Finite Group Databasebeta

← #10949047

Subgroups — #10949047

Order 768.

1150 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5769C211yesmaximal, characteristic, complemented
2C2×C3×#508C211yesmaximal, characteristic, complemented
3#83917C211yesmaximal, characteristic, complemented
4C2×#5193C211yesmaximal, characteristic, complemented
5#72078C211yesmaximal, characteristic, complemented
6#71503C212yesmaximal, complemented
7#71485C212yesmaximal, complemented
8#71484C212yesmaximal, complemented
9#71467C212yesmaximal, complemented
10C2×#4509C211yesmaximal, characteristic, complemented
11#71409C211yesmaximal, characteristic, complemented
12#33526—31nomaximal, Sylow
13C2×#1009—21no—
14C2×C3×#135C2211yescharacteristic
15C2×C3×#135C2211yescharacteristic, complemented
16#5833—21no—
17C2×#949—21no—
18#5769—24no—
19#5769—24no—
20C2×#919C2212yes—
21C2×#914—22no—
22C2×D3×D8C2212yes—
23C22×D24C2211yescharacteristic
24C2×#901C2211yescharacteristic
25C3×#508—24no—
26C2×C3×Q32C2211yescharacteristic, complemented
27C2×C3×SD32C2211yescharacteristic, complemented
28C2×C3×SD32C2211yescharacteristic, complemented
29C2×C3×D16C2211yescharacteristic, complemented
30C22×C3×C16C2211yescharacteristic, complemented
31#5232—21no—
32#5193—24no—
33#5188C2212yescomplemented
34C2×#828C2212yescomplemented
35#5174—22no—
36#5157C2212yescomplemented
37C2×#826C2212yescomplemented
38C2×#818C2211yescharacteristic, complemented
39C2×#812C2211yescharacteristic, complemented
40#4564C2212yescomplemented
41#4562C2212yescomplemented
42#4509C2212yescomplemented
43#4509C2212yescomplemented
44(C2×D3)⋊C16C2212yescomplemented
45(C2×D3)⋊C16C2212yescomplemented
46C2×#576—31no—
47C2×#509—31no—
48C2×#508D311yescharacteristic, complemented
49#3061—31no—
50#2189—32no—
51#2187—32no—
52#2186—32no—
53#2184—32no—
54#2139—31no—
55C2×#357—31no—
56#2113—31no—
57C2×C3×#41—21no—
58C2×C3×#41—21no—
59#1009—28no—
60#1009—44no—
61C2×#239—22no—
62C2×D3×D4—22no—
63C2×D3×D4—42no—
64C2×#235—41no—
65C2×#235D411yescharacteristic
66C22×D12—21no—
67C22×D12—41no—
68C3×#135—22no—
69C3×#135—22no—
70C3×#135—22no—
71C3×#135D414yescomplemented
72C2×C3×Q16C2311yescharacteristic
73C2×C3×Q16C2311yescharacteristic
74C2×C3×SD16—21no—
75C2×C3×SD16—21no—
76C2×C3×D8C2311yescharacteristic
77C2×C3×D8C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79#949—28no—
80#946—22no—
81C2×#215—22no—
82#939—42no—
83#938—22no—
84C2×#213—22no—
85#930—41no—
86C2×C4⋊Dic3—21no—
87#924—41no—
88C2×#208D411yescharacteristic
89#919—24no—
90#919—24no—
91#914—216no—
92D3×D8—24no—
93D3×D8—24no—
94C2×D24—21no—
95C2×D24—22no—
96C2×D24D414yescomplemented
97C2×D24C2312yes—
98#901—22no—
99#901—22no—
100C2×ΓL(1,𝔽25)D412yes—
101C2×D3×C8C2312yes—
102C3×Q32—22no—
103C3×SD32—22no—
104C3×SD32—22no—
105C3×D16—22no—
106C2×C3×C16—22no—
107C2×C3×C16C2311yescharacteristic
108C2×C3×C16C2311yescharacteristic
109#835—22no—
110#832—22no—
111#828—24no—
112#826—24no—
113#818—21no—
114#818C2312yes—
115#812—22no—
116C2×C3⋊C16C2312yes—
117C2×H(5,𝔽2)—61no—
118#576—64no—
119#576—64no—
120C2×AGL(1,ℤ/8)—62no—
121C2×AGL(1,ℤ/8)—61no—
122C2×#135—32no—
123C2×#135C2×D311yescharacteristic
124C2×#135C2×D311yescharacteristic, complemented
125C22×D8—32no—
126C22×D8—31no—
127C2×#131—31no—
128#537—61no—
129#509—64no—
130#508—24no—
131C2×Q32C2×D311yescharacteristic, complemented
132C2×SD32—32no—
133C2×SD32C2×D311yescharacteristic, complemented
