Finite Group Databasebeta

← #34028042

Subgroups — #34028042

Order 18688.

788 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989639C211yesmaximal, characteristic, complemented
2C2×#17989594C211yesmaximal, characteristic
3#22626104C211yesmaximal, characteristic, complemented
4#22626032C211yesmaximal, characteristic, complemented
5#22626028C211yesmaximal, characteristic, complemented
6#22626028C211yesmaximal, characteristic, complemented
7#22625991C211yesmaximal, characteristic, complemented
8#22625989C211yesmaximal, characteristic, complemented
9#22625877C211yesmaximal, characteristic
10#22625876C211yesmaximal, characteristic
11#22625731C211yesmaximal, characteristic, complemented
12#22625672C211yesmaximal, characteristic
13#22625669C211yesmaximal, characteristic
14#22625660C211yesmaximal, characteristic
15#22613208C211yesmaximal, characteristic
16C22×#11924038C2211yescharacteristic
17C22×#11924036C2211yescharacteristic
18#17989639—24no—
19#17989594—24no—
20C2×#14556658C2211yescharacteristic
21C2×#14556658C2211yescharacteristic
22C2×#14556658C2211yescharacteristic
23C2×C4⋊#10596833C2211yescharacteristic
24C2×C4⋊#10596833C2211yescharacteristic
25C2×C4⋊#10596833C2211yescharacteristic
26C2×C4⋊#10596833C2211yescharacteristic
27C2×C4⋊#10596833C2211yescharacteristic
28C2×#14556648C2211yescharacteristic
29C2×#14556648C2211yescharacteristic
30C2×#14556646C2211yescharacteristic
31C2×#14556646C2211yescharacteristic
32C2×#14556646C2211yescharacteristic
33C2×#14556646C2211yescharacteristic
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×#14556556C2211yescharacteristic
47C2×#14556549C2211yescharacteristic
48#17988529C411yescharacteristic
49#17988522C411yescharacteristic
50#17988520C411yescharacteristic
51#17988509C411yescharacteristic
52#17988509C411yescharacteristic
53#17988508C411yescharacteristic
54#17988500C411yescharacteristic
55C4⋊C4×Dic73C2211yescharacteristic
56C4⋊C4×Dic73C2211yescharacteristic
57#17988438C2211yescharacteristic
58#17988434C2211yescharacteristic
59#17988434C2211yescharacteristic
60#398×C73C411yescharacteristic
61C2×#11924038—22no—
62C2×#11924038—22no—
63C2×#11924038—41no—
64C2×#11924036—22no—
65C2×#11924036—22no—
66C2×#11924036—41no—
67C22×#10596833C2311yescharacteristic
68C22×#10596833C2311yescharacteristic
69C22×#10596833C2311yescharacteristic
70C22×#10596833C2311yescharacteristic
71C22×#10596833C2311yescharacteristic
72C22×#10596833C2311yescharacteristic
73C22×#10596833C2311yescharacteristic
74C22×#10596833C2311yescharacteristic
75#14556658—22no—
76#14556658—22no—
77#14556658D414yes—
78C4⋊#10596833—22no—
79C4⋊#10596833—22no—
80C4⋊#10596833—22no—
81C4⋊#10596833—41no—
82C4⋊#10596833—41no—
83#14556648—22no—
84#14556648Q814yescomplemented
