Finite Group Databasebeta

← #34029397

Subgroups — #34029397

Order 18688.

758 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#22626053C211yesmaximal, characteristic
2#22626033C211yesmaximal, characteristic, complemented
3#22626030C211yesmaximal, characteristic, complemented
4#22626007C211yesmaximal, characteristic, complemented
5#22626007C211yesmaximal, characteristic, complemented
6#22626002C211yesmaximal, characteristic, complemented
7#22626002C211yesmaximal, characteristic, complemented
8#22625987C211yesmaximal, characteristic, complemented
9#22625986C211yesmaximal, characteristic, complemented
10#22625911C211yesmaximal, characteristic
11#22625879C211yesmaximal, characteristic, complemented
12#22625879C211yesmaximal, characteristic, complemented
13#22625788C211yesmaximal, characteristic, complemented
14#22625788C211yesmaximal, characteristic, complemented
15#22612452C211yesmaximal, characteristic
16C22×#11924038C2211yescharacteristic
17C2×#14556658C2211yescharacteristic
18C2×#14556658C2211yescharacteristic
19C2×#14556658C2211yescharacteristic
20C2×#14556658C2211yescharacteristic
21C2×#14556658C2211yescharacteristic
22C2×#14556658C2211yescharacteristic
23C2×C4⋊#10596833C2211yescharacteristic
24C2×#14556648C2211yescharacteristic
25C2×#14556648C2211yescharacteristic
26C2×#14556648C2211yescharacteristic
27C2×#14556648C2211yescharacteristic
28C2×#14556648C2211yescharacteristic
29C2×#14556648C2211yescharacteristic
30C2×#14556648C2211yescharacteristic
31C2×#14556646C2211yescharacteristic
32C2×C4×#10596833C2211yescharacteristic
33C2×C4×#10596833C2211yescharacteristic
34#17989453C2211yescharacteristic
35#17989453C2211yescharacteristic
36#17989453C2211yescharacteristic
37#17989453C2211yescharacteristic
38#17989453C2211yescharacteristic
39#17989453C2211yescharacteristic
40#17989453C2211yescharacteristic
41#17989453C2211yescharacteristic
42#17989453C2211yescharacteristic
43#17989453C2211yescharacteristic
44C2×#14556576C411yescharacteristic
45C2×#14556553C2211yescharacteristic
46C2×C42×D73C2211yescharacteristic
47#17988479C411yescharacteristic
48#17988478C411yescharacteristic
49#17988468C411yescharacteristic
50#17988465C411yescharacteristic
51#17988461C411yescharacteristic
52#17988449C411yescharacteristic
53C4⋊C4×Dic73C2211yescharacteristic
54#17988438C2211yescharacteristic
55#17988434C2211yescharacteristic
56#17988434C2211yescharacteristic
57#17988434C2211yescharacteristic
58#387×C73C411yescharacteristic
59C2×#11924038—41no—
60C2×#11924038—41no—
61C2×#11924038—41no—
62C22×#10596833C2311yescharacteristic
63C22×#10596833C2311yescharacteristic
64C22×#10596833C2311yescharacteristic
65C22×#10596833C2311yescharacteristic
66C22×#10596833C2311yescharacteristic
67C22×#10596833C2311yescharacteristic
68C22×#10596833C2311yescharacteristic
69C22×#10596833C2311yescharacteristic
70#14556658—41no—
71#14556658—41no—
72#14556658—41no—
73#14556658—41no—
74#14556658—41no—
75#14556658—41no—
76C4⋊#10596833—41no—
77#14556648—41no—
78#14556648—41no—
79#14556648—41no—
80#14556648—41no—
81#14556648—41no—
82#14556648—41no—
83#14556648—41no—
84#14556646—41no—
85C4×#10596833—41no—
86C4×#10596833—41no—
