C73—22no—
214C23×C73C23×C411yescharacteristic
215C22×D73—21no—
216C22×D73—21no—
217C22×D73—21no—
218C22×D73—21no—
219C22×D73—21no—
220C22×D73—21no—
221C22×D73C4×Q812yes—
222C2×C4×C73—21no—
223C2×C4×C73—21no—
224C2×C4×C73—21no—
225C2×C4×C73—21no—
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×C73—21no—
233C2×C4×C73—21no—
234C2×C4×C73—21no—
235C2×C4×C73—21no—
236C2×C4×C73—21no—
237C2×C4×C73—21no—
238C2×C4×C73—21no—
239C2×C4×C73—21no—
240C2×C4×C73C4×D412yes—
241C2×C4×C73C4×Q812yes—
242C2×C4×C73C2×M1612yes—
243C2×Dic73—21no—
244C2×Dic73—21no—
245C2×Dic73—21no—
246C2×Dic73—21no—
247C2×Dic73—21no—
248C2×Dic73—21no—
249C2×Dic73—21no—
250C2×Dic73—21no—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73C4×D412yes—
257C2×Dic73C2×M1612yes—
258C2×Dic73C2×M1612yes—
259C2×Dic73#13112yes—
260C2×Dic73#13112yes—
261C2×Dic73C22×D411yescharacteristic
262C2×Dic73C22×Q811yescharacteristic
263C2×Dic73C2×#4111yescharacteristic
264C2×Dic73C2×#4111yescharacteristic
265C2×Dic73H(5,𝔽2)11yescharacteristic
266C2×Dic732−1+411yescharacteristic
267C4×D73—22no—
268C4×D73—22no—
269C4×D73—22no—
270C4×D73—22no—
271C4×D73—44no—
272C4×D73—44no—
273Dic146—44no—
274#10596833—22no—
275#10596833—22no—
276#10596833—22no—
277#10596833—22no—
278#10596833—41no—
279#10596833—41no—
280#10596833—41no—
281#10596833—41no—
282C22×C73C2×C4×D411yescharacteristic
283C22×C73C2×C4×Q811yescharacteristic
284C22×C73#52011yescharacteristic
285C22×C73C22×M1611yescharacteristic
286C22×C73C2×#13111yescharacteristic
287C22×C73C2×#13111yescharacteristic
288C22×C73#56811yescharacteristic
289C2×D73—22no—
290C2×D73—22no—
291C2×D73—22no—
292C2×D73—41no—
293C2×D73—41no—
294C2×D73—41no—
295C2×D73—41no—
296C4×C73—22no—
297C4×C73—22no—
298C4×C73—22no—
299C4×C73—41no—
300C4×C73—41no—
301C4×C73—41no—
302C4×C73—41no—
303Dic73—22no—
304Dic73—22no—
305Dic73—41no—
306Dic73—41no—
307Dic73—41no—
308Dic73#43614yes—
309Dic73#54611yescharacteristic
310Dic73D4×Q811yescharacteristic
311Dic73#55211yescharacteristic
312Dic73#55411yescharacteristic
313#22957—731nomaximal, Sylow
314C2×C73#236011yescharacteristic
315C2×C73C2×#43611yescharacteristic
316C2×C73#293311yescharacteristic
317C2×C73#296011yescharacteristic
318C2×C73Q8×M1611yescharacteristic
319C2×C73#299011yescharacteristic
320C2×C73#299311yescharacteristic
321D73—42no—
322C2×#446—731no—
323C2×#423—731no—
324C4×C22⋊Q8—731no—
325#1984—731no—
326#1939—731no—
327#1939—731no—
328#1939—731no—
329#1917—731no—
330#1917—731no—
331#1851—731no—
332#1851—731no—
333C8×C4⋊C4—731no—
334#1753—731no—
335#1753—731no—
336C4×C22⋊C8—731no—
337C73#2295711yescharacteristic, complemented, Sylow
338C22×M16—731no—
339C22×M16—731no—
340C2×C22⋊Q8—731no—
341C2×C4×Q8—731no—
342C22×C42—731no—
343#446—1464no—
344#423—1464no—
345C2×C4⋊C8—731no—
346C2×C4⋊C8—731no—
347C2×C4⋊C8—731no—
348C2×C4⋊C8—731no—
349C2×C4⋊C8—731no—
350C2×C4⋊C8—731no—
351C2×C4⋊C8—731no—
352C2×C4⋊C8—731no—
353C2×C22⋊C8—731no—
