Finite Group Databasebeta

← #33071391

Subgroups — #33071391

Order 18688.

704 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620872C211yesmaximal, characteristic, complemented
2#22619719C211yesmaximal, characteristic, complemented
3#22619042C211yesmaximal, characteristic, complemented
4#22609345C212yesmaximal, complemented
5#22609338C212yesmaximal, complemented
6#22609277C212yesmaximal, complemented
7#22609270C212yesmaximal, complemented
8#22608703C211yesmaximal, characteristic, complemented
9#22608622C211yesmaximal, characteristic, complemented
10C2×#17988331C211yesmaximal, characteristic, complemented
11C73×#3038C211yesmaximal, characteristic
12#17989144C2211yescharacteristic
13#17988959C2212yes—
14#17988933C2212yes—
15#17988894—21no—
16#17988887—21no—
17#17988882C2211yescharacteristic, complemented
18#17988877C2211yescharacteristic, complemented
19#17988872C2211yescharacteristic, complemented
20#17988867C2211yescharacteristic, complemented
21#17988784—21no—
22#17988784—21no—
23#17988738—22no—
24Q8×#10596831C2212yescomplemented
25D4×#10596831—22no—
26D4×#10596831C2212yescomplemented
27#17988640—21no—
28#17988640C2211yescharacteristic
29#17988623—21no—
30C2×C4⋊#10596831—21no—
31C2×C4⋊#10596831C2211yescharacteristic
32C2×C4⋊#10596831C2211yescharacteristic
33#17988583—21no—
34#17988337C2212yescomplemented
35#17988337C2212yescomplemented
36#17988335C2212yescomplemented
37#17988332C2212yescomplemented
38#17988331C2212yes—
39#17988331C2212yes—
40C2×AGL(1,ℤ/8)×C73—21no—
41C2×#127×C73C2211yescharacteristic
42C4×#41×C73C2211yescharacteristic
43#460×C73—21no—
44#460×C73C2212yes—
45#459×C73—21no—
46#459×C73C2212yes—
47#423×C73C2211yescharacteristic
48#418×C73—21no—
49C2×#11924016—22no—
50C2×#11924016—22no—
51C22×#10596831—22no—
52C22×#10596831—21no—
53C22×#10596831—21no—
54#14556591C2311yescharacteristic
55#14556591C2311yescharacteristic
56#14556583—21no—
57#14556570—21no—
58#14556570—21no—
59C4×Dic146—21no—
60C4×Dic146—21no—
61#14556550—21no—
62#14556524—21no—
63#14556524—21no—
64#14556522—21no—
65#14556522—21no—
66C4⋊#10596831—22no—
67C4⋊#10596831—21no—
68C4⋊#10596831—21no—
69C4⋊#10596831D411yescharacteristic
70C4⋊#10596831D411yescharacteristic
71C4⋊#10596831C2311yescharacteristic
72C4⋊#10596831C2311yescharacteristic
73C4⋊#10596831C2312yes—
74#14556515—21no—
75#14556515—21no—
76#14556512—22no—
77#14556512—22no—
78#14556512—21no—
79#14556512—21no—
80C22⋊#10596831—22no—
81C22⋊#10596831D412yes—
82C4×#10596831—22no—
83C4×#10596831C2312yes—
84C2×#41×C73D411yescharacteristic
85C22×D4×C73—21no—
86C22×D4×C73—21no—
87C22×D4×C73—21no—
88AGL(1,ℤ/8)×C73—44no—
89C2×SD16×C73—22no—
90C2×D8×C73—22no—
91C2×M16×C73—21no—
92#127×C73—21no—
93#127×C73—21no—
94#127×C73—21no—
95#127×C73C2312yes—
96C4×Q8×C73D411yescharacteristic
97C4×Q8×C73C2311yescharacteristic
98C4×D4×C73—21no—
99C4×D4×C73—21no—
100C4×D4×C73D411yescharacteristic
101C4×D4×C73C2311yescharacteristic
