Finite Group Databasebeta

← #10704580

Subgroups — #10704580

Order 768.

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

normal subgroups only

This group has 510 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#82230C211yesmaximal, characteristic
2C3×#1464C211yesmaximal, characteristic
3#71572C211yesmaximal, characteristic
4#67455C214yesmaximal, complemented
5#11510—31nomaximal, Sylow
6C23×Dic6C411yescharacteristic
7C2×C3×#117C2211yescharacteristic
8C2×#926C411yescharacteristic
9C3×#432—21no—
10#4973C412yes—
11#4973C2212yes—
12#4943—21no—
13#4943—21no—
14#4929—21no—
15#4929—21no—
16#4926—21no—
17C3×#324C2212yes—
18C3×#323—21no—
19#4528C2212yes—
20#4525—21no—
21#2310—31no—
22#1464—31no—
23#1464D311yescharacteristic
24#1319—34no—
25C23×C3×C4C2×C411yescharacteristic
26C23×Dic3—21no—
27C22×Dic6—21no—
28C22×Dic6—24no—
29C22×Dic6—42no—
30C22×Dic6—41no—
31C22×Dic6—41no—
32C22×Dic6C2×C412yes—
33C3×#117—41no—
34C3×#117—41no—
35C2×C3×C4⋊C4C2×C411yescharacteristic
36C2×C3×C4⋊C4C2311yescharacteristic
37C2×C3×C22⋊C4D411yescharacteristic
38C2×C3×C22⋊C4D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×#217—21no—
42#926—42no—
43C2×C4⋊Dic3—21no—
44C2×C4×Dic3C2×C412yes—
45C3×C4⋊C8—41no—
46C3×C22⋊C8—22no—
47C3×#97—41no—
48C3⋊#95—22no—
49C3⋊#95—22no—
50C3⋊#95—22no—
51C3⋊#95—22no—
52#830—22no—
53C4⋊(C3⋊C8)—41no—
54#803—41no—
55C23×Q8—31no—
56C2×#117—31no—
57C2×#117C2×D311yescharacteristic
58#432—21no—
59#432—61no—
60#391—32no—
61#391—32no—
62#391—61no—
63#391—61no—
64#381—61no—
65#381—61no—
66#324—32no—
67#324C2×D312yes—
68#323—21no—
69#323—61no—
70C24×C3C22⋊C411yescharacteristic
71C22×C3×C4—22no—
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—22no—
75C22×C3×C4C22⋊C411yescharacteristic
76C22×C3×C4C22⋊C411yescharacteristic
77C22×C3×C4C22⋊C411yescharacteristic
78C22×C3×C4C22×C411yescharacteristic
79C22×C3×C4C2×D411yescharacteristic
80C22×C3×C4C2×D411yescharacteristic
81C22×Dic3—24no—
82C22×Dic3—22no—
83C22×Dic3—22no—
84C22×Dic3—42no—
85C22×Dic3—42no—
86C2×Dic6—24no—
87C2×Dic6—22no—
88C2×Dic6—42no—
89C2×Dic6—48no—
90C2×Dic6—48no—
91C2×Dic6—41no—
92C2×Dic6—41no—
93C2×Dic6—42no—
94C2×Dic6—42no—
95C2×Dic6M1614yes—
96C2×C3×C8—42no—
97C3×C4⋊C4—41no—
98C3×C4⋊C4—41no—
99C3×C22⋊C4—41no—
100C3×C22⋊C4—41no—
101C3×C42—21no—
102C3×C42—21no—
103C3×C42—21no—
104C3×C42—21no—
105#217—42no—
106C4⋊Dic3—42no—
107C4×Dic3—42no—
108C2×C3⋊C8—42no—
109C22×Q8—32no—
110C22×Q8—64no—
111C22×Q8—61no—
112C22×Q8—121no—
113C22×Q8—121no—
114C22×Q8—122no—
115C23×C4—61no—
116C23×C4C4×D311yescharacteristic
117#117—41no—
118#117—41no—
119#117—122no—
120C2×C4⋊C4—61no—
121C2×C4⋊C4C4×D311yescharacteristic
122C2×C4⋊C4C22×D311yescharacteristic
123C2×C22⋊C4—61no—
124C2×C22⋊C4D1211yescharacteristic
125C2×C22⋊C4#6711yescharacteristic
126C2×C42—32no—
127C2×C42D1211yescharacteristic
128C2×C42#6711yescharacteristic
129C4⋊C8—41no—
130C4⋊C8—121no—
131C22⋊C8—22no—
132C22⋊C8—62no—
133#97—41no—
134#97—121no—
135#95—62no—
136#95—62no—
137#95—62no—
138#95—62no—
139C23×C3—21no—
140C23×C3—21no—
