Finite Group Databasebeta

← #10683405

Subgroups — #10683405

Order 768.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C4×#869C211yesmaximal, characteristic
2C3×#1767C211yesmaximal, characteristic
3#69393C211yesmaximal, characteristic
4#67478C214yesmaximal
5#7625—31nomaximal, Sylow
6C2×#869C411yescharacteristic
7C2×C4×Dic6—21no—
8C2×C3×C4⋊C8—21no—
9C2×C3×#97—21no—
10C3×C43C2211yescharacteristic
11#4935C411yescharacteristic
12C4×C4⋊Dic3—21no—
13C4×C4⋊Dic3C412yes—
14#4929—21no—
15#4928—21no—
16#4928C2212yes—
17C3×#336C2212yes—
18#4524—21no—
19#4524—21no—
20#4524C2212yes—
21C4×C4⋊Q8—31no—
22#1839—31no—
23#1767D311yescharacteristic
24#1271—34no—
25C22×Dic6—21no—
26C22×C3×C8—22no—
27C2×C3×C42C2×C411yescharacteristic
28C2×C3×C42C2×C411yescharacteristic
29C2×C3×C42D411yescharacteristic
30C2×C3×C42D411yescharacteristic
31C2×C3×C42D411yescharacteristic
32C2×C3×C42D411yescharacteristic
33C2×C3×C42C2311yescharacteristic
34C2×C4⋊Dic3—21no—
35C2×C4⋊Dic3—21no—
36C2×C4⋊Dic3—21no—
37C2×C4⋊Dic3C2×C412yes—
38C2×C4⋊Dic3C2×C412yes—
39C2×Dic3⋊C4—22no—
40C2×C4×Dic3—22no—
41C22×C3⋊C8—22no—
42#869—24no—
43C4×Dic6—42no—
44C4×Dic6—42no—
45C3×C4⋊C8—24no—
46C3×#97—42no—
47C3⋊#95—22no—
48C3⋊#95—22no—
49C2×C4⋊Q8—31no—
50C2×C4×Q8—61no—
51C2×C4⋊C8—21no—
52C2×#97—21no—
53#391—61no—
54#389—32no—
55#389—61no—
56#384—31no—
57C4×C4⋊C4—32no—
58C4×C4⋊C4—61no—
59C43C2×D311yescharacteristic
60#336—32no—
61#336—61no—
62#336—61no—
63#336C2×D312yes—
64C22×C3×C4C22⋊C411yescharacteristic
65C22×C3×C4C22⋊C411yescharacteristic
66C22×C3×C4C22⋊C411yescharacteristic
67C22×C3×C4C22⋊C411yescharacteristic
68C22×C3×C4C22×C411yescharacteristic
69C22×C3×C4C2×D411yescharacteristic
70C22×C3×C4C2×D411yescharacteristic
71C22×Dic3—22no—
72C22×Dic3—22no—
73C2×Dic6—24no—
74C2×Dic6—42no—
75C2×Dic6—42no—
76C2×C3×C8—24no—
77C2×C3×C8—42no—
78C2×C3×C8—42no—
79C3×C42—22no—
80C3×C42—22no—
81C3×C42—22no—
82C3×C42—22no—
83C3×C42—41no—
84C3×C42—41no—
85C3×C42D812yes—
86C3×C42SD1612yes—
87C4⋊Dic3—24no—
88C4⋊Dic3—22no—
89C4⋊Dic3—42no—
90C4⋊Dic3—42no—
91C4⋊Dic3—42no—
92C4⋊Dic3M1614yes—
93Dic3⋊C4—42no—
94Dic3⋊C4—42no—
95C4×Dic3—44no—
96C2×C3⋊C8—42no—
97C2×C3⋊C8—42no—
98C2×C3⋊C8—42no—
99C22×Q8—61no—
100C22×C8—22no—
101C22×C8—62no—
102C4⋊Q8—64no—
103C4×Q8—122no—
104C4×Q8—122no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4—32no—
107C2×C4⋊C4—61no—
108C2×C4⋊C4—61no—
109C2×C4⋊C4—61no—
110C2×C4⋊C4—62no—
111C2×C42—62no—
112C2×C42C4×D311yescharacteristic
113C2×C42C4×D311yescharacteristic
114C2×C42D1211yescharacteristic
115C2×C42D1211yescharacteristic
116C2×C42#6711yescharacteristic
117C2×C42#6711yescharacteristic
118C2×C42C22×D311yescharacteristic
119C4⋊C8—24no—
120#97—42no—
121#95—62no—
122#95—62no—
123C23×C3C2×C22⋊C411yescharacteristic
124C2×C3×C4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4#10112yes—
139C2×C3×C4#10212yes—
140C2×C3×C4#10212yes—
141C2×C3×C4#10412yes—
142C2×C3×C4C2×M1612yes—
143C2×C3×C4C2×D811yescharacteristic
