Finite Group Databasebeta

← #10816045

Subgroups — #10816045

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5077C211yesmaximal, characteristic
2#78742C212yesmaximal, complemented
3#78726C212yesmaximal, complemented
4C3×#1957C211yesmaximal, characteristic, complemented
5#71763C211yesmaximal, characteristic, complemented
6#70218C212yesmaximal, complemented
7#70195C212yesmaximal, complemented
8#69789C212yesmaximal, complemented
9#69592C212yesmaximal, complemented
10#22897—31nomaximal, Sylow
11C2×#1004C2211yescharacteristic
12C2×C3×#127C2211yescharacteristic
13#5798C2211yescharacteristic, complemented
14C2×#932C2212yes—
15C2×#901C2212yes—
16#5504C2212yes—
17C2×C3×#102C2212yes—
18C2×C3×#102C2212yescomplemented
19C2×C3×C4×C8C2211yescharacteristic, complemented
20C3×#384C2211yescharacteristic, complemented
21C2×#833C2211yescharacteristic
22C2×#833C2211yescharacteristic, complemented
23#5215C2212yes—
24#5113C2212yes—
25#5101C2212yescomplemented
26#5077C418yescomplemented
27#4966C2212yes—
28#4964C2212yescomplemented
29C2×#805C2212yes—
30#4785—22no—
31#4772—22no—
32#4771—22no—
33#4770—22no—
34D3×#102—22no—
35#4758—22no—
36#4757—22no—
37#4756—22no—
38#4552C2212yescomplemented
39#4530C2212yescomplemented
40#4523C2212yescomplemented
41C2×#453—31no—
42#2946—32no—
43#2897—32no—
44#1957D311yescharacteristic, complemented
45#1931—32no—
46#1907—32no—
47#1887—31no—
48#1859—32no—
49#1764—32no—
50C22×C3×D4—41no—
51C22×C3×D4C2311yescharacteristic
52C22×C3×D4C2311yescharacteristic
53C22×#67—41no—
54#1004—24no—
55#1004—41no—
56#1004C2×C414yescomplemented
57C2×#237—22no—
58C2×D3×D4—22no—
59C2×#235C2312yes—
60C22×C3×C8C2312yes—
61C3×#127—41no—
62C3×#127C2×C414yescomplemented
63C2×C3×C4⋊C4—41no—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C42C2311yescharacteristic
66C2×#217—41no—
67D4×Dic3—22no—
68D4×Dic3—22no—
69C2×#214—22no—
70#932—24no—
71#932C2×C418yescomplemented
72C2×#213—22no—
73C4×#67—42no—
74#926C2312yes—
75C2×#208C2312yes—
76C2×#208C2312yes—
77#901—24no—
78#901—42no—
79C2×ΓL(1,𝔽25)—22no—
80C2×D3×C8—22no—
81#893—22no—
82#892—22no—
83D3×C4⋊C4—22no—
84#887—22no—
85C3×#102—22no—
86C3×#102—22no—
87C3×#102—22no—
88C3×#102—22no—
89C3×C4×C8—41no—
90#833—41no—
91#833C2×C414yescomplemented
92#832—22no—
93#832—22no—
94#821D412yescomplemented
95#821D412yescomplemented
96#816D412yescomplemented
97#816D412yescomplemented
98#815—22no—
99C8×Dic3—22no—
100#810—22no—
101#809—22no—
102#808—22no—
103#807—22no—
104#805C2×C418yescomplemented
105C2×H(5,𝔽2)—31no—
106C2×AGL(1,ℤ/8)—32no—
107C2×#131—32no—
108C2×#127C2×D311yescharacteristic
109#520—32no—
110#518—31no—
111#453—38no—
112#442—62no—
113#440—62no—
114C4×SD16—62no—
115C4×D8—62no—
116#428—32no—
117C2×#104—32no—
118#418—32no—
119#418—62no—
120#417—32no—
121#417—62no—
122#416—32no—
123#416—62no—
124C2×#102—62no—
125C2×#102C2×D312yes—
126C2×#102C2×D312yescomplemented
127C2×#100—31no—
128C2×#100—31no—
129#405—32no—
130C2×C4×C8C2×D311yescharacteristic, complemented
131#384C2×D311yescharacteristic, complemented
132#343—32no—
133#341—32no—
134#339—32no—
135C24×C3—21no—
136C24×C3—21no—
137C23×D3—41no—
138C2×C3×D4—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—44no—
145C2×C3×D4—82no—
146C2×C3×D4—82no—
147C2×C3×D4C22×C412yes—
148C2×C3×D4C22×C412yes—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4C2411yescharacteristic
152C2×#67—22no—
153C2×#67—42no—
154C2×#67—44no—
155C2×#67—44no—
156C22×Dic3—41no—
157C22×Dic3—41no—
158C22×Dic3—41no—
159#237—28no—
160#237—44no—
161D3×D4—28no—
162D3×D4—44no—
163#235—22no—
164#235—22no—
165#235C22×C414yes—
166#235#4114yes—
167C2×D12C2×D412yes—
168C2×C4×D3—22no—
169C2×C4×D3—42no—
170C2×Dic6C2×D412yes—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8C2×D412yes—
174C2×C3×C8C2×D412yes—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—82no—
178C3×C42—21no—
179C3×C42C22×C412yes—
180#217—22no—
181#217—42no—
182#214—28no—
183#213—28no—
184(C2×D3)⋊C4—42no—
185C4⋊Dic3C2×D412yes—
186C4⋊Dic3C2×D412yes—
187Dic3⋊C4—42no—
188C4×Dic3—22no—
189C4×Dic3—42no—
190#208—22no—
191#208—22no—
192#208—22no—
193#208C22×C414yes—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—22no—
196ΓL(1,𝔽25)—44no—
197D3×C8—44no—
198H(5,𝔽2)—34no—
199H(5,𝔽2)—64no—
200H(5,𝔽2)—121no—
201C2×#41—32no—
202C2×#41—62no—
203C22×D4—41no—
204C22×D4—62no—
205C22×D4—121no—
206C22×D4C22×D311yescharacteristic
207C22×D4C22×D311yescharacteristic
208AGL(1,ℤ/8)—38no—
209AGL(1,ℤ/8)—64no—
210C2×SD16—62no—
211C2×D8—62no—
212#131—64no—
213#131—122no—
214C2×M16—32no—
215C2×M16—32no—
216C2×M16—62no—
217C22×C8—62no—
218C22×C8C22×D312yes—
219#127—41no—
220#127C4×D314yescomplemented