85#14556646—22no—
86#14556646—22no—
87#14556646—22no—
88#14556646—41no—
89C4×#10596833—22no—
90C4×#10596833—41no—
91C22×Dic146C2×C411yescharacteristic
92C22×C4×D73C2311yescharacteristic
93#14556556—22no—
94#14556556—22no—
95#14556549—22no—
96#14556549—22no—
97C2×C4⋊Dic73C2×C411yescharacteristic
98C2×C4⋊Dic73C2×C411yescharacteristic
99C2×C4⋊Dic73C2×C411yescharacteristic
100C2×#11924007C2×C411yescharacteristic
101C2×#11924007C2×C411yescharacteristic
102C2×#11924007C2×C411yescharacteristic
103C2×#11924007C2×C411yescharacteristic
104C2×#11924005C2×C411yescharacteristic
105C2×#11924005C2×C411yescharacteristic
106C2×#11924005C2×C411yescharacteristic
107C2×#11924005C2×C411yescharacteristic
108C2×#11924005C2×C411yescharacteristic
109C2×#11924005C2×C411yescharacteristic
110C2×C4×Dic73C2311yescharacteristic
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
120#14556498C2×C411yescharacteristic
121#14556498C2×C411yescharacteristic
122#14556498C2×C411yescharacteristic
123C2×C4⋊C4×C73C2×C411yescharacteristic
124C2×C4⋊C4×C73C2×C411yescharacteristic
125C2×C4⋊C4×C73C2×C411yescharacteristic
126C2×C4⋊C4×C73C2×C411yescharacteristic
127#95×C73C2×C411yescharacteristic
128#95×C73C2×C411yescharacteristic
129#95×C73C2×C411yescharacteristic
130#11924038—44no—
131#11924036—44no—
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—
152C2×#10596833C2×D412yes—
153C2×#10596833C2×D412yes—
154C2×#10596833C2×Q812yes—
155C2×#10596833C2×Q812yes—
156C23×D73C22×C411yescharacteristic
157C2×Dic146—22no—
158C2×Dic146—41no—
159C2×Dic146—41no—
160C22×Dic73C22×C411yescharacteristic
161C22×Dic73C22×C411yescharacteristic
162C22×Dic73C22×C411yescharacteristic
163C22×Dic73C22×C411yescharacteristic
164C22×Dic73C22×C411yescharacteristic
165C22×Dic73C22×C411yescharacteristic
166C22×Dic73C2411yescharacteristic
167C2×C4×D73—21no—
168C2×C4×D73—21no—
169C2×C4×D73—21no—
170C2×C4×D73—21no—
171C2×C4×D73C2×Q812yes—
172C2×C4×D73#4112yes—
173C4⋊Dic73—22no—
174C4⋊Dic73—41no—
175C4⋊Dic73—41no—
176#11924007—22no—
177#11924007—22no—
178#11924007—41no—
179#11924007—41no—
180#11924005—22no—
181#11924005—22no—
182#11924005—22no—
183#11924005—41no—
184#11924005—41no—
185#11924005—41no—
186C4×Dic73—21no—
187C4×Dic73—21no—
188C4×Dic73—21no—
189C4×Dic73—21no—
190C4×Dic73—21no—
191C4×Dic73—21no—
192C4×Dic73—21no—
193C4×Dic73—21no—
194C4×Dic73—21no—
195C4×Dic73—21no—
196C4×Dic73C2×D412yes—
197C4×Dic73#4112yes—
198C22×C4×C73C22×C411yescharacteristic
199C22×C4×C73C22×C411yescharacteristic
200C22×C4×C73C22×C411yescharacteristic
201C22×C4×C73C22×C411yescharacteristic
202C22×C4×C73C22×C411yescharacteristic
203C22×C4×C73C22×C411yescharacteristic
204C22×C4×C73C22×C411yescharacteristic
205C4⋊C4×C73—22no—
206C4⋊C4×C73—22no—
207C4⋊C4×C73—22no—