102#117×C73—21no—
103#117×C73D411yescharacteristic
104C2×C42×C73—21no—
105C2×C42×C73C2311yescharacteristic
106C4⋊C8×C73—21no—
107C4⋊C8×C73C2312yes—
108#102×C73—22no—
109#102×C73—22no—
110C2×Dic146—21no—
111C2×Dic146—21no—
112#11924016—82no—
113#11924016—82no—
114C2×#10596831—22no—
115C2×#10596831—21no—
116C2×#10596831—21no—
117C2×#10596831—42no—
118C2×#10596831—82no—
119C2×#10596831—81no—
120C2×#10596831C2×D412yes—
121C2×#10596831C2×D412yes—
122C4⋊Dic73—21no—
123C4⋊Dic73—21no—
124C4⋊Dic73—21no—
125C4⋊Dic73—21no—
126#11924007—41no—
127#11924007—41no—
128C22⋊Dic73—41no—
129C4×Dic73—41no—
130C24×C73—41no—
131#41×C73—22no—
132#41×C73SD1614yescomplemented
133C2×Q8×C73C2×D411yescharacteristic
134C2×D4×C73—21no—
135C2×D4×C73—22no—
136C2×D4×C73—22no—
137C2×D4×C73—22no—
138C2×D4×C73—41no—
139C2×D4×C73—41no—
140C2×D4×C73—41no—
141C2×D4×C73—42no—
142C2×D4×C73—42no—
143C2×D4×C73—81no—
144C2×D4×C73—81no—
145C2×D4×C73—82no—
146C2×D4×C73C2×D411yescharacteristic
147C22×C4×C73—21no—
148C22×C4×C73—21no—
149C22×C4×C73—21no—
150C22×C4×C73C2×D411yescharacteristic
151C22×C4×C73C2×D411yescharacteristic
152C22×C4×C73#4111yescharacteristic
153SD16×C73—44no—
154D8×C73—44no—
155M16×C73—42no—
156C2×C8×C73—22no—
157C4⋊C4×C73—21no—
158C4⋊C4×C73—21no—
159C4⋊C4×C73C2×D411yescharacteristic
160C4⋊C4×C73C2×D411yescharacteristic
161C22⋊C4×C73—21no—
162C22⋊C4×C73—21no—
163C22⋊C4×C73—21no—
164C42×C73—21no—
165C42×C73—21no—
166C42×C73—21no—
167C42×C73C2×D411yescharacteristic
168C42×C73C2×D411yescharacteristic
169C42×C73#4111yescharacteristic
170C42×C73C2411yescharacteristic
171C23×C73—21no—
172C23×C73—42no—
173C23×C73—82no—
174C23×C73—82no—
175C23×C73—82no—
176C23×C73#12111yescharacteristic
177Q8×C73—21no—
178Q8×C73C2×SD1612yes—
179D4×C73—24no—
180D4×C73—21no—
181D4×C73—42no—
182D4×C73—42no—
183D4×C73—42no—
184D4×C73—42no—
185D4×C73—82no—
186D4×C73—82no—
187D4×C73—82no—
188D4×C73—82no—
189D4×C73C2×SD1612yes—
190C2×C4×C73—21no—
191C2×C4×C73—21no—
192C2×C4×C73—21no—
193C2×C4×C73—24no—
194C2×C4×C73—21no—
195C2×C4×C73—21no—
196C2×C4×C73—41no—
197C2×C4×C73—41no—
198C2×C4×C73—41no—
199C2×C4×C73—81no—
200C2×C4×C73#12111yescharacteristic
201C2×C4×C73#12111yescharacteristic
202C2×C4×C73#12111yescharacteristic
203C2×C4×C73C2×SD1612yes—
204C2×C4×C73#13512yes—
205C2×C4×C73C22×D411yescharacteristic
206C2×C4×C73C22×D411yescharacteristic
207C2×C4×C73C2×#4111yescharacteristic
208C2×Dic73—41no—
209C2×Dic73—41no—
210Dic146—41no—
211Dic146—41no—
212Dic146—81no—
213C8×C73—42no—
214#10596831—22no—
215#10596831—22no—
216#10596831—41no—
217#10596831—41no—
218#10596831—42no—
219C22×C73—21no—
220C22×C73—21no—
221C22×C73—22no—
222C22×C73—41no—
223C22×C73—41no—
224C22×C73—82no—
225C22×C73—82no—
226C22×C73—82no—
227C22×C73—164no—
228C22×C73#46512yes—
229C22×C73C2×#12111yescharacteristic
230C4×C73—21no—
231C4×C73—22no—
232C4×C73—21no—
233C4×C73—21no—
234C4×C73—41no—
235C4×C73—41no—
236C4×C73#46512yes—