354C2×C22⋊C8—731no—
355C2×#97—731no—
356C2×#97—731no—
357C2×C4×C8—731no—
358C2×C4×C8—731no—
359C2×C4×C8—731no—
360C2×C4×C8—731no—
361#391—731no—
362#389—731no—
363#387—731no—
364#387#6440811yescharacteristic
365#385—731no—
366#385—731no—
367#380—731no—
368C4×C4⋊C4—731no—
369C4×C4⋊C4—731no—
370C4×C4⋊C4—731no—
371C4×C22⋊C4—731no—
372C4×C22⋊C4—731no—
373#336—731no—
374#336—731no—
375#336—731no—
376#336—731no—
377#336—731no—
378#336—731no—
379#336—731no—
380#336—731no—
381#336—731no—
382#336—731no—
383C22×Q8—731no—
384C23×C4—731no—
385C23×C4—731no—
386C23×C4—731no—
387C2×M16—2921no—
388C2×M16—2921no—
389C2×M16—1462no—
390C2×M16—1462no—
391C2×M16—1462no—
392C2×M16—1462no—
393C22×C8—731no—
394C22×C8—731no—
395C22×C8—731no—
396C22×C8—731no—
397C22×C8—731no—
398C22×C8—731no—
399C22×C8—731no—
400C22×C8—731no—
401C22⋊Q8—1464no—
402C4×Q8—1462no—
403C4×Q8—1462no—
404C2×C4⋊C4—731no—
405C2×C4⋊C4—731no—
406C2×C4⋊C4—731no—
407C2×C4⋊C4—731no—
408C2×C4⋊C4—731no—
409C2×C4⋊C4—731no—
410C2×C4⋊C4—731no—
411C2×C4⋊C4C2×#6440811yescharacteristic
412C2×C4⋊C4C2×#6440811yescharacteristic
413C2×C22⋊C4—731no—
414C2×C22⋊C4—731no—
415C2×C22⋊C4—731no—
416C2×C22⋊C4—731no—
417C2×C42—731no—
418C2×C42—731no—
419C2×C42—731no—
420C2×C42—731no—
421C2×C42—731no—
422C2×C42—731no—
423C2×C42—731no—
424C2×C42—1462no—
425C2×C42—1462no—
426C2×C42C2×#6440811yescharacteristic
427C4⋊C8—2921no—
428C4⋊C8—2921no—
429C4⋊C8—734no—
430C4⋊C8—1462no—
431C4⋊C8—1462no—
432C4⋊C8—1462no—
433C4⋊C8—1462no—
434C4⋊C8—1462no—
435C22⋊C8—734no—
436C22⋊C8—1462no—
437#97—1462no—
438#97—1462no—
439C4×C8—2921no—
440C4×C8—2921no—
441C4×C8—1462no—
442C4×C8—1462no—
443#95—731no—
444#95—731no—
445#95—731no—
446#95—731no—
447#95—731no—
448#95—731no—
449#95C2×#6440811yescharacteristic
450#95C2×#6440811yescharacteristic
451#95C2×#6440811yescharacteristic
452#95C2×#6440811yescharacteristic
453C24—731no—
454C2×Q8—1462no—
455C2×Q8—1462no—
456C2×Q8—1462no—
457C22×C4—2921no—
458C22×C4—731no—
459C22×C4—731no—
460C22×C4—731no—
461C22×C4—731no—
462C22×C4—732no—
463C22×C4—732no—
464C22×C4—731no—
465C22×C4—731no—
466C22×C4—731no—
467C22×C4—1462no—
468C22×C4—1462no—
469C22×C4—1462no—
470C22×C4—1462no—
471C22×C4—1461no—
472C22×C4—1461no—
473C22×C4—1461no—
474C22×C4—1461no—
475C22×C4—1462no—
476C22×C4C22×#6440811yescharacteristic
477C22×C4C22×#6440811yescharacteristic
478C22×C4C22×#6440811yescharacteristic
479C22×C4C22×#6440811yescharacteristic
480C22×C4C22×#6440811yescharacteristic
481C22×C4C22×#6440811yescharacteristic
482C22×C4C22×#6440811yescharacteristic
483M16—2924no—
484M16—2924no—
485C2×C8—732no—
486C2×C8—732no—
487C2×C8—732no—
488C2×C8—732no—
489C2×C8—1461no—
490C2×C8—1461no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C2×C8—1461no—
496C2×C8—1461no—
497C2×C8—1461no—
498C2×C8—1461no—
499C2×C8—1461no—
500C2×C8—1461no—
501C2×C8—1461no—
502C2×C8—1461no—
503C2×C8—1461no—
504C2×C8—1461no—
505C2×C8—1461no—
506C2×C8—1461no—
