Finite Group Databasebeta

← #10704826

Subgroups — #10704826

Order 768.

462 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2548C211yesmaximal, characteristic
2#71625C211yesmaximal, characteristic, complemented
3#71624C211yesmaximal, characteristic, complemented
4#67640C214yesmaximal, complemented
5#11567—31nomaximal, Sylow
6C2×C3×#127C411yescharacteristic
7C2×C3×#124C412yes—
8C2×C3×C4×Q8C2211yescharacteristic
9C3×#394—21no—
10C3×#394—21no—
11C3×#390—21no—
12C3×#390—21no—
13C3×#390C2212yes—
14C3×C4×C4⋊C4C411yescharacteristic
15#4926—21no—
16#4925—21no—
17#4528—21no—
18#4528—21no—
19#4528C2212yes—
20#4528C2212yes—
21#2548D311yescharacteristic
22#1467—31no—
23#1461—31no—
24#1318—34no—
25C22×C3×Q8C2×C411yescharacteristic
26C22×C3×D4—21no—
27C22×C3×D4—21no—
28C22×C3×D4C2×C412yes—
29C3×#127—41no—
30C3×#127—41no—
31C3×#124—44no—
32C3×C4×Q8—41no—
33C3×C4×Q8—41no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C4⋊C4C2311yescharacteristic
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C42C2×C411yescharacteristic
43C2×C3×C42C2×C412yes—
44C2×C3×C42D411yescharacteristic
45C2×C3×C42D411yescharacteristic
46C3×#95—21no—
47#830—22no—
48#830—22no—
49C4⋊(C3⋊C8)—41no—
50C4⋊(C3⋊C8)—41no—
51#803—41no—
52C4×C3⋊C8—41no—
53C2×#127Dic311yescharacteristic
54C2×#124Dic312yes—
55C2×C4×Q8C2×D311yescharacteristic
56#432—61no—
57#431—61no—
58#394—21no—
59#394—21no—
60#390—21no—
61#390—21no—
62#390C2×D312yes—
63C4×C4⋊C4Dic311yescharacteristic
64#324—32no—
65#324—32no—
66#324—61no—
67#324—61no—
68C24×C3—22no—
69C2×C3×Q8—22no—
70C2×C3×Q8—22no—
71C2×C3×Q8—41no—
72C2×C3×D4—24no—
73C2×C3×D4—42no—
74C2×C3×D4—42no—
75C2×C3×D4—42no—
76C2×C3×D4—42no—
77C2×C3×D4—42no—
78C2×C3×D4—41no—
79C2×C3×D4—41no—
80C2×C3×D4—82no—
81C22×C3×C4—22no—
82C22×C3×C4C22⋊C411yescharacteristic
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C22⋊C411yescharacteristic
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22×C411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C3×C4⋊C4—41no—
90C3×C4⋊C4—41no—
91C3×C4⋊C4—41no—
92C3×C4⋊C4—42no—
93C3×C22⋊C4—44no—
94C3×C22⋊C4—44no—
95C3×C22⋊C4—82no—
96C3×C22⋊C4—82no—
97C3×C42—21no—
98C3×C42—21no—
99C3×C42—21no—
100C3×C42—21no—
101C3×C42—21no—
102C3×C42—21no—
103C3×C42—42no—
104C2×C3⋊C8—42no—
105C2×C3⋊C8—42no—
106C22×Q8C2×Dic311yescharacteristic
107C22×D4—21no—
108C22×D4—21no—
109C22×D4C2×Dic312yes—
110#127—41no—
111#127—41no—
112#124—44no—
113C4×Q8—41no—
114C4×Q8—41no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4#6711yescharacteristic
117C2×C4⋊C4#6711yescharacteristic
118C2×C4⋊C4C22×D311yescharacteristic
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C42C2×Dic311yescharacteristic
124C2×C42C2×Dic312yes—
125C2×C42#6711yescharacteristic
126C2×C42#6711yescharacteristic
127C4⋊C8—121no—
128C4⋊C8—121no—
129C22⋊C8—62no—
130C22⋊C8—62no—
131#97—121no—
132C4×C8—121no—
133#95—21no—
134C23×C3—42no—
135C23×C3—42no—
136C23×C3—42no—
137C23×C3—42no—
138C23×C3—42no—
139C23×C3—42no—
140C23×C3—42no—
141C23×C3C2×C22⋊C411yescharacteristic
142C3×Q8—44no—
143C3×D4—84no—
144C3×D4—82no—
145C3×D4—84no—
146C3×D4—82no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—22no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
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—42no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4#9911yescharacteristic
165C2×C3×C4#9911yescharacteristic
166C2×C3×C4#10012yes—
167C2×C3×C4#13112yes—
168C3⋊C8—82no—
169C3⋊C8—82no—
170C24—22no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—41no—
174C2×D4—24no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—82no—
183C22×C4—22no—
184C22×C4#21711yescharacteristic
185C22×C4#21711yescharacteristic
186C22×C4#21711yescharacteristic
187C22×C4#21711yescharacteristic
188C22×C4C22×Dic311yescharacteristic
189C22×C4C2×#6711yescharacteristic
190C22×C4C2×#6711yescharacteristic
191C2×C8—122no—
192C2×C8—122no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—42no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C22⋊C4—82no—
200C22⋊C4—82no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—42no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—82no—
211C22×C3—82no—
212C22×C3—82no—
213C22×C3—82no—
214C22×C3—82no—
215C22×C3—82no—
216C22×C3—82no—
217C22×C3#40811yescharacteristic
218C22×C3C2×#9911yescharacteristic
219C22×C3C2×#10011yescharacteristic
220C3×C4—21no—
221C3×C4—22no—
222C3×C4—22no—
223C3×C4—41no—
224C3×C4—41no—
225C3×C4—41no—
226C3×C4—41no—
227C3×C4—82no—
228C3×C4#35512yes—
229C23—42no—
230C23—42no—
231C23—42no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23C2×#21711yescharacteristic
237Q8—44no—
238D4—84no—
239D4—82no—
240D4—84no—
241D4—82no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
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—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4#83312yes—
260C2×C4#83411yescharacteristic
261C2×C4#83411yescharacteristic
262C2×C4#94812yes—
263C8—242no—
264C8—242no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—162no—
268C2×C3#147111yescharacteristic
269C2×C3#212811yescharacteristic
270C2×C3C2×#35511yescharacteristic
271C22—21no—
272C22—21no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22C2×#83311yescharacteristic
281C22C2×#83411yescharacteristic
282C22#523411yescharacteristic
283C4—21no—
284C4—22no—
285C4—22no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—82no—
291C4#454212yes—
292C3#1156711yescharacteristic, complemented, Sylow
293C2—21no—
294C2—21no—
295C2—162no—
296C2#7167611yescharacteristic
297C2C2×#454211yescharacteristic
298C2#7186111yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2548C2
socleC22×C3C2×#99
prime radicalC2×C3×#127C4
cosocleC2×C3×C4⋊C4C23
hypercenter#2548D3
hypocenterC3#11567
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
2. #71625C2semidirect
3. #71624C2semidirect
4. #67640C2semidirect
292. C3#11567semidirect

Maximal subgroups

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

subgroupindex
C3×#25482
#716252
#716242
#676402
#115673