Finite Group Databasebeta

← #34026010

Subgroups — #34026010

Order 18688.

768 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626105C211yesmaximal, characteristic, complemented
2#22626100C211yesmaximal, characteristic, complemented
3#22626007C211yesmaximal, characteristic, complemented
4#22626007C211yesmaximal, characteristic, complemented
5#22626006C211yesmaximal, characteristic, complemented
6#22626002C211yesmaximal, characteristic, complemented
7#22625911C211yesmaximal, characteristic
8#22625907C211yesmaximal, characteristic
9#22625879C211yesmaximal, characteristic, complemented
10#22625876C211yesmaximal, characteristic, complemented
11#22625858C211yesmaximal, characteristic, complemented
12#22625716C211yesmaximal, characteristic, complemented
13#22625672C211yesmaximal, characteristic, complemented
14#22625669C211yesmaximal, characteristic, complemented
15#22612474C211yesmaximal, characteristic
16C22×#11924038C2211yescharacteristic
17C22×#11924036C2211yescharacteristic
18C2×#14556658C2211yescharacteristic
19C2×#14556658C2211yescharacteristic
20C2×#14556658C2211yescharacteristic
21C2×C4⋊#10596833C2211yescharacteristic
22C2×#14556648C2211yescharacteristic
23C2×#14556648C2211yescharacteristic
24C2×#14556648C2211yescharacteristic
25C2×#14556648C2211yescharacteristic
26C2×#14556648C2211yescharacteristic
27C2×#14556648C2211yescharacteristic
28C2×#14556646C2211yescharacteristic
29C2×#14556646C2211yescharacteristic
30C2×#14556646C2211yescharacteristic
31C2×C4×#10596833C2211yescharacteristic
32#17989453C2211yescharacteristic
33#17989453C2211yescharacteristic
34#17989453C2211yescharacteristic
35#17989453C2211yescharacteristic
36#17989453C2211yescharacteristic
37#17989453C2211yescharacteristic
38#17989453C2211yescharacteristic
39#17989453C2211yescharacteristic
40#17989453C2211yescharacteristic
41#17989453C2211yescharacteristic
42#17989453C2211yescharacteristic
43#17989453C2211yescharacteristic
44C2×#14556580C411yescharacteristic
45C2×Q8×Dic73C2211yescharacteristic
46C2×#14556553C2211yescharacteristic
47#17988489C411yescharacteristic
48#17988479C411yescharacteristic
49#17988479C411yescharacteristic
50#17988479C411yescharacteristic
51#17988479C411yescharacteristic
52#17988452C411yescharacteristic
53#17988434C2211yescharacteristic
54#17988434C2211yescharacteristic
55#17988434C2211yescharacteristic
56#17988434C2211yescharacteristic
57C42×Dic73C2211yescharacteristic
58#391×C73C411yescharacteristic
59C2×#11924038—22no—
60C2×#11924038—41no—
61C2×#11924038—41no—
62C2×#11924036—22no—
63C2×#11924036—41no—
64C2×#11924036—41no—
65C22×#10596833C2311yescharacteristic
66C22×#10596833C2311yescharacteristic
67C22×#10596833C2311yescharacteristic
68C22×#10596833C2311yescharacteristic
69C22×#10596833C2311yescharacteristic
70C22×#10596833C2311yescharacteristic
71C22×#10596833C2311yescharacteristic
72C22×#10596833C2311yescharacteristic
73#14556658—22no—
74#14556658—22no—
75#14556658—41no—
76C4⋊#10596833—41no—
77#14556648—22no—
78#14556648—22no—
