Finite Group Databasebeta

← #10950912

Subgroups — #10950912

Order 768.

694 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#466C212yesmaximal, complemented
2C3×#2215C211yesmaximal, characteristic, complemented
3#71467C212yesmaximal, complemented
4#71413C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊C16C211yesmaximal, characteristic, complemented
6#71071C212yesmaximal, complemented
7#69330C212yesmaximal, complemented
8#69199C212yesmaximal, complemented
9#69191C212yesmaximal, complemented
10#33752—31nomaximal, Sylow
11C2×D3×D8C2212yescomplemented
12C22×D3×C8C2211yescharacteristic
13D3×#121—22no—
14C2×C3×D16C2211yescharacteristic, complemented
15C22×C3×C16C2211yescharacteristic, complemented
16C3×#466C2212yescomplemented
17#5157C2212yes—
18C2×#826C2212yescomplemented
19#5115C2212yescomplemented
20C22×C3⋊C16C2211yescharacteristic, complemented
21#5038—22no—
22C2×D48C2211yescharacteristic, complemented
23#4883C2212yes—
24D3×#107C2212yescomplemented
25#4775—22no—
26D3×#102—22no—
27C3×#366C2211yescharacteristic, complemented
28C3×#357C2212yescomplemented
29#4562C2212yescomplemented
30#4509C2212yescomplemented
31(C2×D3)⋊C16C2212yescomplemented
32(C2×D3)⋊C16C2212yescomplemented
33#4504C2211yescharacteristic, complemented
34#4491C2212yescomplemented
35#4489C2212yescomplemented
36C2×#466—32no—
37#2232—32no—
38#2232—32no—
39#2215—31no—
40#2215—32no—
41#2215D311yescharacteristic, complemented
42#2184—32no—
43#2184—32no—
44C2×C22⋊C16—31no—
45C2×D3×D4—22no—
46C2×D3×D4—42no—
47C22×C4×D3D411yescharacteristic
48C2×C3×D8C2312yes—
49C22×C3×C8C2311yescharacteristic
50C3×#121—22no—
51#939—42no—
52#938—22no—
53C2×#213—22no—
54#930—22no—
55C22×C3⋊C8D411yescharacteristic
56D3×D8—42no—
57D3×D8—42no—
58C2×D24C2312yes—
59C2×D3×C8—41no—
60C2×D3×C8—41no—
61C2×D3×C8D412yes—
62C2×D3×C8C2312yes—
63#895—22no—
64D3×C4⋊C4—22no—
65#884—42no—
66D3×C22⋊C4—42no—
67C3×D16—41no—
68C2×C3×C16—41no—
69C2×C3×C16C2311yescharacteristic
70C2×C3×C16C2311yescharacteristic
71C3×#107C2312yes—
72C3×#102—22no—
73#832—22no—
74#826—42no—
75#821—22no—
76#818C2312yes—
77C2×C3⋊C16—41no—
78C2×C3⋊C16C2312yes—
79#809—22no—
80#807—22no—
81D48—41no—
82C22×D8—32no—
83C23×C8—31no—
84C2×#121—62no—
85C2×D16—32no—
86C2×D16—31no—
87C2×D16C2×D311yescharacteristic, complemented
88C22×C16—31no—
89C22×C16C2×D311yescharacteristic, complemented
90#466—32no—
91#466—32no—
92#466—32no—
93#466—62no—
94#466—62no—
95#466C2×D312yescomplemented
96C2×#107—32no—
97C2×#102—62no—
98#366—31no—
99#366—32no—
100#366C2×D311yescharacteristic, complemented
101#357—32no—
102#357—32no—
103#357—32no—
104#357C2×D312yescomplemented
105C22⋊C16—32no—
106C22⋊C16—32no—
107C23×D3—42no—
108C23×D3D811yescharacteristic
109C2×C3×D4—22no—
110C2×C3×D4—42no—
111C22×C3×C4C2×D411yescharacteristic
112C2×#67—42no—
113C2×#67—42no—
114C2×#67—42no—
115C22×Dic3D811yescharacteristic
116D3×D4—44no—
117D3×D4—44no—
118D3×D4—88no—
119C2×D12—22no—
120C2×D12—42no—
121C2×C4×D3—41no—
122C2×C4×D3—41no—
123C2×C4×D3—42no—
124C2×C4×D3D812yes—
125C2×C4×D3C2×D412yes—
126C3×D8—22no—
127C3×D8—22no—
128C2×C3×C8—21no—
129C2×C3×C8—21no—
130C2×C3×C8C2×D411yescharacteristic
131C2×C3×C8C2411yescharacteristic
132C3×C4⋊C4—22no—
133C3×C22⋊C4—42no—
134#217—42no—
135#213—44no—
136(C2×D3)⋊C4—42no—
137(C2×D3)⋊C4—42no—
138C4⋊Dic3—22no—
139Dic3⋊C4—42no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8C2×D412yes—
142D24—22no—
143D24—22no—
144D3×C8—22no—
145D3×C8—22no—
146D3×C8—42no—
147C3×C16—21no—
148C3×C16—21no—
149C3⋊C16—22no—
150C22×D4—62no—
151C22×D4—122no—
152C23×C4—31no—
153C2×D8—32no—
154C2×D8—62no—
155C2×D8—122no—
156C2×D8—122no—
157C2×D8C22×D312yes—
158C22×C8—31no—
159C22×C8—32no—
160C22×C8—32no—
161C22×C8—121no—