237C4×C73C22×SD1611yescharacteristic
238C4×C73C2×#13511yescharacteristic
239C4×C73#57611yescharacteristic
240C4×C73#57711yescharacteristic
241Dic73—81no—
242Dic73—81no—
243#19409—731nomaximal, Sylow
244C2×C73—22no—
245C2×C73—42no—
246C2×C73—162no—
247C2×C73#304511yescharacteristic
248C2×C73C2×#46511yescharacteristic
249C2×C73#306811yescharacteristic
250#3208—731no—
251#3076—731no—
252#3038D7311yescharacteristic
253#2988—731no—
254#1678—732no—
255#1677—732no—
256#1666—732no—
257#1665—732no—
258#1570—731no—
259#1550—731no—
260C2×#331—731no—
261C73#1940911yescharacteristic, complemented, Sylow
262C2×AGL(1,ℤ/8)—21no—
263#533—731no—
264C2×#127C2×D7311yescharacteristic
265C4×#41C2×D7311yescharacteristic
266#492—732no—
267#486—732no—
268#479—1461no—
269#478—1461no—
270#477—731no—
271#476—731no—
272#475—731no—
273#474—731no—
274#460—21no—
275#460—1461no—
276#460—1461no—
277#460C2×D7312yes—
278#459—21no—
279#459C2×D7312yes—
280#450—1462no—
281C8×Q8—732no—
282C8×D4—732no—
283C8×D4—1462no—
284#431—731no—
285#431—1461no—
286#427—1461no—
287#423C2×D7311yescharacteristic
288C2×C4⋊C8—731no—
289C2×C4⋊C8—731no—
290C2×C4⋊C8—1461no—
291#418—21no—
292#416—1461no—
293#334—732no—
294#334—732no—
295#333—732no—
296#332—732no—
297#331—732no—
298#331—732no—
299C2×#41C22⋊D7311yescharacteristic
300C22×D4—21no—
301C22×D4—21no—
302C22×D4—21no—
303AGL(1,ℤ/8)—44no—
304C2×SD16—22no—
305C2×SD16—1462no—
306C2×SD16—1462no—
307C2×D8—22no—
308C2×M16—21no—
309C22×C8—1461no—
310C22×C8—1462no—
311C22×C8—1461no—
312C4⋊Q8—731no—
313C4⋊Q8—731no—
314#127—21no—
315#127—21no—
316#127—21no—
317#127C22×D7312yes—
318#125—1461no—
319C22⋊Q8—1461no—
320C22⋊Q8—1461no—
321C4×Q8—1461no—
322C4×Q8—1461no—
323C4×Q8C22⋊D7311yescharacteristic
324C4×Q8C22×D7311yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22⋊D7311yescharacteristic
328C4×D4C22×D7311yescharacteristic
329#117—21no—
330#117—1461no—
331#117C22⋊D7311yescharacteristic
332C2×C42—21no—
333C2×C42C22×D7311yescharacteristic
334#107—1461no—
335#107—1461no—
336#106—1461no—
337#106—1461no—
338C4⋊C8—21no—
339C4⋊C8—731no—
340C4⋊C8—731no—
341C4⋊C8—732no—
342C4⋊C8—731no—
343C4⋊C8—731no—
344C4⋊C8—1462no—
345C4⋊C8—1461no—
346C4⋊C8—1461no—
347C4⋊C8C22×D7312yes—
348#103—1461no—
349#103—1461no—
350#102—22no—
351#102—22no—
352#102—1462no—
353#102—1462no—
354#102—1461no—
355#102—1461no—
356C22⋊C8—732no—
357C22⋊C8—1462no—
358C4×C8—732no—
359C4×C8—1462no—
360C24—41no—
361#41—22no—
362#41#1192401614yescomplemented
363C2×Q8—1461no—
364C2×Q8—1461no—
365C2×Q8C2×C22⋊D7311yescharacteristic
366C2×D4—21no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—81no—
376C2×D4—81no—
377C2×D4—82no—
378C2×D4C2×C22⋊D7311yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4D4×D7311yescharacteristic
383C22×C4C2×C22⋊D7311yescharacteristic
384C22×C4#1192402711yescharacteristic
385SD16—44no—
386SD16—5842no—
387SD16—5842no—
388D8—44no—
389M16—42no—
390C2×C8—22no—
391C2×C8—2922no—
392C2×C8—5841no—