87C22×D292C2×C411yescharacteristic
88C22×C4×D73C2×C411yescharacteristic
89C22×C4×D73C2×C411yescharacteristic
90C22×C4×D73C2311yescharacteristic
91C22×C4×D73C2311yescharacteristic
92#14556576—24no—
93#14556553—41no—
94#14556553—41no—
95C2×C4⋊Dic73C2×C411yescharacteristic
96C2×#11924007C2×C411yescharacteristic
97C2×#11924007C2×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×C4×Dic73C2×C411yescharacteristic
107C2×C4×Dic73C2311yescharacteristic
108C2×C4×Dic73C2311yescharacteristic
109C2×C4×Dic73C2311yescharacteristic
110C2×C4×Dic73C2311yescharacteristic
111C2×C4×Dic73C2311yescharacteristic
112C42×D73—41no—
113C42×D73—41no—
114#14556498C2×C411yescharacteristic
115#14556498C2×C411yescharacteristic
116#14556498C2×C411yescharacteristic
117#14556498C2×C411yescharacteristic
118#14556498C2×C411yescharacteristic
119#14556498C2×C411yescharacteristic
120C2×C4⋊C4×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—
128C2×#10596833—21no—
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—21no—
152C23×D73C22×C411yescharacteristic
153C23×D73C22×C411yescharacteristic
154C2×D292—22no—
155C2×D292—22no—
156C2×D292—41no—
157C22×Dic73C22×C411yescharacteristic
158C22×Dic73C22×C411yescharacteristic
159C22×Dic73C22×C411yescharacteristic
160C22×Dic73C22×C411yescharacteristic
161C22×Dic73C22×C411yescharacteristic
162C22×Dic73C2411yescharacteristic
163C2×C4×D73—22no—
164C2×C4×D73—22no—
165C2×C4×D73—22no—
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—
176C2×C4×D73—21no—
177C2×C4×D73—21no—
178C2×C4×D73—41no—
179C2×C4×D73—41no—
180C2×C4×D73—41no—
181C4⋊Dic73—22no—
182#11924007—41no—
183#11924007M1614yescomplemented
184#11924005—22no—
185#11924005—22no—
186#11924005—22no—
187#11924005—22no—
188#11924005—22no—
189#11924005—41no—
190#11924005—41no—
191#11924005—41no—
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×Dic73—41no—
203C22×C4×C73C22×C411yescharacteristic
204C22×C4×C73C22×C411yescharacteristic
205C22×C4×C73C22×C411yescharacteristic
206C22×C4×C73C22×C411yescharacteristic
207C22×C4×C73C22×C411yescharacteristic
208C22×C4×C73C22×C411yescharacteristic
209C22×C4×C73C22×C411yescharacteristic
210C4⋊C4×C73—22no—
211C4⋊C4×C73—41no—
212C42×C73—41no—
213C23×C73C23×C411yescharacteristic
214C22×D73—21no—
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#13112yes—
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×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×C73C2×M1612yes—
249C2×Dic73—21no—
250C2×Dic73—21no—
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×Dic73C2×M1612yes—
262C2×Dic73C2×M1612yes—
263C2×Dic73#13112yes—
264C2×Dic73H(5,𝔽2)11yescharacteristic
265C2×Dic73H(5,𝔽2)11yescharacteristic
266C2×Dic73H(5,𝔽2)11yescharacteristic
267C2×Dic73H(5,𝔽2)11yescharacteristic
268C2×Dic732−1+411yescharacteristic
269C2×Dic732−1+411yescharacteristic
270D292—44no—
271C4×D73—44no—
272C4×D73—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—41no—
282#10596833—41no—
283#10596833—41no—
284#10596833—41no—
285#10596833—41no—
286#10596833—41no—
287#10596833—41no—
288#10596833—41no—
289C22×C73#51811yescharacteristic
290C22×C73#52011yescharacteristic
291C22×C73#52011yescharacteristic