79#14556648—41no—
80#14556648—41no—
81#14556648—41no—
82#14556648—41no—
83#14556646—41no—
84#14556646—41no—
85#14556646—41no—
86C4×#10596833—41no—
87C22×D292C2×C411yescharacteristic
88C22×C4×D73C2311yescharacteristic
89C22×C4×D73C2311yescharacteristic
90#14556580—24no—
91Q8×Dic73—41no—
92Q8×Dic73—41no—
93#14556553—41no—
94#14556553—41no—
95C2×C4⋊Dic73C2×C411yescharacteristic
96C2×C4⋊Dic73C2×C411yescharacteristic
97C2×C4⋊Dic73C2×C411yescharacteristic
98C2×#11924005C2×C411yescharacteristic
99C2×#11924005C2×C411yescharacteristic
100C2×#11924005C2×C411yescharacteristic
101C2×#11924005C2×C411yescharacteristic
102C2×#11924005C2×C411yescharacteristic
103C2×#11924005C2×C411yescharacteristic
104C2×#11924005C2×C411yescharacteristic
105C2×#11924005C2×C411yescharacteristic
106C2×#11924005C2×C411yescharacteristic
107C2×#11924005C2×C411yescharacteristic
108C2×C4×Dic73C2×C411yescharacteristic
109C2×C4×Dic73C2×C411yescharacteristic
110C2×C4×Dic73C2×C411yescharacteristic
111C2×C4×Dic73C2311yescharacteristic
112C2×C4×Dic73C2311yescharacteristic
113C2×C4×Dic73C2311yescharacteristic
114C2×C4×Dic73C2311yescharacteristic
115C2×C4×Dic73C2311yescharacteristic
116#14556498C2×C411yescharacteristic
117#14556498C2×C411yescharacteristic
118#14556498C2×C411yescharacteristic
119#14556498C2×C411yescharacteristic
120C22×Q8×C73C2×C411yescharacteristic
121C2×C4⋊C4×C73C2×C411yescharacteristic
122C2×C42×C73C2×C411yescharacteristic
123#95×C73C2×C411yescharacteristic
124#95×C73C2×C411yescharacteristic
125#95×C73C2×C411yescharacteristic
126#95×C73C2×C411yescharacteristic
127#11924038—44no—
128#11924036—44no—
129C2×#10596833—21no—
130C2×#10596833—21no—
131C2×#10596833—21no—
132C2×#10596833—21no—
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#4112yes—
152C2×#10596833#4112yes—
153C23×D73C22×C411yescharacteristic
154C23×D73C22×C411yescharacteristic
155C2×D292—22no—
156C2×D292—22no—
157C2×D292—41no—
158C22×Dic73C22×C411yescharacteristic
159C22×Dic73C22×C411yescharacteristic
160C22×Dic73C22×C411yescharacteristic
161C22×Dic73C22×C411yescharacteristic
162C22×Dic73C22×C411yescharacteristic
163C22×Dic73C2411yescharacteristic
164C2×C4×D73—21no—
165C2×C4×D73—21no—
166C2×C4×D73—21no—
167C2×C4×D73—21no—
168C2×C4×D73—21no—
169C2×C4×D73—21no—
170C2×C4×D73—21no—
171C2×C4×D73—21no—
172C2×C4×D73—21no—
173C2×C4×D73—21no—
174C2×C4×D73—21no—
175C2×C4×D73—21no—
176C4⋊Dic73—41no—
177C4⋊Dic73—41no—
178C4⋊Dic73—41no—
179#11924005—22no—
180#11924005—22no—
181#11924005—22no—
182#11924005—41no—
183#11924005—41no—
184#11924005—41no—
185#11924005—41no—
186#11924005—41no—
187#11924005—41no—
188#11924005M1614yescomplemented
189C4×Dic73—22no—
190C4×Dic73—21no—
191C4×Dic73—21no—
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—41no—
201C4×Dic73—41no—
202C2×Q8×C73—22no—
203C2×Q8×C73—22no—
204C2×Q8×C73—41no—
205C22×C4×C73C22×C411yescharacteristic
206C22×C4×C73C22×C411yescharacteristic
207C22×C4×C73C22×C411yescharacteristic
208C22×C4×C73C22×C411yescharacteristic