208C4⋊C4×C73—22no—
209C23×C73C23×C411yescharacteristic
210C22×D73—21no—
211C22×D73—21no—
212C22×D73—21no—
213C22×D73—21no—
214C22×D73—21no—
215C22×D73—21no—
216C22×D73C4×Q812yes—
217C2×C4×C73—21no—
218C2×C4×C73—21no—
219C2×C4×C73—21no—
220C2×C4×C73—21no—
221C2×C4×C73—21no—
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×C73C4×D412yes—
236C2×C4×C73C4×Q812yes—
237C2×C4×C73#13112yes—
238C2×Dic73—21no—
239C2×Dic73—21no—
240C2×Dic73—21no—
241C2×Dic73—21no—
242C2×Dic73—21no—
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×Dic73C4×D412yes—
256C2×Dic73C22×D411yescharacteristic
257C2×Dic73C22×Q811yescharacteristic
258C2×Dic73C2×#4111yescharacteristic
259C2×Dic73C2×#4111yescharacteristic
260C2×Dic73H(5,𝔽2)11yescharacteristic
261C2×Dic732−1+411yescharacteristic
262C4×D73—22no—
263C4×D73—22no—
264C4×D73—22no—
265C4×D73—22no—
266Dic146—44no—
267#10596833—22no—
268#10596833—22no—
269#10596833—22no—
270#10596833—22no—
271#10596833—41no—
272#10596833—41no—
273#10596833—41no—
274#10596833—41no—
275C22×C73C2×C4×D411yescharacteristic
276C22×C73C2×C4×Q811yescharacteristic
277C22×C73#52011yescharacteristic
278C22×C73C2×#13111yescharacteristic
279C22×C73#56811yescharacteristic
280C22×C73#56811yescharacteristic
281C22×C73#56811yescharacteristic
282C2×D73—22no—
283C2×D73—22no—
284C2×D73—22no—
285C2×D73—41no—
286C2×D73—41no—
287C2×D73—41no—
288C2×D73—41no—
289C4×C73—22no—
290C4×C73—22no—
291C4×C73—22no—
292C4×C73—41no—
293C4×C73—41no—
294C4×C73—41no—
295C4×C73—41no—
296Dic73—22no—
297Dic73—41no—
298Dic73—41no—
299Dic73—41no—
300Dic73—41no—
301Dic73—41no—
302Dic73#54611yescharacteristic
303Dic73D4×Q811yescharacteristic
304Dic73#55211yescharacteristic
305Dic73#55411yescharacteristic
306#22961—731nomaximal, Sylow
307C2×C73#236011yescharacteristic
308C2×C73#293011yescharacteristic
309C2×C73#293511yescharacteristic
310C2×C73#296111yescharacteristic
311C2×C73#296211yescharacteristic
312C2×C73#299211yescharacteristic
313C2×C73#299511yescharacteristic
314D73—42no—
315C2×#446—731no—
316C2×#424—731no—
317#2396—731no—
318#1985—731no—
319#1940—731no—
320#1939—731no—
321#1939—731no—
322#1917—731no—
323#1917—731no—
324#1851—731no—
325#1850—731no—
326#1771—731no—
327#1753—731no—
328#1752—731no—
329#1750—731no—
330C73#2296111yescharacteristic, complemented, Sylow
331C22×M16—731no—
332C23×C8—731no—
333C2×C4×Q8—731no—
334C2×#117—731no—
335#446—1464no—
336#424—1464no—
337C2×C4⋊C8—731no—
338C2×C4⋊C8—731no—
339C2×C4⋊C8—731no—
340C2×C4⋊C8—731no—
341C2×C4⋊C8—731no—
342C2×C4⋊C8—731no—
343C2×C4⋊C8—731no—
344C2×C4⋊C8—731no—
345C2×C22⋊C8—731no—
346C2×C22⋊C8—731no—
347C2×#97—731no—
348C2×#97—731no—
349C2×#97—731no—
350C2×#97—731no—
351C2×C4×C8—731no—
352C2×C4×C8—731no—
353#400—731no—
354#399—731no—
355#398—731no—