144C2×C3×C4C2×SD1611yescharacteristic
145C2×C3×C4#13712yes—
146C2×Dic3—24no—
147C2×Dic3—42no—
148C2×Dic3—42no—
149C2×Dic3—42no—
150C2×Dic3—42no—
151C2×Dic3—42no—
152Dic6—48no—
153C3×C8—44no—
154C3⋊C8—82no—
155C2×Q8—64no—
156C2×Q8—122no—
157C2×Q8—122no—
158C22×C4—62no—
159C22×C4—62no—
160C22×C4(C2×D3)⋊C411yescharacteristic
161C22×C4(C2×D3)⋊C411yescharacteristic
162C22×C4(C2×D3)⋊C411yescharacteristic
163C22×C4(C2×D3)⋊C411yescharacteristic
164C22×C4C2×C4×D311yescharacteristic
165C22×C4C2×D1211yescharacteristic
166C22×C4C2×#6711yescharacteristic
167C2×C8—24no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—122no—
171C2×C8—122no—
172C2×C8—122no—
173C4⋊C4—34no—
174C4⋊C4—64no—
175C4⋊C4—62no—
176C4⋊C4—122no—
177C4⋊C4—122no—
178C4⋊C4—122no—
179C4⋊C4—122no—
180C4⋊C4—122no—
181C42—22no—
182C42—22no—
183C42—22no—
184C42—22no—
185C42—41no—
186C42—41no—
187C42—124no—
188C42#21312yes—
189C42#21512yes—
190C22×C3#40711yescharacteristic
191C22×C3#41011yescharacteristic
192C22×C3C2×#10111yescharacteristic
193C22×C3C2×#10211yescharacteristic
194C22×C3#41911yescharacteristic
195C22×C3C2×#10411yescharacteristic
196C22×C3#42111yescharacteristic
197C3×C4—22no—
198C3×C4—22no—
199C3×C4—22no—
200C3×C4—22no—
201C3×C4—21no—
202C3×C4—21no—
203C3×C4—22no—
204C3×C4—22no—
205Dic3—44no—
206Dic3—82no—
207C23C2×(C2×D3)⋊C411yescharacteristic
208Q8—128no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—64no—
224C2×C4—122no—
225C2×C4—122no—
226C2×C4—122no—
227C2×C4—122no—
228C2×C4—122no—
229C2×C4#80512yes—
230C2×C4#80912yes—
231C2×C4#80912yes—
232C2×C4#82412yes—
233C2×C4D3×M1612yes—
234C2×C4#90912yes—
235C2×C4C2×#21311yescharacteristic
236C2×C4C2×#21511yescharacteristic
237C8—44no—
238C8—242no—
239C2×C3#146711yescharacteristic
240C2×C3#148611yescharacteristic
241C2×C3#149111yescharacteristic
242C2×C3#150211yescharacteristic
243C2×C3#151511yescharacteristic
244C2×C3#153111yescharacteristic
245C2×C3#154611yescharacteristic
246C22C2×#80511yescharacteristic
247C22C2×#80911yescharacteristic
248C22#500611yescharacteristic
249C22#512011yescharacteristic
250C22#512611yescharacteristic
251C22C2×#82411yescharacteristic
252C22#513911yescharacteristic
253C4—21no—
254C4—21no—
255C4—22no—
256C4—22no—
257C4—22no—
258C4—22no—
259C4—22no—
260C4—22no—
261C4—124no—
262C4—242no—
263C3#762511yescharacteristic, complemented, Sylow
264C2#6946911yescharacteristic
265C2#6980111yescharacteristic
266C2#6985711yescharacteristic
267C2#7049611yescharacteristic
268C2#7050611yescharacteristic
269C2#7083011yescharacteristic
270C2#7083711yescharacteristic
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1767C2
socle, prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#1767D3
hypocenterC3#7625
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
263. C3#7625semidirect

Maximal subgroups

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

subgroupindex
C4×#8692
C3×#17672
#693932
#674782
#76253