209C22×C4×C73C22×C411yescharacteristic
210C22×C4×C73C22×C411yescharacteristic
211C22×C4×C73C22×C411yescharacteristic
212C4⋊C4×C73—41no—
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×D73—21no—
222C22×D73—21no—
223C22×D73—21no—
224C22×D73—21no—
225C22×D73—21no—
226C22×D73—21no—
227C22×D73—21no—
228C22×D73C2×M1612yes—
229Q8×C73—44no—
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×C73—21no—
241C2×C4×C73—21no—
242C2×C4×C73—21no—
243C2×C4×C73—21no—
244C2×C4×C73—21no—
245C2×C4×C73—21no—
246C2×C4×C73—21no—
247C2×C4×C73—21no—
248C2×C4×C73—21no—
249C2×C4×C73C2×M1612yes—
250C2×C4×C73#13112yes—
251C2×Dic73—21no—
252C2×Dic73—21no—
253C2×Dic73—21no—
254C2×Dic73—21no—
255C2×Dic73—21no—
256C2×Dic73—21no—
257C2×Dic73—21no—
258C2×Dic73—21no—
259C2×Dic73—21no—
260C2×Dic73—21no—
261C2×Dic73—21no—
262C2×Dic73—21no—
263C2×Dic73—21no—
264C2×Dic73—21no—
265C2×Dic73C2×M1612yes—
266C2×Dic73C2×#4111yescharacteristic
267C2×Dic73C2×#4111yescharacteristic
268C2×Dic73H(5,𝔽2)11yescharacteristic
269C2×Dic73H(5,𝔽2)11yescharacteristic
270C2×Dic73H(5,𝔽2)11yescharacteristic
271C2×Dic732−1+411yescharacteristic
272D292—44no—
273C4×D73—41no—
274C4×D73—41no—
275C4×D73—41no—
276C4×D73—41no—
277C4×D73—41no—
278C4×D73—41no—
279C4×D73—41no—
280C4×D73—41no—
281#10596833—22no—
282#10596833—22no—
283#10596833—41no—
284#10596833—41no—
285#10596833—41no—
286#10596833—41no—
287#10596833—41no—
288#10596833—41no—
289C22×C73C4×#4111yescharacteristic
290C22×C73#51811yescharacteristic
291C22×C73#52011yescharacteristic
292C22×C73C22×M1611yescharacteristic
293C22×C73C2×#13111yescharacteristic
294C22×C73#56811yescharacteristic
295C22×C73#56811yescharacteristic
296C2×D73—22no—
297C2×D73—22no—
298C2×D73—22no—
299C2×D73—41no—
300C2×D73—41no—
301C2×D73—41no—
302C2×D73—41no—
303C2×D73—41no—
304C2×D73—41no—
305C2×D73—41no—
306C2×D73—41no—
307C2×D73—41no—
308C2×D73—41no—
309C2×D73—41no—
310C4×C73—22no—
311C4×C73—22no—
312C4×C73—41no—
313C4×C73—41no—
314C4×C73—41no—
315C4×C73—41no—
316C4×C73—41no—
317Dic73—22no—
318Dic73—41no—
319Dic73—41no—
320Dic73—41no—
321Dic73—41no—
322Dic73#53811yescharacteristic
323Dic73#53811yescharacteristic
324Dic73#54211yescharacteristic
325Dic73#54211yescharacteristic
326#22391—731nomaximal, Sylow
327C2×C73#233711yescharacteristic
328C2×C73#296911yescharacteristic
329C2×C73#297011yescharacteristic
330C2×C73#297411yescharacteristic
331C2×C73#298011yescharacteristic
332C2×C73#298911yescharacteristic
333C2×C73#299311yescharacteristic
334D73—42no—
335D73—81no—
336C4×#124—731no—
337#1985—731no—
338#1984—731no—
339#1924—731no—
340#1924—731no—
341#1924—731no—
342#1923—731no—
343#1870—731no—
344#1869—731no—
345#1851—731no—
346#1850—731no—
347#1839—731no—
348#1767—731no—
349#1753—731no—
350#1752—731no—
351C73#2239111yescharacteristic, complemented, Sylow
352C22×M16—731no—
353C23×C8—731no—
354C2×#124—731no—
355C2×C4×Q8—731no—
356C2×C4×D4—731no—
357C2×C4⋊C8—731no—
358C2×C4⋊C8—731no—
359C2×C4⋊C8—731no—
360C2×C4⋊C8—731no—
361C2×C22⋊C8—731no—
362C2×C22⋊C8—731no—