162C22×C8—121no—
163C22×C8C22×D311yescharacteristic
164#121—22no—
165#121—62no—
166#121—62no—
167#121—62no—
168#121—122no—
169#121—122no—
170C2×C4⋊C4—62no—
171C2×C22⋊C4—122no—
172D16—41no—
173D16—121no—
174D16—122no—
175C2×C16—32no—
176C2×C16—41no—
177C2×C16—121no—
178C2×C16C22×D311yescharacteristic
179C2×C16C22×D311yescharacteristic
180#107—32no—
181#107—62no—
182#107C22×D312yes—
183#102—22no—
184#102—62no—
185#102—62no—
186#102—62no—
187C23×C3—42no—
188C23×C3C2×D811yescharacteristic
189C22×D3—41no—
190C22×D3—41no—
191C22×D3—42no—
192C22×D3—84no—
193C22×D3—84no—
194C22×D3—84no—
195C22×D3D1614yescomplemented
196C22×D3C2×D812yes—
197C3×D4—42no—
198C3×D4—42no—
199C3×D4—84no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—42no—
203C2×C3×C4C2×D811yescharacteristic
204C2×C3×C4C22×D411yescharacteristic
205#67—84no—
206#67—84no—
207#67—84no—
208C2×Dic3—41no—
209C2×Dic3—42no—
210C2×Dic3C2×D812yes—
211D12—42no—
212D12—42no—
213D12—84no—
214C4×D3—22no—
215C4×D3—22no—
216C4×D3—42no—
217C4×D3—84no—
218C3×C8—21no—
219C3×C8H(5,𝔽2)11yescharacteristic
220C3×C8H(5,𝔽2)11yescharacteristic
221C3⋊C8—22no—
222C24—31no—
223C24—122no—
224C2×D4—22no—
225C2×D4—42no—
226C2×D4—62no—
227C2×D4—122no—
228C2×D4—122no—
229C2×D4—124no—
230C2×D4—124no—
231C2×D4—122no—
232C2×D4—122no—
233C2×D4—248no—
234C22×C4—31no—
235C22×C4—32no—
236C22×C4—32no—
237C22×C4—121no—
238C22×C4—121no—
239C22×C4—122no—
240C22×C4D3×D411yescharacteristic
241D8—22no—
242D8—22no—
243D8—62no—
244D8—62no—
245D8—124no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—32no—
249C2×C8—62no—
250C2×C8—62no—
251C2×C8—122no—
252C2×C8—121no—
253C2×C8D3×D411yescharacteristic
254C2×C8C23×D311yescharacteristic
255C4⋊C4—22no—
256C4⋊C4—62no—
257C4⋊C4—122no—
258C22⋊C4—42no—
259C22⋊C4—122no—
260C22⋊C4—122no—
261C22⋊C4—122no—
262C16—21no—
263C16—21no—
264C16—62no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—82no—
268C22×C3—82no—
269C22×C3—82no—
270C22×C3C2×D1612yes—
271C22×C3C22×D811yescharacteristic
272C2×D3—22no—
273C2×D3—22no—
274C2×D3—44no—
275C2×D3—82no—
276C2×D3—82no—
277C2×D3—82no—
278C2×D3—168no—
279C2×D3C2×D1614yes—
280C3×C4—21no—
281C3×C4—82no—
282C3×C4#53411yescharacteristic
283C3×C4#57511yescharacteristic
284Dic3—22no—
285Dic3—82no—
286C23—32no—
287C23—34no—
288C23—42no—
289C23—121no—
290C23—121no—
291C23—122no—
292C23—244no—
293C23—244no—
294C23—244no—
295C23D3×D811yescharacteristic
296D4—42no—
297D4—42no—
298D4—84no—
299D4—122no—
300D4—122no—
301D4—244no—
302D4—244no—
303D4—244no—
304D4—244no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—32no—
308C2×C4—42no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—122no—
312C2×C4—121no—
313C2×C4—122no—
314C2×C4—244no—
315C2×C4D3×D811yescharacteristic
316C2×C4C2×D3×D411yescharacteristic
317C8—21no—
318C8—62no—
319C8#100411yescharacteristic
320C8#100911yescharacteristic
321C2×C3—22no—
322C2×C3—162no—
323C2×C3#318611yescharacteristic
324C2×C3C22×D1611yescharacteristic
325C2×C3#341411yescharacteristic
326D3—24no—
327D3—162no—
328C22—21no—
329C22—21no—
330C22—34no—
331C22—62no—
332C22—62no—
333C22—82no—
334C22—82no—
335C22—82no—
336C22—124no—
337C22—242no—
338C22—242no—
339C22—242no—
340C22—488no—
341C22D3×D1612yes—
342C22C2×D3×D811yescharacteristic
343C4—21no—
344C4—62no—
345C4—82no—
346C4—242no—
347C4#560511yescharacteristic
348C4#576811yescharacteristic
349C3#3375211yescharacteristic, complemented, Sylow
350C2—22no—
351C2—64no—
352C2—162no—
353C2—482no—
354C2#8071911yescharacteristic
355C2C2×D3×D1611yescharacteristic
356C2#8203011yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2215C2
socleC22×C3C22×D8
hypercenter#2215D3
hypocenterC3#33752
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2215

