Finite Group Databasebeta

← #33071396

Subgroups — #33071396

Order 18688.

758 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620846C211yesmaximal, characteristic, complemented
2#22619411C211yesmaximal, characteristic, complemented
3#22619042C211yesmaximal, characteristic, complemented
4#22609377C212yesmaximal, complemented
5#22609370C212yesmaximal, complemented
6#22609289C212yesmaximal, complemented
7#22609284C212yesmaximal, complemented
8#22608705C211yesmaximal, characteristic, complemented
9#22608608C211yesmaximal, characteristic, complemented
10C2×#17988330C211yesmaximal, characteristic, complemented
11C73×#3035C211yesmaximal, characteristic, complemented
12C2×#14556616—21no—
13C2×#14556588C2211yescharacteristic
14#17988963C2212yes—
15#17988935C2212yes—
16#17988787—21no—
17#17988787C2212yes—
18#17988782—21no—
19#17988782—21no—
20#17988782—21no—
21#17988782C2212yes—
22#17988720—22no—
23Q8×#10596831C2212yescomplemented
24D4×#10596831—22no—
25D4×#10596831C2212yescomplemented
26#17988640—21no—
27#17988640C2211yescharacteristic
28#17988611C2211yescharacteristic
29C2×C4⋊#10596831—21no—
30C2×C4⋊#10596831C2211yescharacteristic
31#17988591—21no—
32#17988339C2212yescomplemented
33#17988339C2212yescomplemented
34#17988333C2212yescomplemented
35#17988331C2212yescomplemented
36#17988330C2212yes—
37#17988330C2212yes—
38C2×#135×C73—21no—
39#532×C73C2211yescharacteristic
40C4×#41×C73C2211yescharacteristic
41#464×C73—21no—
42#463×C73—21no—
43#462×C73C2211yescharacteristic, complemented
44#461×C73C2211yescharacteristic, complemented
45#460×C73C2211yescharacteristic, complemented
46#459×C73C2211yescharacteristic, complemented
47C2×C4⋊C8×C73C2211yescharacteristic
48#416×C73—21no—
49C22×D292—21no—
50C22×D292—21no—
51C22×D292—21no—
52#14556616—44no—
53C2×#11924015—22no—
54C2×#11924013—22no—
55C2×#11924013—22no—
56C2×#11924013—22no—
57C2×#11924012—21no—
58C22×#10596831—22no—
59C22×#10596831—21no—
60#14556588—21no—
61#14556588—21no—
62#14556588—21no—
63#14556588C2312yes—
64C4⋊#10596831—22no—
65C4⋊#10596831—21no—
66C4⋊#10596831—21no—
67C4⋊#10596831D411yescharacteristic
68C4⋊#10596831D411yescharacteristic
69C4⋊#10596831C2312yes—
70C4⋊#10596831C2312yes—
71#14556511—22no—
72#14556511—22no—
73#14556511—22no—
74#14556511—22no—
75C22⋊#10596831—22no—
76C22⋊#10596831D412yes—
77C4×#10596831—22no—
78C4×#10596831C2312yes—
79C2×#41×C73—21no—
80C2×#41×C73D411yescharacteristic
81#135×C73—44no—
82C2×Q16×C73—21no—
83C2×SD16×C73—21no—
84C2×SD16×C73—21no—
85C2×D8×C73—21no—
86C22×C8×C73—21no—
87C4⋊Q8×C73C2311yescharacteristic
88#127×C73C2311yescharacteristic
89#124×C73—21no—
90#121×C73—21no—
91#121×C73—21no—
92C4×Q8×C73D411yescharacteristic
93C4×Q8×C73C2311yescharacteristic
94C4×D4×C73—21no—
95C4×D4×C73—21no—
96C4×D4×C73—21no—
97C4×D4×C73—21no—
98C4×D4×C73D411yescharacteristic
99C4×D4×C73C2311yescharacteristic
100#117×C73—21no—
101#117×C73D411yescharacteristic
102C2×C42×C73—21no—
103C2×C42×C73C2311yescharacteristic
104C4⋊C8×C73—21no—
105C4⋊C8×C73C2311yescharacteristic
106C4⋊C8×C73C2311yescharacteristic
107#103×C73—21no—
108#103×C73—21no—
109#102×C73—21no—
110#102×C73—21no—
111C23×D73—41no—
112C2×D292—22no—