363C2×C22⋊C8—731no—
364C2×C22⋊C8—731no—
365C2×C22⋊C8—731no—
366C2×C22⋊C8—731no—
367C2×#97—731no—
368C2×#97—731no—
369C2×#97—731no—
370C2×C4×C8—731no—
371#391#6440811yescharacteristic
372#390—731no—
373#388—731no—
374#388—731no—
375#388—731no—
376#388—731no—
377#382—731no—
378C4×C22⋊C4—731no—
379C4×C22⋊C4—731no—
380C4×C22⋊C4—731no—
381C4×C22⋊C4—731no—
382C43—731no—
383#336—731no—
384#336—731no—
385#336—731no—
386#336—731no—
387#336—731no—
388#336—731no—
389#336—731no—
390#336—731no—
391#336—731no—
392#336—731no—
393#336—731no—
394#336—731no—
395C22×Q8C2×#6440811yescharacteristic
396C22×D4—731no—
397C23×C4—731no—
398C23×C4—731no—
399C2×M16—2921no—
400C2×M16—2921no—
401C2×M16—1462no—
402C22×C8—2921no—
403C22×C8—2921no—
404C22×C8—731no—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8—731no—
408C22×C8—731no—
409C22×C8—731no—
410C22×C8—731no—
411C22×C8—731no—
412C22×C8—1462no—
413#124—1464no—
414C4×Q8—2921no—
415C4×Q8—2921no—
416C4×D4—2921no—
417C4×D4—2921no—
418C2×C4⋊C4—731no—
419C2×C4⋊C4—731no—
420C2×C4⋊C4—731no—
421C2×C4⋊C4C2×#6440811yescharacteristic
422C2×C22⋊C4—731no—
423C2×C22⋊C4—731no—
424C2×C22⋊C4—731no—
425C2×C22⋊C4—731no—
426C2×C22⋊C4—731no—
427C2×C22⋊C4—731no—
428C2×C22⋊C4—731no—
429C2×C22⋊C4—731no—
430C2×C22⋊C4—731no—
431C2×C22⋊C4—731no—
432C2×C42—731no—
433C2×C42—731no—
434C2×C42—731no—
435C2×C42—731no—
436C2×C42—731no—
437C2×C42—731no—
438C2×C42—731no—
439C2×C42—731no—
440C2×C42C2×#6440811yescharacteristic
441C4⋊C8—2921no—
442C4⋊C8—2921no—
443C4⋊C8—1462no—
444C4⋊C8—1462no—
445C22⋊C8—2921no—
446C22⋊C8—2921no—
447C22⋊C8—2921no—
448C22⋊C8—2921no—
449C22⋊C8—1462no—
450C22⋊C8—1462no—
451#97—2921no—
452#97—2921no—
453#97—2921no—
454C4×C8—2921no—
455#95—731no—
456#95—731no—
457#95—731no—
458#95—731no—
459#95C2×#6440811yescharacteristic
460#95C2×#6440811yescharacteristic
461#95C2×#6440811yescharacteristic
462#95C2×#6440811yescharacteristic
463C24—731no—
464C24—731no—
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—41no—
468C2×D4—2921no—
469C2×D4—1462no—
470C2×D4—1462no—
471C22×C4—731no—
472C22×C4—731no—
473C22×C4—731no—
474C22×C4—731no—
475C22×C4—731no—
476C22×C4—731no—
477C22×C4—1461no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1461no—
482C22×C4—1461no—
483C22×C4—1461no—
484C22×C4—1461no—
485C22×C4—1461no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4C22×#6440811yescharacteristic
490C22×C4C22×#6440811yescharacteristic
491C22×C4C22×#6440811yescharacteristic
492C22×C4C22×#6440811yescharacteristic
493C22×C4C22×#6440811yescharacteristic
494C22×C4C22×#6440811yescharacteristic
495C22×C4C22×#6440811yescharacteristic
496M16—2924no—
497C2×C8—2924no—
498C2×C8—732no—
499C2×C8—732no—
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—
509C2×C8—1461no—
510C2×C8—1461no—
511C2×C8—1461no—
512C2×C8—1461no—
513C2×C8—1461no—
514C2×C8—1461no—
515C2×C8—1461no—
516C2×C8—1461no—
517C2×C8—1461no—
518C2×C8—1461no—
519C2×C8—1461no—
520C2×C8—1461no—
521C2×C8—1461no—
522C4⋊C4—41no—
523C4⋊C4—2921no—
524C4⋊C4—2921no—