134C2×SD32C2×D311yescharacteristic, complemented
135C2×D16—32no—
136C2×D16C2×D311yescharacteristic, complemented
137C2×M32—31no—
138C22×C16C2×D311yescharacteristic, complemented
139#468—62no—
140#467—32no—
141#466—32no—
142C2×#107—31no—
143#417—61no—
144#358—32no—
145#357—32no—
146#357—32no—
147#357—32no—
148C22⋊C16—32no—
149C22⋊C16—32no—
150C23×D3—41no—
151C23×D3—42no—
152C3×#41—24no—
153C3×#41—42no—
154C3×#41—42no—
155C3×#41—42no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—41no—
161C2×C3×D4—41no—
162C22×C3×C4—41no—
163C22×C3×C4—41no—
164C22×C3×C4C2×D411yescharacteristic
165C2×#67—42no—
166C2×#67—42no—
167C2×#67D812yes—
168C22×Dic3—41no—
169#239—28no—
170#239—44no—
171D3×D4—28no—
172D3×D4—44no—
173D3×D4—416no—
174#235—22no—
175#235—22no—
176#235—48no—
177C2×D12—24no—
178C2×D12—22no—
179C2×D12—48no—
180C2×D12—42no—
181C2×D12—41no—
182C2×D12—42no—
183C2×D12D811yescharacteristic
184C2×C4×D3—42no—
185C2×C4×D3C2×D412yes—
186C2×Dic6D811yescharacteristic
187C3×Q16—21no—
188C3×Q16—21no—
189C3×Q16—21no—
190C3×Q16C2×D412yes—
191C3×SD16—24no—
192C3×SD16—42no—
193C3×D8—21no—
194C3×D8—21no—
195C3×D8—21no—
196C3×D8C2×D412yes—
197C2×C3×C8C2×D411yescharacteristic
198C2×C3×C8C2×D412yes—
199C2×C3×C8C2×D412yes—
200C2×C3×C8C2411yescharacteristic
201#217—42no—
202#215—28no—
203#213—28no—
204(C2×D3)⋊C4—42no—
205C4⋊Dic3—22no—
206C4⋊Dic3—41no—
207Dic3⋊C4—42no—
208#208—22no—
209C2×C3⋊C8C2×D412yes—
210D24—24no—
211D24—22no—
212D24C2×D414yes—
213ΓL(1,𝔽25)—24no—
214D3×C8—22no—
215D3×C8—22no—
216C3×C16—21no—
217C3×C16—21no—
218C3⋊C16—22no—
219H(5,𝔽2)—68no—
220H(5,𝔽2)—124no—
221C2×#41—21no—
222C2×#41—21no—
223C2×#41—31no—
224C2×#41—62no—
225C2×#41—121no—
226C22×D4—62no—
227C22×D4—61no—
228C22×D4—121no—
229C22×D4—122no—
230AGL(1,ℤ/8)—616no—
231AGL(1,ℤ/8)—68no—
232#135—22no—
233#135—22no—
234#135—22no—
235#135—64no—
236#135—64no—
237#135#6714yescomplemented
238C2×Q16C22×D311yescharacteristic
239C2×Q16C22×D311yescharacteristic
240C2×SD16—21no—
241C2×SD16—21no—
242C2×SD16—62no—
243C2×D8—34no—
244C2×D8—32no—
245C2×D8—62no—
246C2×D8—64no—
247C2×D8—64no—
248C2×D8—61no—
249C2×D8—62no—
250C2×D8C22×D311yescharacteristic
251C2×D8C22×D311yescharacteristic
252#131—62no—
253#131—62no—
254C2×M16—31no—
255C2×M16—32no—
256C22×C8—32no—
257C22×C8C22×D311yescharacteristic
258C22⋊Q8—62no—
259C22⋊Q8—121no—
260#121—62no—
261#121—121no—
262#121—122no—
263C2×C4⋊C4—61no—
264Q32—22no—
265SD32—22no—
266SD32—22no—
267SD32—64no—
268D16—22no—
269D16—64no—
270M32—62no—
271C2×C16—22no—
272C2×C16—32no—
273C2×C16C22×D311yescharacteristic
274C2×C16C22×D311yescharacteristic
275#107—32no—
276#107—61no—
277#103—62no—
278#102—62no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3C2×D811yescharacteristic
282C22×D3—48no—
283C22×D3—42no—
284C22×D3—44no—
285C22×D3—162no—
286C22×D3—162no—
287C22×D3—161no—
288C22×D3—161no—
289C22×D3C2×D812yes—
290C3×Q8—22no—
291C3×Q8—41no—
292C3×Q8—41no—
293C3×Q8—41no—
294C3×D4—22no—
295C3×D4—44no—
296C3×D4—41no—
297C3×D4—41no—
298C3×D4—41no—
299C3×D4—82no—
300C2×C3×C4—41no—
301C2×C3×C4—44no—
302C2×C3×C4—41no—
303C2×C3×C4—82no—
304C2×C3×C4#12012yes—
305C2×C3×C4C2×D811yescharacteristic
306C2×C3×C4AGL(1,ℤ/8)12yes—
307C2×C3×C4C22×D411yescharacteristic
308#67—24no—
309#67—48no—
310#67—162no—
311C2×Dic3—42no—
312C2×Dic3—161no—
313C2×Dic3C2×D812yes—
314D12—24no—
315D12—22no—
316D12—48no—
317D12—42no—
318D12—44no—
319C4×D3—22no—
320C4×D3—22no—
321C4×D3—48no—
322Dic6—22no—
