Finite Group Databasebeta

← #10705779

Subgroups — #10705779

Order 768.

482 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#82476C211yesmaximal, characteristic
2C3×#1467C211yesmaximal, characteristic, complemented
3C4×#836C211yesmaximal, characteristic, complemented
4#67456C214yesmaximal, complemented
5#11781—31nomaximal, Sylow
6C2×C3×C4×Q8C2211yescharacteristic
7C2×#947—21no—
8C2×#874C411yescharacteristic
9C3×#432—21no—
10C2×#836—21no—
11#4991C412yes—
12#4991C2212yes—
13#4988—21no—
14#4939—21no—
15#4939—21no—
16C4×C4⋊Dic3C411yescharacteristic
17C4×#208—21no—
18C3×#324—21no—
19C3×#324C2212yes—
20#4530C2212yes—
21#2548—31no—
22C4×#101—31no—
23#1467D311yescharacteristic, complemented
24#1345—34no—
25C22×C3×Q8D411yescharacteristic
26C22×D12—21no—
27C22×D12—21no—
28C22×D12C2×C412yes—
29C3×C4×Q8—41no—
30C3×C4×Q8—41no—
31C2×C3×C4⋊C4C2×C411yescharacteristic
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C4⋊C4C2311yescharacteristic
34C2×C3×C42C2×C411yescharacteristic
35C2×C3×C42D411yescharacteristic
36C2×C3×C42D411yescharacteristic
37#947—44no—
38C2×(C2×D3)⋊C4—22no—
39C2×(C2×D3)⋊C4—22no—
40C2×(C2×D3)⋊C4—22no—
41C2×(C2×D3)⋊C4—22no—
42C2×C4⋊Dic3—21no—
43C2×C4×Dic3—21no—
44C2×#208—22no—
45#874—41no—
46#874—41no—
47C3×C4⋊C8—41no—
48C3×C22⋊C8—22no—
49C3×#97—41no—
50#836—24no—
51C3⋊#95C2×C412yes—
52#803—41no—
53C4×C3⋊C8—41no—
54C2×#127—31no—
55C2×#124—61no—
56C2×C4×Q8C2×D311yescharacteristic
57#432—21no—
58C2×#101—61no—
59C4×M16—61no—
60#394—32no—
61#394—32no—
62#390—61no—
63#390—61no—
64#390—61no—
65C4×C4⋊C4—31no—
66#343—32no—
67#324—21no—
68#324C2×D312yes—
69C23×D3—22no—
70C2×C3×Q8—22no—
71C2×C3×Q8—22no—
72C2×C3×Q8—41no—
73C22×C3×C4C22⋊C411yescharacteristic
74C22×C3×C4C22⋊C411yescharacteristic
75C22×C3×C4C22⋊C411yescharacteristic
76C22×C3×C4C22⋊C411yescharacteristic
77C22×C3×C4C22×C411yescharacteristic
78C22×C3×C4C2×D411yescharacteristic
79C22×C3×C4C2×D411yescharacteristic
80C22×Dic3—22no—
81C2×D12—42no—
82C2×D12—42no—
83C2×D12—42no—
84C2×D12—42no—
85C2×D12—42no—
86C2×D12—41no—
87C2×D12—41no—
88C2×D12—42no—
89C2×D12—82no—
90C2×C3×C8—42no—
91C3×C4⋊C4—41no—
92C3×C4⋊C4—41no—
93C3×C4⋊C4—41no—
94C3×C42—21no—
95C3×C42—21no—
96C3×C42—21no—
97C3×C42—21no—
98C3×C42—21no—
99C3×C42—21no—
100(C2×D3)⋊C4—44no—
101(C2×D3)⋊C4—44no—
102(C2×D3)⋊C4—82no—
103(C2×D3)⋊C4—82no—
104C4⋊Dic3—42no—
105C4×Dic3—42no—
106#208—24no—
107#208—42no—
108C2×C3⋊C8—42no—
109C22×Q8D1211yescharacteristic
110C22×D4—32no—
111C22×D4—61no—
112C22×D4—61no—
113C2×M16—62no—
114#127—121no—
115#127—121no—
116#124—124no—
117C4×Q8—41no—
118C4×Q8—41no—
119C2×C4⋊C4—61no—
120C2×C4⋊C4C4×D311yescharacteristic
121C2×C4⋊C4#6711yescharacteristic
122C2×C4⋊C4C22×D311yescharacteristic
123C2×C22⋊C4—62no—
124C2×C22⋊C4—62no—
125C2×C22⋊C4—62no—
126C2×C22⋊C4—62no—
127C2×C42—61no—
128C2×C42C4×D311yescharacteristic
129C2×C42D1211yescharacteristic
130C2×C42#6711yescharacteristic
131C4⋊C8—41no—
132#101—64no—
133C22⋊C8—22no—
134#97—41no—
135#97—121no—
136C4×C8—121no—
137#95—32no—
138C23×C3C2×C22⋊C411yescharacteristic
139C22×D3—42no—
140C22×D3—42no—
141C22×D3—42no—