221C4×Q8—62no—
222C4×D4—62no—
223C4×D4—62no—
224C4×D4—62no—
225C4×D4—122no—
226#117—32no—
227#117—62no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—62no—
230C2×C4⋊C4C22×D312yes—
231C2×C22⋊C4—121no—
232C2×C42C22×D311yescharacteristic
233#107—62no—
234#106—62no—
235#104—38no—
236#103—32no—
237#103—32no—
238#103—62no—
239#102—22no—
240#102—22no—
241#102—22no—
242#102—22no—
243#102—32no—
244#102—32no—
245#102—62no—
246#102—62no—
247#102—62no—
248#100—34no—
249#100—121no—
250#97—62no—
251C4×C8—41no—
252C4×C8—62no—
253C23×C3—22no—
254C23×C3—22no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—81no—
260C23×C3—81no—
261C23×C3—81no—
262C23×C3—81no—
263C23×C3H(5,𝔽2)11yescharacteristic
264C22×D3—22no—
265C22×D3—44no—
266C22×D3—161no—
267C22×D3—161no—
268C3×D4—24no—
269C3×D4—24no—
270C3×D4—42no—
271C3×D4—42no—
272C3×D4—88no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—42no—
277C2×C3×C4—82no—
278C2×C3×C4C23×C411yescharacteristic
279C2×C3×C4C22×D412yes—
280C2×C3×C4C2×#4112yes—
281C2×C3×C4H(5,𝔽2)11yescharacteristic
282#67—24no—
283#67—24no—
284#67—48no—
285C2×Dic3—22no—
286C2×Dic3—22no—
287C2×Dic3—22no—
288C2×Dic3—44no—
289C2×Dic3—161no—
290C2×Dic3—161no—
291D12—22no—
292D12C4×D414yes—
293C4×D3—24no—
294C4×D3—24no—
295C4×D3—162no—
296Dic6—22no—
297Dic6C4×D414yes—
298C3×C8—22no—
299C3×C8—22no—
300C3⋊C8—24no—
301C3⋊C8—42no—
302C24—21no—
303C24—21no—
304C24—121no—
305#41—34no—
306#41—34no—
307#41—62no—
308#41—62no—
309#41—68no—
310#41—124no—
311C2×Q8—32no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—32no—
319C2×D4—44no—
320C2×D4—68no—
321C2×D4—62no—
322C2×D4—82no—
323C2×D4—82no—
324C2×D4—124no—
325C2×D4—122no—
326C2×D4—124no—
327C2×D4—124no—
328C2×D4C2×C4×D312yes—
329C2×D4C2×C4×D312yes—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—62no—
333C22×C4—122no—
334C22×C4—121no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4C23×D311yescharacteristic
338SD16—68no—
339D8—68no—
340M16—34no—
341M16—62no—
342M16—62no—
343M16—62no—
344M16—124no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—62no—
348C2×C8—62no—
349C2×C8—124no—
350C2×C8D3×D412yes—
351C2×C8D3×D412yes—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—32no—
355C4⋊C4—32no—
356C4⋊C4—82no—
357C4⋊C4—122no—
358C22⋊C4—62no—
359C22⋊C4—122no—
360C22⋊C4—122no—
361C42—21no—
362C42—62no—
363C42—122no—
364C42C2×C4×D312yes—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—42no—
368C22×C3—42no—
369C22×C3—82no—
370C22×C3—82no—
371C22×C3—81no—
372C22×C3—81no—
373C22×C3—81no—
374C22×C3—81no—
375C22×C3—161no—
376C22×C3—161no—
377C22×C3#51811yescharacteristic
378C22×C3D4211yescharacteristic
379C22×C3#55111yescharacteristic
380C2×D3—24no—
381C2×D3—42no—
382C2×D3—42no—
383C2×D3—164no—
384C3×C4—24no—
385C3×C4—82no—
386C3×C4C2×C4×D412yes—
387C3×C4#54612yes—
388Dic3—24no—
389Dic3—42no—
390Dic3—42no—
391C23—22no—
392C23—22no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—62no—
398C23—81no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—124no—
403C23—481no—
404C23—481no—
405C23#100411yescharacteristic
406Q8—34no—
407Q8—62no—
408D4—24no—
409D4—24no—
410D4—34no—
411D4—42no—
412D4—42no—
413D4—64no—
414D4—64no—
415D4—62no—
416D4—88no—
417D4—128no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—42no—
422C2×C4—62no—
423C2×C4—64no—
424C2×C4—64no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—82no—
428C2×C4—124no—
429C2×C4—482no—
430C2×C4—481no—
431C2×C4—481no—
432C2×C4C22×C4×D311yescharacteristic
433C2×C4C2×D3×D412yes—
434C2×C4#100411yescharacteristic
435C2×C4D3×#4112yes—
436C8—22no—
437C8—22no—
438C8—64no—
439C8—122no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—82no—
443C2×C3—82no—
444C2×C3#235811yescharacteristic
445C2×C3#329111yescharacteristic
446C2×C3#329211yescharacteristic
447D3—44no—
448C22—21no—
449C22—21no—
450C22—42no—
451C22—42no—
452C22—64no—
453C22—82no—
454C22—82no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—122no—
460C22—122no—
461C22—161no—
462C22—161no—
463C22—484no—
464C22#548611yescharacteristic
465C22#567911yescharacteristic
466C22#571711yescharacteristic
467C4—24no—
468C4—64no—
469C4—82no—
470C4—122no—
471C4—122no—
472C4C4×D3×D412yes—
473C4#561012yes—
474C3#2289711yescharacteristic, complemented, Sylow
475C2—21no—
476C2—21no—
477C2—82no—
478C2—82no—
479C2—124no—
480C2#7643211yescharacteristic
481C2#8177411yescharacteristic
482C2#8179111yescharacteristic
subgroupG/H
centerC22#5717
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1957C2
socleC22×C3D42
prime radicalC2×#5077C2
cosocleC22×C3×C4C24
hypercenter#1957D3
hypocenterC3#22897
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1957