113C2×D292—22no—
114C2×D292—22no—
115C2×D292—42no—
116C2×D292—42no—
117C2×D292—41no—
118C2×D292—41no—
119C2×D292—41no—
120C2×D292—81no—
121C2×D292—81no—
122C2×D292—82no—
123#11924015—44no—
124#11924013—44no—
125#11924013—82no—
126#11924013—82no—
127#11924012—42no—
128C2×#10596831—22no—
129C2×#10596831—22no—
130C2×#10596831—82no—
131C2×#10596831—81no—
132C2×#10596831C2×D412yes—
133C2×#10596831C2×D412yes—
134#41×C73—22no—
135#41×C73—42no—
136#41×C73—42no—
137#41×C73D814yescomplemented
138C2×Q8×C73—21no—
139C2×Q8×C73C2×D411yescharacteristic
140C2×D4×C73—21no—
141C2×D4×C73—21no—
142C2×D4×C73—21no—
143C2×D4×C73—21no—
144C2×D4×C73—41no—
145C2×D4×C73C2×D411yescharacteristic
146C22×C4×C73—21no—
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—41no—
150C22×C4×C73C2×D411yescharacteristic
151C22×C4×C73C2×D411yescharacteristic
152C22×C4×C73#4111yescharacteristic
153Q16×C73—42no—
154SD16×C73—42no—
155SD16×C73—42no—
156D8×C73—42no—
157C2×C8×C73—21no—
158C2×C8×C73—21no—
159C2×C8×C73—42no—
160C4⋊C4×C73—21no—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—21no—
163C4⋊C4×C73—21no—
164C4⋊C4×C73C2×D411yescharacteristic
165C4⋊C4×C73C2×D411yescharacteristic
166C22⋊C4×C73—21no—
167C22⋊C4×C73—21no—
168C22⋊C4×C73—21no—
169C22⋊C4×C73—41no—
170C22⋊C4×C73—41no—
171C42×C73—21no—
172C42×C73—21no—
173C42×C73—21no—
174C42×C73C2×D411yescharacteristic
175C42×C73C2×D411yescharacteristic
176C42×C73#4111yescharacteristic
177C42×C73C2411yescharacteristic
178C23×C73—21no—
179C23×C73—41no—
180C23×C73#12111yescharacteristic
181C22×D73—42no—
182C22×D73—82no—
183C22×D73—82no—
184C22×D73—82no—
185Q8×C73—21no—
186Q8×C73—41no—
187Q8×C73—41no—
188Q8×C73C2×D812yes—
189D4×C73—24no—
190D4×C73—21no—
191D4×C73—41no—
192D4×C73—41no—
193D4×C73—41no—
194D4×C73—41no—
195D4×C73—81no—
196D4×C73—82no—
197D4×C73C2×D812yes—
198C2×C4×C73—21no—
199C2×C4×C73—21no—
200C2×C4×C73—21no—
201C2×C4×C73—24no—
202C2×C4×C73—21no—
203C2×C4×C73—21no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73—41no—
207C2×C4×C73—41no—
208C2×C4×C73—81no—
209C2×C4×C73—82no—
210C2×C4×C73#12111yescharacteristic
211C2×C4×C73#12111yescharacteristic
212C2×C4×C73#12111yescharacteristic
213C2×C4×C73C2×D812yes—
214C2×C4×C73#13512yes—
215C2×C4×C73C22×D411yescharacteristic
216C2×C4×C73C22×D411yescharacteristic
217C2×C4×C73C2×#4111yescharacteristic
218D292—42no—
219D292—42no—
220D292—42no—
221D292—42no—
222D292—82no—
223D292—82no—
224D292—82no—
225D292—82no—
226C8×C73—41no—
227C8×C73—41no—
228#10596831—22no—
229#10596831—22no—
230#10596831—42no—
231#10596831—42no—
232C22×C73—21no—
233C22×C73—21no—
234C22×C73—22no—
235C22×C73—41no—
236C22×C73—41no—
237C22×C73—81no—
238C22×C73—81no—
239C22×C73—81no—
240C22×C73#46612yes—
241C22×C73C2×#12111yescharacteristic
242C2×D73—82no—
243C2×D73—82no—
244C2×D73—82no—
245C2×D73—164no—
246C4×C73—21no—
247C4×C73—22no—
248C4×C73—21no—
249C4×C73—21no—
250C4×C73—41no—
251C4×C73—41no—
252C4×C73—81no—
253C4×C73#46612yes—
254C4×C73C22×D811yescharacteristic
255C4×C73C2×#13511yescharacteristic
256C4×C73#57611yescharacteristic
257C4×C73#57711yescharacteristic
258#19410—731nomaximal, Sylow
259C2×C73—22no—
260C2×C73—42no—
