Finite Group Databasebeta

← #10950936

Subgroups — #10950936

Order 768.

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

normal subgroups only

This group has 590 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×#467C212yesmaximal
2C3×#2216C211yesmaximal, characteristic, complemented
3#71502C212yesmaximal, complemented
4#71416C211yesmaximal, characteristic, complemented
5C2×(C2×D3)⋊C16C211yesmaximal, characteristic
6#71081C212yesmaximal, complemented
7#69331C212yesmaximal, complemented
8#69232C212yesmaximal, complemented
9#69225C212yesmaximal, complemented
10#33755—31nomaximal, Sylow
11C2×D3×Q16C2212yes—
12C22×D3×C8C2211yescharacteristic
13D3×C22⋊Q8—22no—
14C2×C3×Q32C2211yescharacteristic, complemented
15C22×C3×C16C2211yescharacteristic
16C3×#467C2212yes—
17#5188C2212yes—
18C2×#829C2212yescomplemented
19#5116C2212yes—
20C22×C3⋊C16C2211yescharacteristic
21#5051—22no—
22C2×Dic24C2211yescharacteristic, complemented
23#4887C2212yes—
24D3×#107C2212yes—
25#4813—22no—
26D3×#103—22no—
27C3×#366C2211yescharacteristic, complemented
28C3×#358C2212yescomplemented
29#4564C2212yescomplemented
30(C2×D3)⋊C16C2212yes—
31(C2×D3)⋊C16C2212yes—
32#4504C2211yescharacteristic, complemented
33#4503C2212yescomplemented
34#4492C2212yescomplemented
35#4489C2212yescomplemented
36C2×#467—32no—
37#2236—32no—
38#2236—32no—
39#2216—31no—
40#2216—32no—
41#2216D311yescharacteristic, complemented
42#2188—32no—
43#2188—32no—
44C2×C22⋊C16—31no—
45C2×D3×Q8—22no—
46C22×C4×D3D411yescharacteristic
47C2×C3×Q16C2312yes—
48C22×C3×C8C2311yescharacteristic
49C3×C22⋊Q8—22no—
50#946—22no—
51C2×#216—22no—
52#924—22no—
53C22×C3⋊C8D411yescharacteristic
54D3×Q16—42no—
55D3×Q16—42no—
56C2×Dic12C2312yes—
57C2×D3×C8—41no—
58C2×D3×C8—41no—
59C2×D3×C8D412yes—
60C2×D3×C8C2312yes—
61#897—22no—
62#896—42no—
63D3×C4⋊C4—22no—
64D3×C4⋊C4—42no—
65D3×C22⋊C4—42no—
66#878—42no—
67C3×Q32—41no—
68C2×C3×C16—41no—
69C2×C3×C16C2311yescharacteristic
70C2×C3×C16C2311yescharacteristic
71C3×#107C2312yes—
72C3×#103—22no—
73#835—22no—
74#829—42no—
75#818C2312yes—
76#816—22no—
77C2×C3⋊C16—41no—
78C2×C3⋊C16C2312yes—
79#810—22no—
80#807—22no—
81Dic24—41no—
82C22×Q16—32no—
83C23×C8—31no—
84C2×C22⋊Q8—62no—
85C2×Q32—32no—
86C2×Q32—31no—
87C2×Q32C2×D311yescharacteristic, complemented
88C22×C16—31no—
89C22×C16C2×D311yescharacteristic
90#467—32no—
91#467—32no—
92#467—32no—
93#467—62no—
94#467—62no—
95#467C2×D312yes—
96C2×#107—32no—
97C2×#103—62no—
98#366—31no—
99#366—32no—
100#366C2×D311yescharacteristic, complemented
101#358—32no—
102#358—32no—
103#358—32no—
104#358C2×D312yescomplemented
105C22⋊C16—32no—
106C22⋊C16—32no—
107C23×D3D811yescharacteristic
108C2×C3×Q8—22no—
109C22×C3×C4C2×D411yescharacteristic
110C22×Dic3D811yescharacteristic
111D3×Q8—44no—
112D3×Q8—44no—
113C2×C4×D3—41no—
114C2×C4×D3—41no—
115C2×C4×D3—42no—
116C2×C4×D3—42no—
117C2×C4×D3D812yes—
118C2×C4×D3C2×D412yes—
119C2×Dic6—22no—
120C2×Dic6—42no—
121C3×Q16—22no—
122C3×Q16—22no—
123C2×C3×C8—21no—
124C2×C3×C8—21no—
125C2×C3×C8C2×D411yescharacteristic
126C2×C3×C8C2411yescharacteristic
127C3×C4⋊C4—22no—
128C3×C4⋊C4—42no—
129C3×C22⋊C4—42no—
130#217—42no—
131#216—44no—
132(C2×D3)⋊C4—42no—
133(C2×D3)⋊C4—42no—
134C4⋊Dic3—22no—
135C4⋊Dic3—42no—
136Dic3⋊C4—42no—
137Dic3⋊C4—42no—
138Dic3⋊C4—42no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8C2×D412yes—
141Dic12—22no—
142Dic12—22no—
143D3×C8—22no—
144D3×C8—22no—
145D3×C8—42no—
146C3×C16—21no—
147C3×C16—21no—
148C3⋊C16—22no—
149C22×Q8—62no—
150C23×C4—31no—
151C2×Q16—32no—
152C2×Q16—62no—
153C2×Q16—122no—
154C2×Q16—122no—
155C2×Q16C22×D312yes—
156C22×C8—31no—
157C22×C8—32no—
158C22×C8—32no—
159C22×C8—121no—
160C22×C8—121no—
161C22×C8C22×D311yescharacteristic
162C22⋊Q8—22no—
163C22⋊Q8—62no—
164C22⋊Q8—62no—
165C22⋊Q8—62no—
166C22⋊Q8—122no—
167C22⋊Q8—122no—
168C2×C4⋊C4—62no—
169C2×C4⋊C4—122no—
170C2×C22⋊C4—122no—
171Q32—41no—
172Q32—121no—
173Q32—122no—
174C2×C16—32no—