292C22×C73C22×M1611yescharacteristic
293C22×C73C2×#13111yescharacteristic
294C22×C73C2×#13111yescharacteristic
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—41no—
312C4×C73—41no—
313C4×C73—41no—
314C4×C73—41no—
315C4×C73—41no—
316C4×C73—41no—
317Dic73—22no—
318Dic73—22no—
319Dic73—22no—
320Dic73—41no—
321Dic73—41no—
322Dic73#56011yescharacteristic
323Dic73#56211yescharacteristic
324Dic73#56211yescharacteristic
325Dic73#56311yescharacteristic
326#23320—731nomaximal, Sylow
327C2×C73#237611yescharacteristic
328C2×C73#297411yescharacteristic
329C2×C73#297611yescharacteristic
330C2×C73#298911yescharacteristic
331C2×C73#299111yescharacteristic
332C2×C73#299211yescharacteristic
333C2×C73#299411yescharacteristic
334D73—42no—
335D73—81no—
336C4×#123—731no—
337#1954—731no—
338#1940—731no—
339#1939—731no—
340#1924—731no—
341#1924—731no—
342#1923—731no—
343#1923—731no—
344#1916—731no—
345#1916—731no—
346#1870—731no—
347#1851—731no—
348#1851—731no—
349#1797—731no—
350#1797—731no—
351C73#2332011yescharacteristic, complemented, Sylow
352C22×M16—731no—
353C2×#123—731no—
354C2×C4×D4—731no—
355C22×C42—731no—
356C2×C4⋊C8—731no—
357C2×C4⋊C8—731no—
358C2×C4⋊C8—731no—
359C2×C4⋊C8—731no—
360C2×C4⋊C8—731no—
361C2×C4⋊C8—731no—
362C2×C4⋊C8—731no—
363C2×C22⋊C8—731no—
364C2×C22⋊C8—731no—
365C2×C22⋊C8—731no—
366C2×C22⋊C8—731no—
367C2×C22⋊C8—731no—
368C2×C22⋊C8—731no—
369C2×C22⋊C8—731no—
370C2×#97—731no—
371C2×C4×C8—731no—
372C2×C4×C8—731no—
373#388—731no—
374#388—731no—
375#387—731no—
376#387#6440811yescharacteristic
377#386—731no—
378#385—731no—
379#381—731no—
380C4×C4⋊C4—731no—
381C4×C4⋊C4—731no—
382C4×C22⋊C4—731no—
383C4×C22⋊C4—731no—
384C4×C22⋊C4—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×D4—731no—
396C23×C4—731no—
397C23×C4—731no—
398C23×C4—731no—
399C23×C4—731no—
400C2×M16—2921no—
401C2×M16—2921no—
402C2×M16—2921no—
403C22×C8—731no—
404C22×C8—731no—
405C22×C8—731no—
406C22×C8—731no—
407C22×C8—731no—
408C22×C8—731no—
409C22×C8—731no—
410C22×C8—731no—
411#123—1464no—
412C4×D4—2921no—
413C4×D4—2921no—
414C2×C4⋊C4—731no—
415C2×C4⋊C4—731no—
416C2×C4⋊C4—731no—
417C2×C4⋊C4C2×#6440811yescharacteristic
418C2×C4⋊C4C2×#6440811yescharacteristic
419C2×C22⋊C4—731no—
420C2×C22⋊C4—731no—
421C2×C22⋊C4—731no—
422C2×C22⋊C4—731no—
423C2×C22⋊C4—731no—
424C2×C22⋊C4—731no—
425C2×C22⋊C4—731no—
426C2×C22⋊C4—731no—
427C2×C42—2921no—
428C2×C42—2921no—
429C2×C42—731no—
430C2×C42—731no—
431C2×C42—731no—
432C2×C42—731no—
433C2×C42—731no—
434C2×C42—731no—
435C2×C42C2×#6440811yescharacteristic
436C4⋊C8—2921no—
437C4⋊C8—2921no—
438C4⋊C8—2921no—
439C4⋊C8—2921no—
440C4⋊C8—2921no—
441C4⋊C8—2921no—
442C4⋊C8—2921no—
443C22⋊C8—2921no—
444C22⋊C8—2921no—
445C22⋊C8—2921no—
446C22⋊C8—2921no—
447C22⋊C8—2921no—
448C22⋊C8—2921no—
449C22⋊C8—2921no—
450#97—2921no—
451C4×C8—2921no—
452C4×C8—2921no—
453#95—731no—
454#95—731no—
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×D4—2921no—
466C2×D4—1462no—
467C2×D4—1462no—
468C22×C4—2921no—
469C22×C4—2921no—
470C22×C4—2921no—
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—1462no—
486C22×C4—1461no—
487C22×C4—1461no—
488C22×C4—1461no—
489C22×C4—1461no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4C22×#6440811yescharacteristic