525C4⋊C4—2921no—
526C22⋊C4—2921no—
527C22⋊C4—2921no—
528C22⋊C4—2921no—
529C22⋊C4—2921no—
530C22⋊C4—2921no—
531C22⋊C4—2921no—
532C22⋊C4—734no—
533C22⋊C4—1462no—
534C22⋊C4—1462no—
535C22⋊C4—1462no—
536C42—22no—
537C42—2921no—
538C42—2921no—
539C42—1461no—
540C42—1461no—
541C42—1461no—
542C42—1461no—
543C42—1461no—
544C42—1461no—
545C42—1461no—
546C42—1461no—
547C42—1461no—
548C42—1461no—
549C42—1462no—
550C23—732no—
551C23—1461no—
552C23—1461no—
553C23—1461no—
554C23—1461no—
555C23—1461no—
556C23—1461no—
557C23—1461no—
558C23—1461no—
559C23—1461no—
560C23—1461no—
561C23—1461no—
562C23—1461no—
563C23—1461no—
564C23C23×#6440811yescharacteristic
565Q8—44no—
566D4—2924no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2921no—
594C2×C4—731no—
595C2×C4—731no—
596C2×C4—731no—
597C2×C4—731no—
598C2×C4—731no—
599C2×C4—731no—
600C2×C4—732no—
601C2×C4—1461no—
602C2×C4—1461no—
603C2×C4—1461no—
604C2×C4—1461no—
605C2×C4—1461no—
606C2×C4—1461no—
607C2×C4—1461no—
608C2×C4—1461no—
609C2×C4—1461no—
610C2×C4—1461no—
611C2×C4—1461no—
612C2×C4—1461no—
613C2×C4—1461no—
614C2×C4—1461no—
615C2×C4D73⋊M1612yes—
616C2×C4#1455667812yes—
617C8—2921no—
618C8—2921no—
619C8—2921no—
620C8—2921no—
621C8—2921no—
622C8—2921no—
623C8—1462no—
624C8—1462no—
625C22—2921no—
626C22—2921no—
627C22—2921no—
628C22—2921no—
629C22—2921no—
630C22—2921no—
631C22—2921no—
632C22—2921no—
633C22—2921no—
634C22—2921no—
635C22—2921no—
636C22—1462no—
637C22—1462no—
638C22—1462no—
639C22#41×#6440811yescharacteristic
640C22#1798965411yescharacteristic
641C22#1798965611yescharacteristic
642C22C2×D73⋊M1611yescharacteristic
643C22C2×#1455667811yescharacteristic
644C22#1798966711yescharacteristic
645C22#1798966711yescharacteristic
646C4—22no—
647C4—22no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—2921no—
654C4—2921no—
655C4—2921no—
656C4—2921no—
657C4—731no—
658C4—731no—
659C4—731no—
660C4—731no—
661C4—1462no—
662C2—2922no—
663C2—5841no—
664C2#2262643211yescharacteristic
665C2#2262676411yescharacteristic
666C2#2262677311yescharacteristic
667C2#2262679011yescharacteristic
668C2#2262681811yescharacteristic
669C2#2262685511yescharacteristic
670C2#2262687611yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
Fitting#391×C73C4
prime radicalC22×D292C2×C4
cosocleC22×Dic73C24
hypercenter#391#64408
hypocenterC73#22391
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #22626105C2semidirect
2. #22626100C2semidirect
3. #22626007C2semidirect
4. #22626007C2semidirect
5. #22626006C2semidirect
6. #22626002C2semidirect
9. #22625879C2semidirect
10. #22625876C2semidirect
11. #22625858C2semidirect
12. #22625716C2semidirect
13. #22625672C2semidirect
14. #22625669C2semidirect
188. #11924005M16semidirect
351. C73#22391semidirect

Maximal subgroups

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

subgroupindex
#226261052
#226261002
#226260072
#226260062
#226260022
#226259112
#226259072
#226258792
#226258762
#226258582
#226257162
#226256722
#226256692
#226124742
#2239173