261C2×C73—161no—
262C2×C73#304511yescharacteristic
263C2×C73C2×#46611yescharacteristic
264C2×C73#306811yescharacteristic
265D73—162no—
266#3205—731no—
267#3038—731no—
268#3035D7311yescharacteristic, complemented
269#2988—731no—
270#1682—732no—
271#1681—732no—
272#1668—732no—
273#1667—732no—
274#1571—731no—
275#1547—731no—
276C2×#331—731no—
277C73#1941011yescharacteristic, complemented, Sylow
278C2×AGL(1,ℤ/8)—1461no—
279C2×#135—21no—
280#532C2×D7311yescharacteristic
281C2×#127—731no—
282C4×#41C2×D7311yescharacteristic
283#493—732no—
284#486—732no—
285#464—21no—
286#463—21no—
287#462C2×D7311yescharacteristic, complemented
288#461C2×D7311yescharacteristic, complemented
289#460—732no—
290#460—1461no—
291#460C2×D7311yescharacteristic, complemented
292#459—732no—
293#459—1461no—
294#459—1461no—
295#459—1461no—
296#459C2×D7311yescharacteristic, complemented
297#447—1462no—
298C8×Q8—732no—
299C8×D4—732no—
300C8×D4—1462no—
301#431—731no—
302#431—1461no—
303#423—731no—
304C2×C4⋊C8—731no—
305C2×C4⋊C8—1461no—
306C2×C4⋊C8C2×D7311yescharacteristic
307#418—1461no—
308#416—21no—
309#335—732no—
310#335—732no—
311#332—732no—
312#331—732no—
313#331—732no—
314#331—732no—
315C2×#41—21no—
316C2×#41C22⋊D7311yescharacteristic
317C22×D4—1461no—
318C22×D4—1461no—
319C22×D4—1461no—
320AGL(1,ℤ/8)—2924no—
321#135—44no—
322C2×Q16—21no—
323C2×SD16—21no—
324C2×SD16—21no—
325C2×SD16—1462no—
326C2×D8—21no—
327C2×D8—1462no—
328C2×D8—1462no—
329C2×D8—1462no—
330C2×M16—1461no—
331C22×C8—21no—
332C22×C8—1462no—
333C22×C8—1461no—
334C4⋊Q8C22×D7311yescharacteristic
335#127—732no—
336#127—1461no—
337#127—1461no—
338#127—1461no—
339#127C22×D7311yescharacteristic
340#124—21no—
341#121—21no—
342#121—21no—
343C4×Q8C22⋊D7311yescharacteristic
344C4×Q8C22×D7311yescharacteristic
345C4×D4—21no—
346C4×D4—21no—
347C4×D4—21no—
348C4×D4—21no—
349C4×D4C22⋊D7311yescharacteristic
350C4×D4C22×D7311yescharacteristic
351#117—21no—
352#117C22⋊D7311yescharacteristic
353C2×C42—21no—
354C2×C42C22×D7311yescharacteristic
355C4⋊C8—21no—
356C4⋊C8—731no—
357C4⋊C8—731no—
358C4⋊C8—732no—
359C4⋊C8—732no—
360C4⋊C8—1462no—
361C4⋊C8—1461no—
362C4⋊C8—1461no—
363C4⋊C8C22×D7311yescharacteristic
364C4⋊C8C22×D7311yescharacteristic
365#103—21no—
366#103—21no—
367#102—21no—
368#102—21no—
369#102—1462no—
370#102—1462no—
371#102—1462no—
372#102—1462no—
373C22⋊C8—732no—
374C22⋊C8—1462no—
375C4×C8—732no—
376C4×C8—1462no—
377C24—2921no—
378#41—22no—
379#41—42no—
380#41—42no—
381#41#1192401314yescomplemented
382C2×Q8—21no—
383C2×Q8C2×C22⋊D7311yescharacteristic
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—41no—
389C2×D4—2921no—
390C2×D4—2921no—
391C2×D4—2921no—
392C2×D4—2922no—
393C2×D4—2922no—
394C2×D4—5841no—
395C2×D4—5841no—
396C2×D4—5842no—
397C2×D4—1462no—
398C2×D4—1462no—
399C2×D4—1462no—
400C2×D4C2×C22⋊D7311yescharacteristic
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—41no—
405C22×C4D4×D7311yescharacteristic
406C22×C4C2×C22⋊D7311yescharacteristic
407C22×C4#1192402711yescharacteristic
408Q16—42no—
409SD16—42no—
410SD16—42no—
411SD16—2924no—
412D8—42no—
413D8—2924no—
414D8—5842no—
415D8—5842no—
416M16—2922no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—42no—