175C2×C16—41no—
176C2×C16—121no—
177C2×C16C22×D311yescharacteristic
178C2×C16C22×D311yescharacteristic
179#107—32no—
180#107—62no—
181#107C22×D312yes—
182#103—22no—
183#103—62no—
184#103—62no—
185#103—62no—
186C23×C3C2×D811yescharacteristic
187C22×D3—41no—
188C22×D3—41no—
189C22×D3Q3214yescomplemented
190C22×D3C2×D812yes—
191C3×Q8—42no—
192C3×Q8—42no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4C2×D811yescharacteristic
198C2×C3×C4C22×D411yescharacteristic
199C2×Dic3—41no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3C2×D812yes—
203C4×D3—22no—
204C4×D3—22no—
205C4×D3—42no—
206C4×D3—84no—
207C4×D3—84no—
208Dic6—42no—
209Dic6—42no—
210Dic6—84no—
211C3×C8—21no—
212C3×C8H(5,𝔽2)11yescharacteristic
213C3×C8H(5,𝔽2)11yescharacteristic
214C3⋊C8—22no—
215C24—31no—
216C2×Q8—22no—
217C2×Q8—62no—
218C2×Q8—122no—
219C2×Q8—124no—
220C2×Q8—124no—
221C22×C4—31no—
222C22×C4—32no—
223C22×C4—32no—
224C22×C4—121no—
225C22×C4—121no—
226C22×C4—122no—
227C22×C4—122no—
228C22×C4D3×D411yescharacteristic
229Q16—22no—
230Q16—22no—
231Q16—62no—
232Q16—62no—
233Q16—124no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—32no—
237C2×C8—62no—
238C2×C8—62no—
239C2×C8—122no—
240C2×C8—121no—
241C2×C8D3×D411yescharacteristic
242C2×C8C23×D311yescharacteristic
243C4⋊C4—22no—
244C4⋊C4—42no—
245C4⋊C4—62no—
246C4⋊C4—122no—
247C4⋊C4—122no—
248C4⋊C4—122no—
249C4⋊C4—122no—
250C22⋊C4—42no—
251C22⋊C4—122no—
252C22⋊C4—122no—
253C22⋊C4—122no—
254C16—21no—
255C16—21no—
256C16—62no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3C2×Q3212yes—
260C22×C3C22×D811yescharacteristic
261C2×D3—22no—
262C2×D3—22no—
263C2×D3—44no—
264C2×D3C2×Q3214yes—
265C3×C4—21no—
266C3×C4—82no—
267C3×C4—82no—
268C3×C4#53411yescharacteristic
269C3×C4#57511yescharacteristic
270Dic3—22no—
271Dic3—82no—
272Dic3—82no—
273C23—32no—
274C23—34no—
275C23—121no—
276C23—121no—
277C23D3×D811yescharacteristic
278Q8—42no—
279Q8—42no—
280Q8—122no—
281Q8—122no—
282Q8—244no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—32no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—62no—
289C2×C4—62no—
290C2×C4—122no—
291C2×C4—121no—
292C2×C4—122no—
293C2×C4—122no—
294C2×C4—244no—
295C2×C4—244no—
296C2×C4D3×D811yescharacteristic
297C2×C4C2×D3×D411yescharacteristic
298C8—21no—
299C8—62no—
300C8#100411yescharacteristic
301C8#100911yescharacteristic
302C2×C3—22no—
303C2×C3#318611yescharacteristic
304C2×C3C22×Q3211yescharacteristic
305C2×C3#341411yescharacteristic
306D3—24no—
307C22—21no—
308C22—21no—
309C22—34no—
310C22—62no—
311C22—62no—
312C22—124no—
313C22D3×Q3212yes—
314C22C2×D3×D811yescharacteristic
315C4—21no—
316C4—62no—
317C4—82no—
318C4—82no—
319C4—242no—
320C4—242no—
321C4#560511yescharacteristic
322C4#576811yescharacteristic
323C3#3375511yescharacteristic, complemented, Sylow
324C2—22no—
325C2—64no—
326C2#8071911yescharacteristic
327C2C2×D3×Q3211yescharacteristic
328C2#8203011yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2216C2
socleC22×C3C22×D8
prime radicalC23×D3D8
hypercenter#2216D3
hypocenterC3#33755
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2216

Complements

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

NH ≅ G/Ntype
2. C3×#2216C2semidirect
3. #71502C2semidirect
4. #71416C2semidirect
6. #71081C2semidirect
7. #69331C2semidirect
8. #69232C2semidirect
9. #69225C2semidirect
14. C2×C3×Q32C22semidirect
18. C2×#829C22semidirect
22. C2×Dic24C22semidirect
27. C3×#366C22semidirect
28. C3×#358C22semidirect
29. #4564C22semidirect
32. #4504C22semidirect
33. #4503C22semidirect
34. #4492C22semidirect
35. #4489C22semidirect
41. #2216D3semidirect
87. C2×Q32C2×D3semidirect
100. #366C2×D3semidirect
104. #358C2×D3semidirect
189. C22×D3Q32semidirect
323. C3#33755semidirect

Maximal subgroups

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

subgroupindex
D3×#4672
C3×#22162
#715022
#714162
C2×(C2×D3)⋊C162
#710812
#693312
#692322
#692252
#337553