Finite Group Databasebeta

← #10705402

Subgroups — #10705402

Order 768.

452 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2621C211yesmaximal, characteristic
2#71587C211yesmaximal, characteristic
3#71571C211yesmaximal, characteristic
4#67650C214yesmaximal, complemented
5#11700—31nomaximal, Sylow
6C2×C3×C4⋊Q8C411yescharacteristic
7C2×C3×C22⋊Q8—21no—
8C2×C3×#117C2211yescharacteristic
9C22×#208—21no—
10C3×#398—21no—
11C3×#398C2212yes—
12C3×#397—21no—
13C3×#397C412yes—
14C3×#391—21no—
15C3×#382C411yescharacteristic
16#4925—21no—
17#4530—21no—
18#4530C2212yes—
19#4528C2212yes—
20#4525—21no—
21#2621D311yescharacteristic
22#1742—31no—
23#1458—31no—
24#1346—34no—
25C22×C3×Q8—21no—
26C23×C3×C4D411yescharacteristic
27C3×C4⋊Q8—41no—
28C3×C4⋊Q8—41no—
29C3×C22⋊Q8—42no—
30C3×C22⋊Q8—42no—
31C3×#117—41no—
32C3×#117—41no—
33C2×C3×C4⋊C4—21no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4C2×C412yes—
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C4⋊C4C2311yescharacteristic
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4C2×C411yescharacteristic
41C2×C3×C22⋊C4D411yescharacteristic
42C2×C3×C42C2×C411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×#208—24no—
45C2×#208—22no—
46C2×#208—41no—
47C2×#208—42no—
48C22×C3⋊C8—41no—
49C3×#95—21no—
50C3×#95—22no—
51C3×#95—22no—
52C3×#95C2×C412yes—
53#830—22no—
54C4⋊(C3⋊C8)—41no—
55C4×C3⋊C8—41no—
56C22×M16—61no—
57C2×C4⋊Q8Dic311yescharacteristic
58C2×C22⋊Q8—21no—
59C2×#117C2×D311yescharacteristic
60#431—61no—
61#398—21no—
62#398C2×D312yes—
63#397—21no—
64#397Dic312yes—
65#391—21no—
66#382Dic311yescharacteristic
67#343—32no—
68#343—61no—
69#324—32no—
70#323—61no—
71C24×C3C22⋊C411yescharacteristic
72C2×C3×Q8—41no—
73C2×C3×Q8—41no—
74C2×C3×Q8—42no—
75C2×C3×Q8—42no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4C22⋊C411yescharacteristic
82C22×C3×C4C22⋊C411yescharacteristic
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4D812yes—
85C22×C3×C4SD1612yes—
86C22×C3×C4C22×C411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C3×C4⋊C4—41no—
90C3×C4⋊C4—41no—
91C3×C4⋊C4—42no—
92C3×C4⋊C4—42no—
93C3×C4⋊C4—42no—
94C3×C4⋊C4—42no—
95C3×C4⋊C4—42no—
96C3×C22⋊C4—41no—
97C3×C22⋊C4—41no—
98C3×C22⋊C4—44no—
99C3×C42—21no—
100C3×C42—21no—
101C3×C42—21no—
102C3×C42—21no—
103#208—24no—
104#208—44no—
105#208—42no—
106C2×C3⋊C8—42no—
107C2×C3⋊C8—44no—
108C2×C3⋊C8—42no—
109C22×Q8—21no—
110C23×C4#6711yescharacteristic
111C2×M16—64no—
112C2×M16—62no—
113C2×M16—121no—
114C2×M16—122no—
115C22×C8—121no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118C22⋊Q8—42no—
119C22⋊Q8—42no—
120#117—41no—
121#117—41no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4C2×Dic312yes—
126C2×C4⋊C4#6711yescharacteristic
127C2×C4⋊C4C22×D311yescharacteristic
128C2×C22⋊C4—22no—
129C2×C22⋊C4C2×Dic311yescharacteristic
130C2×C22⋊C4#6711yescharacteristic
131C2×C42C2×Dic311yescharacteristic
132C2×C42#6711yescharacteristic
133C4⋊C8—121no—
134C22⋊C8—62no—
135C4×C8—121no—
136#95—21no—
137#95—22no—
138#95—22no—
139#95C2×Dic312yes—
140C23×C3—21no—
141C23×C3—21no—
142C23×C3—41no—
143C23×C3—41no—
144C23×C3#10212yes—
145C23×C3C2×C22⋊C411yescharacteristic
146C3×Q8—82no—
147C3×Q8—82no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
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—41no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4#9911yescharacteristic
169C2×C3×C4#9911yescharacteristic
170C2×C3×C4#10212yes—
171C2×C3×C4#10412yes—
172C2×C3×C4C2×D811yescharacteristic
173C2×C3×C4C2×SD1611yescharacteristic
174C3⋊C8—44no—
175C3⋊C8—82no—
176C24#21711yescharacteristic
177C2×Q8—42no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—42no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4#21312yes—
187C22×C4#21412yes—
188C22×C4#21711yescharacteristic
189C22×C4#21711yescharacteristic
190C22×C4#21711yescharacteristic
191C22×C4C22×Dic311yescharacteristic
192C22×C4C2×#6711yescharacteristic
193C22×C4C2×#6711yescharacteristic
194M16—64no—
195M16—124no—
196M16—122no—
197C2×C8—122no—
198C2×C8—124no—
199C2×C8—122no—
200C4⋊C4—41no—
201C4⋊C4—41no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—44no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—21no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—22no—
217C22×C3—42no—
218C22×C3—41no—
219C22×C3—41no—
220C22×C3—81no—
221C22×C3C2×#9911yescharacteristic
222C22×C3C2×#10211yescharacteristic
223C22×C3C2×#10411yescharacteristic
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—41no—
227C3×C4—41no—
228C3×C4—41no—
229C3×C4—41no—
230C3×C4—82no—
231C3×C4—82no—
232C3×C4#32712yes—
233C3×C4#32712yes—
234C23—21no—
235C23—21no—
236C23—41no—
237C23—41no—
238C23#83212yes—
239C23C2×#21711yescharacteristic
240Q8—82no—
241Q8—82no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4#83212yes—
263C2×C4#83411yescharacteristic
264C2×C4#83411yescharacteristic
265C2×C4#83712yes—
266C2×C4C2×#21311yescharacteristic
267C2×C4C2×#21411yescharacteristic
268C8—124no—
269C8—242no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—42no—
273C2×C3C2×#32711yescharacteristic
274C2×C3#212311yescharacteristic
275C2×C3#213511yescharacteristic
276C22—21no—
277C22—21no—
278C22—22no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22C2×#83211yescharacteristic
284C22C2×#83411yescharacteristic
285C22C2×#83711yescharacteristic
286C4—21no—
287C4—21no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—82no—
293C4—82no—
294C4#453712yes—
295C4#453712yes—
296C3#1170011yescharacteristic, complemented, Sylow
297C2—21no—
298C2—21no—
299C2—42no—
300C2C2×#453711yescharacteristic
301C2#7187011yescharacteristic
302C2#7187311yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2621C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#2621D3
hypocenterC3#11700
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2621

Complements

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

NH ≅ G/Ntype
4. #67650C2semidirect
296. C3#11700semidirect

Maximal subgroups

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

subgroupindex
C3×#26212
#715872
#715712
#676502
#117003