393C2×C8—5842no—
394C2×C8—732no—
395C2×C8—732no—
396C2×C8—1462no—
397C2×C8—1461no—
398C2×C8—1461no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—2921no—
402C4⋊C4—2921no—
403C4⋊C4—1461no—
404C4⋊C4—1461no—
405C4⋊C4—1461no—
406C4⋊C4—1461no—
407C4⋊C4C2×C22⋊D7311yescharacteristic
408C4⋊C4C2×C22⋊D7311yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—2921no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—2921no—
417C42D4×D7311yescharacteristic
418C42C2×C22⋊D7311yescharacteristic
419C42#1192402711yescharacteristic
420C42C23×D7311yescharacteristic
421C23—21no—
422C23—42no—
423C23—82no—
424C23—82no—
425C23—82no—
426C23#1455656311yescharacteristic
427Q8—21no—
428Q8—2921no—
429Q8—2921no—
430Q8—5841no—
431Q8C2×#1192401612yes—
432D4—24no—
433D4—21no—
434D4—42no—
435D4—42no—
436D4—42no—
437D4—42no—
438D4—82no—
439D4—82no—
440D4—82no—
441D4—82no—
442D4C2×#1192401612yes—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—24no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—81no—
453C2×C4—2921no—
454C2×C4—2921no—
455C2×C4#1455656311yescharacteristic
456C2×C4#1455656311yescharacteristic
457C2×C4#1455656311yescharacteristic
458C2×C4C2×#1192401612yes—
459C2×C4#1455661112yes—
460C2×C4C2×D4×D7311yescharacteristic
461C2×C4C22×C22⋊D7311yescharacteristic
462C2×C4C2×#1192402711yescharacteristic
463C8—42no—
464C8—2922no—
465C8—2921no—
466C8—2921no—
467C8—1462no—
468C8—1462no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—164no—
478C22#1798881412yes—
479C22C2×#1455656311yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—5841no—
487C4—5841no—
488C4#1798881412yes—
489C4C22×#1192401611yescharacteristic
490C4C2×#1455661111yescharacteristic
491C4#1798939111yescharacteristic
492C4#1798939711yescharacteristic
493C2—22no—
494C2—42no—
495C2—162no—
496C2#2261947411yescharacteristic
497C2C2×#1798881411yescharacteristic
498C2#2261965611yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
Fitting, prime radicalC73×#3038C2
socleC22×C73C2×#121
hypercenter#3038D73
hypocenterC73#19409
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #22620872C2semidirect
2. #22619719C2semidirect
3. #22619042C2semidirect
4. #22609345C2semidirect
5. #22609338C2semidirect
6. #22609277C2semidirect
7. #22609270C2semidirect
8. #22608703C2semidirect
9. #22608622C2semidirect
10. C2×#17988331C2semidirect
17. #17988882C22semidirect
18. #17988877C22semidirect
19. #17988872C22semidirect
20. #17988867C22semidirect
24. Q8×#10596831C22semidirect
26. D4×#10596831C22semidirect
34. #17988337C22semidirect
35. #17988337C22semidirect
36. #17988335C22semidirect
37. #17988332C22semidirect
132. #41×C73SD16semidirect
261. C73#19409semidirect
362. #41#11924016semidirect

Maximal subgroups

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

subgroupindex
#226208722
#226197192
#226190422
#226093452
#226093382
#226092772
#226092702
#226087032
#226086222
C2×#179883312
C73×#30382
#1940973