507C2×C8—1461no—
508C2×C8—1461no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—734no—
512C4⋊C4—1462no—
513C4⋊C4—1462no—
514C4⋊C4—1462no—
515C4⋊C4—1462no—
516C4⋊C4—1462no—
517C4⋊C4—1462no—
518C22⋊C4—2921no—
519C22⋊C4—2921no—
520C22⋊C4—1462no—
521C22⋊C4—1462no—
522C42—22no—
523C42—732no—
524C42—732no—
525C42—1461no—
526C42—1461no—
527C42—1461no—
528C42—1461no—
529C42—1461no—
530C42—1461no—
531C42—1461no—
532C42—1462no—
533C42—1461no—
534C42—1461no—
535C42—1461no—
536C23—732no—
537C23—1461no—
538C23—1461no—
539C23—1461no—
540C23—1461no—
541C23—1461no—
542C23—1461no—
543C23C23×#6440811yescharacteristic
544Q8—2924no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—2924no—
564C2×C4—2924no—
565C2×C4—731no—
566C2×C4—731no—
567C2×C4—732no—
568C2×C4—731no—
569C2×C4—731no—
570C2×C4—731no—
571C2×C4—731no—
572C2×C4—732no—
573C2×C4—732no—
574C2×C4—732no—
575C2×C4—732no—
576C2×C4—1462no—
577C2×C4—1462no—
578C2×C4—1462no—
579C2×C4—1462no—
580C2×C4—1461no—
581C2×C4—1461no—
582C2×C4—1461no—
583C2×C4—1461no—
584C2×C4—1461no—
585C2×C4—1461no—
586C2×C4—1461no—
587C2×C4—1461no—
588C2×C4—1461no—
589C2×C4—1461no—
590C2×C4—1461no—
591C2×C4—1461no—
592C2×C4—1461no—
593C2×C4D4×#6440812yes—
594C2×C4Q8×#6440812yes—
595C2×C4D73⋊M1612yes—
596C8—2921no—
597C8—2921no—
598C8—2921no—
599C8—2921no—
600C8—1462no—
601C8—1462no—
602C8—1462no—
603C8—1462no—
604C22—2921no—
605C22—2921no—
606C22—2921no—
607C22—2921no—
608C22—1462no—
609C22—1462no—
610C22—1462no—
611C22C2×D4×#6440811yescharacteristic
612C22C2×Q8×#6440811yescharacteristic
613C22#1798965611yescharacteristic
614C22C2×D73⋊M1611yescharacteristic
615C22D73⋊#13111yescharacteristic
616C22D73⋊#13111yescharacteristic
617C22#1798966611yescharacteristic
618C4—22no—
619C4—22no—
620C4—22no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—2921no—
626C4—2921no—
627C4—2921no—
628C4—734no—
629C4—731no—
630C4—731no—
631C4—731no—
632C4—731no—
633C4—1462no—
634C4—1462no—
635C2—2922no—
636C2#2262645511yescharacteristic
637C2D73⋊#43611yescharacteristic
638C2#2262666511yescharacteristic
639C2#2262672111yescharacteristic
640C2#2262673311yescharacteristic
641C2#2262686011yescharacteristic
642C2#2262687511yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
Fitting#387×C73C4
prime radicalC23×D73C22×C4
cosocleC22×Dic73C24
hypercenter#387#64408
hypocenterC73#22957
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C2×#17989639C2semidirect
3. #22626102C2semidirect
4. #22626030C2semidirect
5. #22626030C2semidirect
6. #22626029C2semidirect
7. #22625991C2semidirect
8. #22625990C2semidirect
11. #22625723C2semidirect
84. #14556648Q8semidirect
186. #11924007M16semidirect
337. C73#22957semidirect

Maximal subgroups

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

subgroupindex
C2×#179896392
C2×#179895922
#226261022
#226260302
#226260292
#226259912
#226259902
#226258792
#226258772
#226257232
#226256722
C4×#145566482
#226124652
#2295773