141C23×C3—41no—
142C23×C3—41no—
143C23×C3#10012yes—
144C23×C3C2×C22⋊C411yescharacteristic
145C2×C3×C4—24no—
146C2×C3×C4—24no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4#9911yescharacteristic
160C2×C3×C4#9911yescharacteristic
161C2×C3×C4C2×M1612yes—
162C2×Dic3—24no—
163C2×Dic3—48no—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—42no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169Dic6—44no—
170Dic6—44no—
171Dic6—44no—
172Dic6—44no—
173C3×C8—82no—
174C3⋊C8—82no—
175C24(C2×D3)⋊C411yescharacteristic
176C2×Q8—34no—
177C2×Q8—64no—
178C2×Q8—62no—
179C2×Q8—128no—
180C2×Q8—128no—
181C2×Q8—122no—
182C2×Q8—122no—
183C2×Q8—122no—
184C2×Q8—121no—
185C2×Q8—121no—
186C22×C4—22no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—64no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4—122no—
194C22×C4—122no—
195C22×C4(C2×D3)⋊C411yescharacteristic
196C22×C4(C2×D3)⋊C411yescharacteristic
197C22×C4(C2×D3)⋊C411yescharacteristic
198C22×C4C2×C4×D311yescharacteristic
199C22×C4C2×D1211yescharacteristic
200C22×C4C2×#6711yescharacteristic
201C2×C8—42no—
202C2×C8—122no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—122no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—122no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—122no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—22no—
217C22×C3—42no—
218C22×C3—42no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3#40711yescharacteristic
222C22×C3C2×#9911yescharacteristic
223C22×C3C2×#10011yescharacteristic
224C3×C4—21no—
225C3×C4—21no—
226C3×C4—22no—
227C3×C4—41no—
228C3×C4—41no—
229C3×C4—41no—
230C3×C4—41no—
231Dic3—44no—
232Dic3—82no—
233C23—21no—
234C23—21no—
235C23—41no—
236C23—41no—
237C23#82312yes—
238C23C2×(C2×D3)⋊C411yescharacteristic
239Q8—124no—
240Q8—124no—
241Q8—124no—
242Q8—124no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—64no—
258C2×C4—128no—
259C2×C4—122no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4#80611yescharacteristic
265C2×C4#80611yescharacteristic
266C2×C4D3×M1612yes—
267C8—82no—
268C8—242no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—42no—
272C2×C3#146211yescharacteristic
273C2×C3#212911yescharacteristic
274C2×C3#213511yescharacteristic
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—41no—
282C22C2×#80611yescharacteristic
283C22#512611yescharacteristic
284C22C2×#82311yescharacteristic
285C4—21no—
286C4—21no—
287C4—22no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—124no—
293C4—242no—
294C3#1151011yescharacteristic, complemented, Sylow
295C2—21no—
296C2—21no—
297C2—42no—
298C2#6996711yescharacteristic
299C2#7005911yescharacteristic
300C2#7015011yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1464C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1464D3
hypocenterC3#11510
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
4. #67455C2semidirect
294. C3#11510semidirect

Maximal subgroups

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

subgroupindex
#822302
C3×#14642
#715722
#674552
#115103