356#398#6440811yescharacteristic
357#396—731no—
358#396—731no—
359#395—731no—
360#393—731no—
361C4×C4⋊C4—731no—
362C4×C4⋊C4—731no—
363C4×C4⋊C4—731no—
364C4×C22⋊C4—731no—
365C4×C22⋊C4—731no—
366#336—731no—
367#336—731no—
368#336—731no—
369#336—731no—
370#336—731no—
371#336—731no—
372#336—731no—
373#336—731no—
374#336—731no—
375#336—731no—
376C22×Q8—731no—
377C23×C4—731no—
378C2×M16—2921no—
379C2×M16—1462no—
380C2×M16—1462no—
381C22×C8—2921no—
382C22×C8—731no—
383C22×C8—731no—
384C22×C8—731no—
385C22×C8—731no—
386C22×C8—731no—
387C22×C8—731no—
388C22×C8—731no—
389C22×C8—731no—
390C22×C8—1462no—
391C22×C8—1462no—
392C4×Q8—1462no—
393C4×Q8—1462no—
394#117—1462no—
395#117—1462no—
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⋊C4C2×#6440811yescharacteristic
404C2×C4⋊C4C2×#6440811yescharacteristic
405C2×C4⋊C4C2×#6440811yescharacteristic
406C2×C4⋊C4C2×#6440811yescharacteristic
407C2×C22⋊C4—731no—
408C2×C22⋊C4—731no—
409C2×C22⋊C4—731no—
410C2×C22⋊C4—731no—
411C2×C22⋊C4—731no—
412C2×C22⋊C4—731no—
413C2×C42—731no—
414C2×C42—731no—
415C2×C42—731no—
416C2×C42—731no—
417C2×C42—731no—
418C2×C42—731no—
419C4⋊C8—2921no—
420C4⋊C8—2921no—
421C4⋊C8—734no—
422C4⋊C8—1462no—
423C4⋊C8—1462no—
424C4⋊C8—1462no—
425C4⋊C8—1462no—
426C4⋊C8—1462no—
427C22⋊C8—734no—
428C22⋊C8—1462no—
429#97—2921no—
430#97—1462no—
431#97—1462no—
432#97—1462no—
433C4×C8—2921no—
434C4×C8—1462no—
435#95—731no—
436#95—731no—
437#95—731no—
438#95—731no—
439#95—731no—
440#95—731no—
441#95—731no—
442#95C2×#6440811yescharacteristic
443#95C2×#6440811yescharacteristic
444#95C2×#6440811yescharacteristic
445C24—731no—
446C2×Q8—2921no—
447C2×Q8—2921no—
448C2×Q8—1462no—
449C22×C4—731no—
450C22×C4—731no—
451C22×C4—732no—
452C22×C4—732no—
453C22×C4—731no—
454C22×C4—731no—
455C22×C4—731no—
456C22×C4—731no—
457C22×C4—731no—
458C22×C4—1461no—
459C22×C4—1461no—
460C22×C4—1461no—
461C22×C4—1461no—
462C22×C4C22×#6440811yescharacteristic
463C22×C4C22×#6440811yescharacteristic
464C22×C4C22×#6440811yescharacteristic
465C22×C4C22×#6440811yescharacteristic
466C22×C4C22×#6440811yescharacteristic
467C22×C4C22×#6440811yescharacteristic
468C22×C4C22×#6440811yescharacteristic
469M16—2924no—
470C2×C8—2924no—
471C2×C8—732no—
472C2×C8—732no—
473C2×C8—732no—
474C2×C8—732no—
475C2×C8—1461no—
476C2×C8—1461no—
477C2×C8—1461no—
478C2×C8—1461no—
479C2×C8—1461no—
480C2×C8—1461no—
481C2×C8—1461no—
482C2×C8—1461no—
483C2×C8—1461no—
484C2×C8—1461no—
485C2×C8—1461no—
486C2×C8—1461no—
487C2×C8—1461no—
488C2×C8—1461no—
489C2×C8—1461no—
490C2×C8—1461no—
491C2×C8—1461no—
492C2×C8—1461no—
493C2×C8—1461no—
494C2×C8—1461no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—2921no—
500C4⋊C4—2921no—
501C4⋊C4—2921no—
502C4⋊C4—2921no—
503C4⋊C4—1462no—
504C4⋊C4—1462no—
