Finite Group Databasebeta

← #10713875

Subgroups — #10713875

Order 768.

420 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C3×#2567C211yesmaximal, characteristic
2#71637C211yesmaximal, characteristic
3C4×#836C211yesmaximal, characteristic
4#67644C214yesmaximal
5#13302—31nomaximal, Sylow
6C2×C3×C4⋊Q8C411yescharacteristic
7C2×C3×C4×Q8C2211yescharacteristic
8C3×#395—21no—
9C3×#395—21no—
10C3×#391C2212yes—
11C3×#389—21no—
12C3×#389—21no—
13C3×#384C412yes—
14C3×C4×C4⋊C4C411yescharacteristic
15C2×#836C2212yes—
16C2×#836C2212yes—
17#4924—21no—
18C4×#208—21no—
19#4530—21no—
20#4530—21no—
21#2567D311yescharacteristic
22#1801—31no—
23C4×#101—31no—
24#1394—34no—
25C22×C3×Q8—21no—
26C22×C3×Q8C2311yescharacteristic
27C3×C4⋊Q8—41no—
28C3×C4⋊Q8—41no—
29C3×C4×Q8—41no—
30C3×C4×Q8—41no—
31C2×C3×C4⋊C4—21no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4C2×C412yes—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C42C2×C411yescharacteristic
40C2×C3×C42C2×C412yes—
41C2×C3×C42D411yescharacteristic
42C2×C3×C42D411yescharacteristic
43C2×#208—22no—
44C2×#208—22no—
45C3×#95—21no—
46C3×#95—22no—
47C3×#95—22no—
48#836—22no—
49#836—24no—
50#836Q814yescomplemented
51C4⋊(C3⋊C8)—41no—
52C4⋊(C3⋊C8)—41no—
53#803—41no—
54C4×C3⋊C8—41no—
55C2×C4⋊Q8Dic311yescharacteristic
56C2×C4×Q8C2×D311yescharacteristic
57#423—61no—
58C2×#101—32no—
59C2×#101—32no—
60C4×M16—61no—
61#395—21no—
62#395—21no—
63#391C2×D312yes—
64#389—21no—
65#389—21no—
66#384Dic312yes—
67C4×C4⋊C4Dic311yescharacteristic
68#343—61no—
69#343—61no—
70C2×C3×Q8—41no—
71C2×C3×Q8—41no—
72C2×C3×Q8—42no—
73C2×C3×Q8—82no—
74C2×C3×Q8C2×Q812yes—
75C2×C3×Q8#4112yes—
76C22×C3×C4—22no—
77C22×C3×C4—22no—
78C22×C3×C4C22⋊C411yescharacteristic
79C22×C3×C4C22⋊C411yescharacteristic
80C22×C3×C4C22⋊C411yescharacteristic
81C22×C3×C4C22⋊C411yescharacteristic
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C3×C4⋊C4—24no—
86C3×C4⋊C4—41no—
87C3×C4⋊C4—41no—
88C3×C4⋊C4—41no—
89C3×C4⋊C4—42no—
90C3×C4⋊C4—41no—
91C3×C4⋊C4—41no—
92C3×C4⋊C4—82no—
93C3×C4⋊C4—82no—
94C3×C42—21no—
95C3×C42—21no—
96C3×C42—21no—
97C3×C42—21no—
98C3×C42—21no—
99C3×C42—21no—
100C3×C42—42no—
101#208—24no—
102#208—42no—
103#208—42no—
104#208—42no—
105C2×C3⋊C8—42no—
106C2×C3⋊C8—42no—
107C22×Q8—21no—
108C22×Q8C22×D311yescharacteristic
109C2×M16—62no—
110C2×M16—62no—
111C4⋊Q8—41no—
112C4⋊Q8—41no—
113C4×Q8—41no—
114C4×Q8—41no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4C2×Dic312yes—
120C2×C4⋊C4C2×Dic311yescharacteristic
121C2×C4⋊C4#6711yescharacteristic
122C2×C4⋊C4#6711yescharacteristic
123C2×C42C2×Dic311yescharacteristic
124C2×C42C2×Dic312yes—
125C2×C42#6711yescharacteristic
126C2×C42#6711yescharacteristic
127C4⋊C8—121no—
128C4⋊C8—121no—
129#101—34no—
130#101—62no—
131#101—64no—
132#97—121no—
133C4×C8—121no—
134#95—21no—
135#95—22no—
136#95—22no—
137C23×C3C2×C22⋊C411yescharacteristic
138C3×Q8—42no—
139C3×Q8—42no—
140C3×Q8—84no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—22no—
148C2×C3×C4—24no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—41no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4#9911yescharacteristic
161C2×C3×C4#9911yescharacteristic
162C2×C3×C4C4×Q812yes—
163C2×C3×C4C22⋊Q812yes—
164C2×C3×C4#12411yescharacteristic
165C2×C3×C4C4⋊Q811yescharacteristic
166C3⋊C8—44no—
167C3⋊C8—82no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—42no—
171C2×Q8—82no—
172C2×Q8D3×Q812yes—
173C2×Q8#23912yes—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4#21711yescharacteristic
177C22×C4#21711yescharacteristic
178C22×C4#21711yescharacteristic
179C22×C4#21711yescharacteristic
180C22×C4C22×Dic311yescharacteristic
181C22×C4C2×#6711yescharacteristic
182C22×C4C2×#6711yescharacteristic
183M16—64no—
184M16—122no—
185M16—122no—
186M16—122no—
187C2×C8—122no—
188C2×C8—122no—
189C4⋊C4—24no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—42no—
205C22×C3—21no—
206C22×C3—21no—
207C22×C3#39111yescharacteristic
208C22×C3C2×#9911yescharacteristic
209C22×C3#41011yescharacteristic
210C3×C4—21no—
211C3×C4—22no—
212C3×C4—21no—
213C3×C4—22no—
214C3×C4—21no—
215C3×C4—44no—
216C3×C4—41no—
217C3×C4—41no—
218C3×C4—41no—
219C3×C4—82no—
220C3×C4#35612yes—
221C23C2×#21711yescharacteristic
222Q8—42no—
223Q8—42no—
224Q8—84no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4#83411yescharacteristic
245C2×C4#83411yescharacteristic
246C2×C4#94411yescharacteristic
247C2×C4Q8×Dic312yes—
248C2×C4#94612yes—
249C2×C4#94711yescharacteristic
250C8—124no—
251C8—242no—
252C2×C3—21no—
253C2×C3—21no—
254C2×C3#199611yescharacteristic
255C2×C3C2×#35611yescharacteristic
256C2×C3#213411yescharacteristic
257C22—21no—
258C22—21no—
259C22C2×#83411yescharacteristic
260C22#522911yescharacteristic
261C22#524011yescharacteristic
262C4—22no—
263C4—21no—
264C4—21no—
265C4—22no—
266C4—21no—
267C4—44no—
268C4—41no—
269C4—41no—
270C4—41no—
271C4—82no—
272C4#454812yes—
273C3#1330211yescharacteristic, complemented, Sylow
274C2—21no—
275C2—21no—
276C2#7170111yescharacteristic
277C2C2×#454811yescharacteristic
278C2#7186211yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC22×Q8C22×D3
FittingC3×#2567C2
socleC22×C3C2×#99
prime radicalC23×C3C2×C22⋊C4
cosocleC22×C3×Q8C23
hypercenter#2567D3
hypocenterC3#13302
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
50. #836Q8semidirect
273. C3#13302semidirect

Maximal subgroups

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

subgroupindex
C3×#25672
#716372
C4×#8362
#676442
#133023