Complements

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

NH ≅ G/Ntype
1. D3×#466C2semidirect
2. C3×#2215C2semidirect
3. #71467C2semidirect
4. #71413C2semidirect
5. C2×(C2×D3)⋊C16C2semidirect
6. #71071C2semidirect
7. #69330C2semidirect
8. #69199C2semidirect
9. #69191C2semidirect
11. C2×D3×D8C22semidirect
14. C2×C3×D16C22semidirect
15. C22×C3×C16C22semidirect
16. C3×#466C22semidirect
18. C2×#826C22semidirect
19. #5115C22semidirect
20. C22×C3⋊C16C22semidirect
22. C2×D48C22semidirect
24. D3×#107C22semidirect
27. C3×#366C22semidirect
28. C3×#357C22semidirect
29. #4562C22semidirect
30. #4509C22semidirect
31. (C2×D3)⋊C16C22semidirect
32. (C2×D3)⋊C16C22semidirect
33. #4504C22semidirect
34. #4491C22semidirect
35. #4489C22semidirect
41. #2215D3semidirect
87. C2×D16C2×D3semidirect
89. C22×C16C2×D3semidirect
95. #466C2×D3semidirect
100. #366C2×D3semidirect
104. #357C2×D3semidirect
195. C22×D3D16semidirect
349. C3#33752semidirect

Maximal subgroups

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

subgroupindex
D3×#4662
C3×#22152
#714672
#714132
C2×(C2×D3)⋊C162
#710712
#693302
#691992
#691912
#337523