505C4⋊C4—1462no—
506C22⋊C4—2921no—
507C22⋊C4—2921no—
508C22⋊C4—2921no—
509C22⋊C4—1462no—
510C22⋊C4—1462no—
511C22⋊C4—1462no—
512C42—732no—
513C42—732no—
514C42—1461no—
515C42—1461no—
516C42—1461no—
517C42—1461no—
518C42—1461no—
519C42—1461no—
520C42—1461no—
521C42—1461no—
522C42—1461no—
523C42—1461no—
524C23—732no—
525C23—1461no—
526C23—1461no—
527C23—1461no—
528C23—1461no—
529C23—1461no—
530C23—1461no—
531C23C23×#6440811yescharacteristic
532Q8—2924no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—731no—
552C2×C4—731no—
553C2×C4—731no—
554C2×C4—731no—
555C2×C4—731no—
556C2×C4—731no—
557C2×C4—732no—
558C2×C4—1462no—
559C2×C4—1462no—
560C2×C4—1462no—
561C2×C4—1462no—
562C2×C4—1461no—
563C2×C4—1461no—
564C2×C4—1461no—
565C2×C4—1461no—
566C2×C4—1461no—
567C2×C4—1461no—
568C2×C4—1461no—
569C2×C4—1461no—
570C2×C4—1461no—
571C2×C4—1461no—
572C2×C4—1461no—
573C2×C4—1461no—
574C2×C4—1461no—
575C2×C4—1461no—
576C2×C4—1461no—
577C2×C4—1461no—
578C2×C4—1461no—
579C2×C4D4×#6440812yes—
580C2×C4Q8×#6440812yes—
581C2×C4#1455667812yes—
582C8—2921no—
583C8—2921no—
584C8—2921no—
585C8—2921no—
586C8—1462no—
587C8—1462no—
588C8—1462no—
589C8—1462no—
590C22—2921no—
591C22—2921no—
592C22—2921no—
593C22—2921no—
594C22—1462no—
595C22—1462no—
596C22—1462no—
597C22C2×D4×#6440811yescharacteristic
598C22C2×Q8×#6440811yescharacteristic
599C22#1798965611yescharacteristic
600C22C2×#1455667811yescharacteristic
601C22#1798966611yescharacteristic
602C22#1798966611yescharacteristic
603C22#1798966711yescharacteristic
604C4—22no—
605C4—22no—
606C4—22no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—41no—
611C4—2921no—
612C4—2921no—
613C4—2921no—
614C4—2921no—
615C4—2921no—
616C4—731no—
617C4—731no—
618C4—731no—
619C4—731no—
620C4—1462no—
621C2—2922no—
622C2#2262645511yescharacteristic
623C2#2262665111yescharacteristic
624C2#2262667311yescharacteristic
625C2#2262672311yescharacteristic
626C2#2262672511yescharacteristic
627C2#2262686611yescharacteristic
628C2#2262688911yescharacteristic
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
Fitting#398×C73C4
prime radicalC23×D73C22×C4
cosocleC22×Dic73C24
hypercenter#398#64408
hypocenterC73#22961
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#17989639C2semidirect
3. #22626104C2semidirect
4. #22626032C2semidirect
5. #22626028C2semidirect
6. #22626028C2semidirect
7. #22625991C2semidirect
8. #22625989C2semidirect
11. #22625731C2semidirect
84. #14556648Q8semidirect
330. C73#22961semidirect

Maximal subgroups

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

subgroupindex
C2×#179896392
C2×#179895942
#226261042
#226260322
#226260282
#226259912
#226259892
#226258772
#226258762
#226257312
#226256722
#226256692
#226256602
#226132082
#2296173