142C22×D3—42no—
143C22×D3—42no—
144C22×D3—42no—
145C22×D3—42no—
146C3×Q8—44no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—22no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4—41no—
161C2×C3×C4#9911yescharacteristic
162C2×C3×C4#9911yescharacteristic
163C2×C3×C4#10412yes—
164C2×C3×C4AGL(1,ℤ/8)12yes—
165C2×Dic3—42no—
166C2×Dic3—42no—
167C2×Dic3—42no—
168D12—82no—
169D12—84no—
170D12—84no—
171D12—82no—
172C3×C8—82no—
173C3⋊C8—44no—
174C24—62no—
175C2×Q8—22no—
176C2×Q8—22no—
177C2×Q8—41no—
178C2×D4—122no—
179C2×D4—122no—
180C2×D4—122no—
181C2×D4—122no—
182C2×D4—122no—
183C2×D4—121no—
184C2×D4—121no—
185C2×D4—122no—
186C2×D4—242no—
187C22×C4—62no—
188C22×C4(C2×D3)⋊C411yescharacteristic
189C22×C4(C2×D3)⋊C411yescharacteristic
190C22×C4(C2×D3)⋊C411yescharacteristic
191C22×C4(C2×D3)⋊C411yescharacteristic
192C22×C4C2×C4×D311yescharacteristic
193C22×C4C2×D1211yescharacteristic
194C22×C4C2×#6711yescharacteristic
195M16—64no—
196M16—122no—
197C2×C8—42no—
198C2×C8—122no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—122no—
203C22⋊C4—124no—
204C22⋊C4—124no—
205C22⋊C4—242no—
206C22⋊C4—242no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—122no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3C2×#9911yescharacteristic
217C22×C3#41811yescharacteristic
218C22×C3C2×#10411yescharacteristic
219C2×D3—82no—
220C2×D3—82no—
221C2×D3—82no—
222C2×D3—82no—
223C2×D3—82no—
224C2×D3—82no—
225C2×D3—82no—
226C3×C4—21no—
227C3×C4—22no—
228C3×C4—22no—
229C3×C4—41no—
230C3×C4—41no—
231C3×C4—41no—
232C3×C4—41no—
233C3×C4#35512yes—
234Dic3—82no—
235C23—122no—
236C23—122no—
237C23—122no—
238C23—122no—
239C23—122no—
240C23—122no—
241C23—122no—
242C23C2×(C2×D3)⋊C411yescharacteristic
243Q8—44no—
244D4—244no—
245D4—242no—
246D4—244no—
247D4—242no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4#80611yescharacteristic
266C2×C4#80611yescharacteristic
267C2×C4#82512yes—
268C2×C4#90812yes—
269C8—82no—
270C8—124no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3#150711yescharacteristic
274C2×C3#212211yescharacteristic
275C2×C3C2×#35511yescharacteristic
276D3—162no—
277C22—21no—
278C22—21no—
279C22—242no—
280C22—242no—
281C22—242no—
282C22—242no—
283C22—242no—
284C22—242no—
285C22—242no—
286C22C2×#80611yescharacteristic
287C22#513111yescharacteristic
288C22C2×#82511yescharacteristic
289C4—22no—
290C4—22no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—242no—
297C4#447712yes—
298C3#1178111yescharacteristic, complemented, Sylow
299C2—21no—
300C2—21no—
301C2—482no—
302C2#6997711yescharacteristic
303C2#7005711yescharacteristic
304C2C2×#447711yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1467C2
socleC22×C3C2×#99
prime radicalC2×#874C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1467D3
hypocenterC3#11781
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. C3×#1467C2semidirect
3. C4×#836C2semidirect
4. #67456C2semidirect
23. #1467D3semidirect
298. C3#11781semidirect

Maximal subgroups

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

subgroupindex
#824762
C3×#14672
C4×#8362
#674562
#117813