Finite Group Databasebeta

← #10705376

Subgroups — #10705376

Order 768.

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

normal subgroups only

This group has 468 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#71592C211yesmaximal, characteristic
3#71571C211yesmaximal, characteristic
4#67649C214yesmaximal
5#11694—31nomaximal, Sylow
6C2×C3×C4⋊Q8C411yescharacteristic, complemented
7C2×C3×C22⋊Q8—21no—
8C2×C3×#117C2211yescharacteristic
9C2×#926—21no—
10C3×#398—21no—
11C3×#398C412yescomplemented
12C3×#397—21no—
13C3×#397C2212yes—
14C3×#391—21no—
15C3×#382C411yescharacteristic, complemented
16#4925—21no—
17#4528—21no—
18#4526—21no—
19#4526C2212yes—
20#4525C2212yes—
21#2621D311yescharacteristic
22#1740—31no—
23#1458—31no—
24#1344—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⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4D411yescharacteristic
41C2×C3×C22⋊C4C2311yescharacteristic
42C2×C3×C42C2×C411yescharacteristic
43C2×C3×C42D411yescharacteristic
44C2×#217—22no—
45#926—24no—
46#926—42no—
47C2×C4⋊Dic3—41no—
48C2×C4×Dic3—41no—
49C3×#95—21no—
50C3×#95—22no—
51C3×#95—22no—
52C3×#95C2×C412yes—
53#830—22no—
54C4⋊(C3⋊C8)—41no—
55C4×C3⋊C8—41no—
56C2×C4⋊Q8Dic311yescharacteristic, complemented
57C2×C22⋊Q8—21no—
58C2×#117—61no—
59C2×#117C2×D311yescharacteristic
60#431—61no—
61#398—21no—
62#398Dic312yescomplemented
63#397—21no—
64#397C2×D312yes—
65#391—21no—
66#382Dic311yescharacteristic, complemented
67#342—32no—
68#342—61no—
69#324—61no—
70#323—32no—
71C24×C3C2×D411yescharacteristic
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×C4C22⋊C411yescharacteristic
85C22×C3×C4D812yes—
86C22×C3×C4SD1612yes—
87C22×C3×C4C22×C411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C22×Dic3—42no—
90C3×C4⋊C4—41no—
91C3×C4⋊C4—41no—
92C3×C4⋊C4—42no—
93C3×C4⋊C4—42no—
94C3×C4⋊C4—42no—
95C3×C4⋊C4—44no—
96C3×C22⋊C4—41no—
97C3×C22⋊C4—41no—
98C3×C22⋊C4—42no—
99C3×C22⋊C4—42no—
100C3×C42—21no—
101C3×C42—21no—
102C3×C42—21no—
103C3×C42—21no—
104#217—24no—
105#217—42no—
106C4⋊Dic3—44no—
107C4×Dic3—44no—
108C2×C3⋊C8—42no—
109C22×Q8—21no—
110C23×C4#6711yescharacteristic
111C4⋊Q8—41no—
112C4⋊Q8—41no—
113C22⋊Q8—42no—
114C22⋊Q8—42no—
115#117—41no—
116#117—41no—
117#117—64no—
118#117—122no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—121no—
123C2×C4⋊C4C2×Dic312yes—
124C2×C4⋊C4C2×Dic311yescharacteristic
125C2×C4⋊C4#6711yescharacteristic
126C2×C22⋊C4—22no—
127C2×C22⋊C4—62no—
128C2×C22⋊C4#6711yescharacteristic
129C2×C22⋊C4C22×D311yescharacteristic
130C2×C42—121no—
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×C3C2×C22⋊C411yescharacteristic
145C23×C3C2×D811yescharacteristic
146C23×C3C2×SD1611yescharacteristic
147C3×Q8—82no—
148C3×Q8—82no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—41no—
160C2×C3×C4—41no—
161C2×C3×C4—41no—
162C2×C3×C4—41no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4#9911yescharacteristic
170C2×C3×C4#9911yescharacteristic
171C2×C3×C4#10212yes—
172C2×C3×C4#10212yes—
173C2×C3×C4#10412yes—
174C2×Dic3—44no—
175C2×Dic3—82no—
176C2×Dic3—82no—
177C3⋊C8—82no—
178C24C2×#6711yescharacteristic
179C2×Q8—42no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—42no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—122no—
189C22×C4#21312yes—
190C22×C4#21412yes—
191C22×C4#21711yescharacteristic
192C22×C4#21711yescharacteristic
193C22×C4#21711yescharacteristic
194C22×C4#21711yescharacteristic
195C22×C4C22×Dic311yescharacteristic
196C22×C4C2×#6711yescharacteristic
197C2×C8—122no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—44no—
204C4⋊C4—124no—
205C22⋊C4—41no—
206C22⋊C4—41no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—64no—
210C22⋊C4—122no—
211C42—21no—
212C42—21no—
213C42—21no—
214C42—21no—
215C42—124no—
216C22×C3—21no—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—21no—
220C22×C3—41no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—81no—
225C22×C3C2×#9911yescharacteristic
226C22×C3C2×#10211yescharacteristic
227C22×C3C2×#10411yescharacteristic
228C3×C4—22no—
229C3×C4—41no—
230C3×C4—41no—
231C3×C4—41no—
232C3×C4—41no—
233C3×C4—82no—
234C3×C4—82no—
235C3×C4#32712yes—
236C3×C4#32712yes—
237Dic3—84no—
238C23—21no—
239C23—21no—
240C23—41no—
241C23—41no—
242C23C2×#21311yescharacteristic
243C23C2×#21411yescharacteristic
244C23C2×#21711yescharacteristic
245Q8—82no—
246Q8—82no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—124no—
268C2×C4—242no—
269C2×C4—242no—
270C2×C4#83212yes—
271C2×C4#83212yes—
272C2×C4#83411yescharacteristic
273C2×C4#83411yescharacteristic
274C2×C4#83712yes—
275C8—242no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3C2×#32711yescharacteristic
281C2×C3#212311yescharacteristic
282C2×C3#212911yescharacteristic
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—81no—
292C22C2×#83211yescharacteristic
293C22C2×#83411yescharacteristic
294C22C2×#83711yescharacteristic
295C4—22no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—82no—
301C4—82no—
302C4—244no—
303C4#453712yes—
304C4#453712yes—
305C3#1169411yescharacteristic, complemented, Sylow
306C2—21no—
307C2—21no—
308C2—41no—
309C2—41no—
310C2C2×#453711yescharacteristic
311C2#7187011yescharacteristic
312C2#7187111yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#2621C2
socleC22×C3C2×#99
prime radicalC24×C3C2×D4
cosocleC2×C3×C22⋊C4C23
hypercenter#2621D3
hypocenterC3#11694
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
6. C2×C3×C4⋊Q8C4semidirect
11. C3×#398C4semidirect
15. C3×#382C4semidirect
56. C2×C4⋊Q8Dic3semidirect
62. #398Dic3semidirect
66. #382Dic3semidirect
305. C3#11694semidirect

Maximal subgroups

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

subgroupindex
C3×#26212
#715922
#715712
#676492
#116943