493C22×C4C22×#6440811yescharacteristic
494C22×C4C22×#6440811yescharacteristic
495C22×C4C22×#6440811yescharacteristic
496C22×C4C22×#6440811yescharacteristic
497C22×C4C22×#6440811yescharacteristic
498C22×C4C22×#6440811yescharacteristic
499M16—2924no—
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—
522C2×C8—1461no—
523C2×C8—1461no—
524C4⋊C4—22no—
525C4⋊C4—41no—
526C4⋊C4—2921no—
527C4⋊C4—734no—
528C4⋊C4—1462no—
529C22⋊C4—2921no—
530C22⋊C4—2921no—
531C22⋊C4—2921no—
532C22⋊C4—1462no—
533C22⋊C4—1462no—
534C22⋊C4—1462no—
535C22⋊C4—1462no—
536C22⋊C4—1462no—
537C42—41no—
538C42—2921no—
539C42—1461no—
540C42—1461no—
541C42—1461no—
542C42—1461no—
543C42—1461no—
544C42—1461no—
545C42—1461no—
546C42—1461no—
547C42—1461no—
548C42—1461no—
549C23—732no—
550C23—1461no—
551C23—1461no—
552C23—1461no—
553C23—1461no—
554C23—1461no—
555C23—1461no—
556C23—1461no—
557C23—1461no—
558C23—1461no—
559C23—1461no—
560C23—1461no—
561C23—1461no—
562C23—1461no—
563C23C23×#6440811yescharacteristic
564D4—2924no—
565C2×C4—21no—
566C2×C4—21no—
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—2921no—
586C2×C4—2921no—
587C2×C4—2921no—
588C2×C4—2921no—
589C2×C4—2921no—
590C2×C4—2921no—
591C2×C4—2921no—
592C2×C4—2921no—
593C2×C4—2924no—
594C2×C4—2924no—
595C2×C4—731no—
596C2×C4—731no—
597C2×C4—731no—
598C2×C4—731no—
599C2×C4—731no—
600C2×C4—731no—
601C2×C4—732no—
602C2×C4—732no—
603C2×C4—732no—
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×C4—1461no—
616C2×C4D73⋊M1612yes—
617C8—2921no—
618C8—2921no—
619C8—2921no—
620C8—2921no—
621C8—2921no—
622C8—2921no—
623C8—2921no—
624C8—2921no—
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#1798965411yescharacteristic
640C22#1798965611yescharacteristic
641C22#1798965611yescharacteristic
642C22C2×D73⋊M1611yescharacteristic
643C22D73⋊#13111yescharacteristic
644C22D73⋊#13111yescharacteristic
645C22#1798966711yescharacteristic
646C4—22no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—2921no—
654C4—2921no—
655C4—731no—
656C4—731no—
657C4—731no—
658C4—731no—
659C4—1462no—
660C4—1462no—
661C4—1462no—
662C2—2922no—
663C2—5841no—
664C2#2262647111yescharacteristic
665C2#2262678811yescharacteristic
666C2#2262680211yescharacteristic
667C2#2262685511yescharacteristic
668C2#2262686411yescharacteristic
669C2#2262687111yescharacteristic
670C2#2262688011yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
Fitting#387×C73C4
prime radicalC22×D292C2×C4
cosocleC22×Dic73C24
hypercenter#387#64408
hypocenterC73#23320
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #22626033C2semidirect
3. #22626030C2semidirect
4. #22626007C2semidirect
5. #22626007C2semidirect
6. #22626002C2semidirect
7. #22626002C2semidirect
8. #22625987C2semidirect
9. #22625986C2semidirect
11. #22625879C2semidirect
12. #22625879C2semidirect
13. #22625788C2semidirect
14. #22625788C2semidirect
183. #11924007M16semidirect
351. C73#23320semidirect

Maximal subgroups

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

subgroupindex
#226260532
#226260332
#226260302
#226260072
#226260022
#226259872
#226259862
#226259112
#226258792
#226257882
#226124522
#2332073