323C3×C8#12012yes—
324C3×C8C22×D411yescharacteristic
325C3×C8C22×D411yescharacteristic
326C3⋊C8—22no—
327C24—122no—
328C24—121no—
329#41—24no—
330#41—42no—
331#41—42no—
332#41—42no—
333#41—62no—
334#41—62no—
335#41—68no—
336#41—128no—
337#41—124no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—31no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—32no—
344C2×D4—31no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—68no—
348C2×D4—64no—
349C2×D4—62no—
350C2×D4—122no—
351C2×D4—128no—
352C2×D4—122no—
353C2×D4—124no—
354C2×D4—1216no—
355C2×D4—122no—
356C2×D4—121no—
357C2×D4—122no—
358C22×C4—32no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—122no—
362C22×C4—121no—
363C22×C4D3×D411yescharacteristic
364Q16—21no—
365Q16—21no—
366Q16—21no—
367Q16C2×#6712yes—
368SD16—24no—
369SD16—42no—
370SD16—68no—
371D8—21no—
372D8—21no—
373D8—21no—
374D8—34no—
375D8—68no—
376D8—64no—
377D8—62no—
378D8C2×#6712yes—
379M16—62no—
380M16—64no—
381C2×C8—32no—
382C2×C8—62no—
383C2×C8—62no—
384C2×C8D3×D411yescharacteristic
385C2×C8D3×D412yes—
386C2×C8C2×#6712yes—
387C2×C8C23×D311yescharacteristic
388C4⋊C4—62no—
389C4⋊C4—122no—
390C4⋊C4—121no—
391C22⋊C4—122no—
392C22⋊C4—122no—
393C16—21no—
394C16—21no—
395C16—62no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—42no—
399C22×C3—81no—
400C22×C3—81no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3#44712yes—
405C22×C3C22×D811yescharacteristic
406C2×D3—22no—
407C2×D3—22no—
408C2×D3—22no—
409C2×D3—44no—
410C2×D3—82no—
411C2×D3—82no—
412C2×D3—168no—
413C2×D3—164no—
414C3×C4—42no—
415C3×C4—81no—
416C3×C4#44712yes—
417C3×C4C2×#12011yescharacteristic
418C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
419Dic3—22no—
420Dic3—82no—
421C23—32no—
422C23—41no—
423C23—41no—
424C23—128no—
425C23—122no—
426C23—124no—
427C23—481no—
428C23—481no—
429C23—482no—
430C23—482no—
431C23D3×D811yescharacteristic
432Q8—22no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436Q8—62no—
437D4—22no—
438D4—44no—
439D4—41no—
440D4—41no—
441D4—41no—
442D4—64no—
443D4—62no—
444D4—64no—
445D4—82no—
446D4—128no—
447D4—128no—
448D4—124no—
449D4—122no—
450D4—482no—
451C2×C4—32no—
452C2×C4—41no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—82no—
458C2×C4—128no—
459C2×C4—122no—
460C2×C4—481no—
461C2×C4D3×D811yescharacteristic
462C2×C4#91112yes—
463C2×C4#93712yes—
464C2×C4C2×D3×D411yescharacteristic
465C8—62no—
466C8#93712yes—
467C8C2×D3×D411yescharacteristic
468C8C22×#6711yescharacteristic
469C2×C3—22no—
470C2×C3—82no—
471C2×C3—161no—
472C2×C3C2×#44711yescharacteristic
473C2×C3#341511yescharacteristic
474C2×C3#341611yescharacteristic
475D3—42no—
476D3—84no—
477C22—21no—
478C22—21no—
479C22—42no—
480C22—62no—
481C22—62no—
482C22—62no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—124no—
489C22—242no—
490C22—242no—
491C22—484no—
492C22—488no—
493C22#515612yes—
494C22C2×D3×D811yescharacteristic
495C4—42no—
496C4—62no—
497C4—81no—
498C4—242no—
499C4#515612yes—
500C4C2×#91111yescharacteristic
501C4C2×#93711yescharacteristic
502C3#3352611yescharacteristic, complemented, Sylow
503C2—22no—
504C2—82no—
505C2—122no—
506C2—161no—
507C2—244no—
508C2#8203111yescharacteristic
509C2#8203211yescharacteristic
510C2C2×#515611yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×#508C2
socleC22×C3C22×D8
hypercenterC2×#508D3
hypocenterC3#33526
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#508