420C2×C8—5841no—
421C2×C8—5842no—
422C2×C8—732no—
423C2×C8—732no—
424C2×C8—1462no—
425C2×C8—1462no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4C2×C22⋊D7311yescharacteristic
431C4⋊C4C2×C22⋊D7311yescharacteristic
432C22⋊C4—21no—
433C22⋊C4—21no—
434C22⋊C4—21no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42D4×D7311yescharacteristic
441C42C2×C22⋊D7311yescharacteristic
442C42#1192402711yescharacteristic
443C42C23×D7311yescharacteristic
444C23—21no—
445C23—41no—
446C23—2922no—
447C23—5842no—
448C23—5842no—
449C23—5842no—
450C23#1455656311yescharacteristic
451Q8—21no—
452Q8—41no—
453Q8—41no—
454Q8C2×#1192401312yes—
455D4—24no—
456D4—21no—
457D4—41no—
458D4—41no—
459D4—41no—
460D4—41no—
461D4—81no—
462D4—82no—
463D4—2922no—
464D4—2922no—
465D4—2922no—
466D4—2922no—
467D4—5842no—
468D4—5842no—
469D4—5842no—
470D4—5842no—
471D4C2×#1192401312yes—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—24no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4#1455656311yescharacteristic
485C2×C4#1455656311yescharacteristic
486C2×C4#1455656311yescharacteristic
487C2×C4C2×#1192401312yes—
488C2×C4#1455661112yes—
489C2×C4C2×D4×D7311yescharacteristic
490C2×C4C22×C22⋊D7311yescharacteristic
491C2×C4C2×#1192402711yescharacteristic
492C8—41no—
493C8—41no—
494C8—2922no—
495C8—2922no—
496C8—1462no—
497C8—1462no—
498C22—21no—
499C22—21no—
500C22—22no—
501C22—41no—
502C22—41no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—5842no—
507C22—5842no—
508C22—5842no—
509C22—11684no—
510C22#1798882112yes—
511C22C2×#1455656311yescharacteristic
512C4—21no—
513C4—21no—
514C4—21no—
515C4—22no—
516C4—41no—
517C4—41no—
518C4—81no—
519C4#1798882112yes—
520C4C22×#1192401311yescharacteristic
521C4C2×#1455661111yescharacteristic
522C4#1798938911yescharacteristic
523C4#1798939411yescharacteristic
524C2—22no—
525C2—42no—
526C2—161no—
527C2—11682no—
528C2#2261946811yescharacteristic
529C2C2×#1798882111yescharacteristic
530C2#2261966111yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3035C2
socleC22×C73C2×#121
hypercenter#3035D73
hypocenterC73#19410
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
1. #22620846C2semidirect
2. #22619411C2semidirect
3. #22619042C2semidirect
4. #22609377C2semidirect
5. #22609370C2semidirect
6. #22609289C2semidirect
7. #22609284C2semidirect
8. #22608705C2semidirect
9. #22608608C2semidirect
10. C2×#17988330C2semidirect
11. C73×#3035C2semidirect
23. Q8×#10596831C22semidirect
25. D4×#10596831C22semidirect
32. #17988339C22semidirect
33. #17988339C22semidirect
34. #17988333C22semidirect
35. #17988331C22semidirect
43. #462×C73C22semidirect
44. #461×C73C22semidirect
45. #460×C73C22semidirect
46. #459×C73C22semidirect
137. #41×C73D8semidirect
268. #3035D73semidirect
277. C73#19410semidirect
287. #462C2×D73semidirect
288. #461C2×D73semidirect
291. #460C2×D73semidirect
296. #459C2×D73semidirect
381. #41#11924013semidirect

Maximal subgroups

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

subgroupindex
#226208462
#226194112
#226190422
#226093772
#226093702
#226092892
#226092842
#226087052
#226086082
C2×#179883302
C73×#30352
#1941073