Complements

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

NH ≅ G/Ntype
2. #78742C2semidirect
3. #78726C2semidirect
4. C3×#1957C2semidirect
5. #71763C2semidirect
6. #70218C2semidirect
7. #70195C2semidirect
8. #69789C2semidirect
9. #69592C2semidirect
13. #5798C22semidirect
18. C2×C3×#102C22semidirect
19. C2×C3×C4×C8C22semidirect
20. C3×#384C22semidirect
22. C2×#833C22semidirect
25. #5101C22semidirect
26. #5077C4semidirect
28. #4964C22semidirect
38. #4552C22semidirect
39. #4530C22semidirect
40. #4523C22semidirect
44. #1957D3semidirect
56. #1004C2×C4semidirect
62. C3×#127C2×C4semidirect
71. #932C2×C4semidirect
91. #833C2×C4semidirect
94. #821D4semidirect
95. #821D4semidirect
96. #816D4semidirect
97. #816D4semidirect
104. #805C2×C4semidirect
126. C2×#102C2×D3semidirect
130. C2×C4×C8C2×D3semidirect
131. #384C2×D3semidirect
220. #127C4×D3semidirect
474. C3#22897semidirect

Maximal subgroups

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

subgroupindex
C2×#50772
#787422
#787262
C3×#19572
#717632
#702182
#701952
#697892
#695922
#228973