Complements

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

NH ≅ G/Ntype
1. C2×#5769C2semidirect
2. C2×C3×#508C2semidirect
3. #83917C2semidirect
4. C2×#5193C2semidirect
5. #72078C2semidirect
6. #71503C2semidirect
7. #71485C2semidirect
8. #71484C2semidirect
9. #71467C2semidirect
10. C2×#4509C2semidirect
11. #71409C2semidirect
15. C2×C3×#135C22semidirect
26. C2×C3×Q32C22semidirect
27. C2×C3×SD32C22semidirect
28. C2×C3×SD32C22semidirect
29. C2×C3×D16C22semidirect
30. C22×C3×C16C22semidirect
33. #5188C22semidirect
34. C2×#828C22semidirect
36. #5157C22semidirect
37. C2×#826C22semidirect
38. C2×#818C22semidirect
39. C2×#812C22semidirect
40. #4564C22semidirect
41. #4562C22semidirect
42. #4509C22semidirect
43. #4509C22semidirect
44. (C2×D3)⋊C16C22semidirect
45. (C2×D3)⋊C16C22semidirect
48. C2×#508D3semidirect
71. C3×#135D4semidirect
96. C2×D24D4semidirect
124. C2×#135C2×D3semidirect
131. C2×Q32C2×D3semidirect
133. C2×SD32C2×D3semidirect
134. C2×SD32C2×D3semidirect
136. C2×D16C2×D3semidirect
138. C22×C16C2×D3semidirect
237. #135#67semidirect
502. C3#33526semidirect

Maximal subgroups

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

subgroupindex
C2×#57692
C2×C3×#5082
#839172
C2×#51932
#720782
#715032
#714852
#714842
#714672
C2×#45092
#714092
#335263