Finite Group Databasebeta

← #10704827

Subgroups — #10704827

Order 768.

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

normal subgroups only

This group has 462 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#82476C211yesmaximal, characteristic
2C3×#1467C211yesmaximal, characteristic, complemented
3#71624C211yesmaximal, characteristic, complemented
4#67454C214yesmaximal, complemented
5#11567—31nomaximal, Sylow
6C2×C3×C4×Q8C2211yescharacteristic
7C2×#947C412yes—
8C2×#874C411yescharacteristic
9C3×#432—21no—
10#4991—21no—
11#4991—21no—
12#4988C2212yes—
13#4939—21no—
14#4939—21no—
15C4×C4⋊Dic3C411yescharacteristic
16#4925—21no—
17C3×#324—21no—
18C3×#324C2212yes—
19#4528—21no—
20#4528C2212yes—
21#2548—31no—
22#1467D311yescharacteristic, complemented
23#1461—31no—
24#1318—34no—
25C22×C3×Q8C2×C411yescharacteristic
26C22×D12—21no—
27C22×D12—21no—
28C22×D12C2×C412yes—
29C3×C4×Q8—41no—
30C3×C4×Q8—41no—
31C2×C3×C4⋊C4D411yescharacteristic
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C4⋊C4C2311yescharacteristic
34C2×C3×C42C2×C411yescharacteristic
35C2×C3×C42D411yescharacteristic
36C2×C3×C42D411yescharacteristic
37#947—44no—
38C2×(C2×D3)⋊C4—22no—
39C2×(C2×D3)⋊C4—22no—
40C2×(C2×D3)⋊C4—22no—
41C2×(C2×D3)⋊C4—22no—
42C2×C4⋊Dic3—21no—
43C2×C4×Dic3C2×C412yes—
44#874—41no—
45#874—41no—
46C3×C4⋊C8—41no—
47C3×C22⋊C8—22no—
48C3×#97—41no—
49C3⋊#95—21no—
50#830—22no—
51C4⋊(C3⋊C8)—41no—
52C4×C3⋊C8—41no—
53C2×#127—31no—
54C2×#124—32no—
55C2×C4×Q8C2×D311yescharacteristic
56#432—21no—
57#431—61no—
58#394—61no—
59#394—61no—
60#390—32no—
61#390—61no—
62#390—61no—
63C4×C4⋊C4—31no—
64#324—21no—
65#324—32no—
66#324—61no—
67#324C2×D312yes—
68C23×D3—22no—
69C2×C3×Q8—22no—
70C2×C3×Q8—22no—
71C2×C3×Q8—41no—
72C22×C3×C4C22⋊C411yescharacteristic
73C22×C3×C4C22⋊C411yescharacteristic
74C22×C3×C4C22⋊C411yescharacteristic
75C22×C3×C4C22⋊C411yescharacteristic
76C22×C3×C4C22×C411yescharacteristic
77C22×C3×C4C2×D411yescharacteristic
78C22×C3×C4C2×D411yescharacteristic
79C22×Dic3—22no—
80C2×D12—24no—
81C2×D12—42no—
82C2×D12—42no—
83C2×D12—42no—
84C2×D12—42no—
85C2×D12—42no—
86C2×D12—41no—
87C2×D12—41no—
88C2×D12—82no—
89C2×C3×C8—42no—
90C3×C4⋊C4—41no—
91C3×C4⋊C4—41no—
92C3×C4⋊C4—41no—
93C3×C42—21no—
94C3×C42—21no—
95C3×C42—21no—
96C3×C42—21no—
97C3×C42—21no—
98C3×C42—21no—
99(C2×D3)⋊C4—44no—
100(C2×D3)⋊C4—44no—
101(C2×D3)⋊C4—82no—
102(C2×D3)⋊C4—82no—
103C4⋊Dic3—42no—
104C4×Dic3—42no—
105C2×C3⋊C8—42no—
106C22×Q8C4×D311yescharacteristic
107C22×D4—32no—
108C22×D4—61no—
109C22×D4—61no—
110#127—121no—
111#127—121no—
112#124—124no—
113C4×Q8—41no—
114C4×Q8—41no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4D1211yescharacteristic
117C2×C4⋊C4#6711yescharacteristic
118C2×C4⋊C4C22×D311yescharacteristic
119C2×C22⋊C4—62no—
120C2×C22⋊C4—62no—
121C2×C22⋊C4—62no—
122C2×C22⋊C4—62no—
123C2×C42—32no—
124C2×C42C4×D311yescharacteristic
125C2×C42D1211yescharacteristic
126C2×C42#6711yescharacteristic
127C4⋊C8—41no—
128C4⋊C8—121no—
129C22⋊C8—22no—
130C22⋊C8—62no—
131#97—41no—
132C4×C8—121no—
133#95—61no—
134C23×C3C2×C22⋊C411yescharacteristic
135C22×D3—42no—
136C22×D3—42no—
137C22×D3—42no—
138C22×D3—42no—
139C22×D3—42no—
140C22×D3—42no—
141C22×D3—42no—
142C3×Q8—44no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—22no—
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—41no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4#9911yescharacteristic
158C2×C3×C4#9911yescharacteristic
159C2×C3×C4#10012yes—
160C2×C3×C4#13112yes—
161C2×Dic3—42no—
162C2×Dic3—42no—
163C2×Dic3—42no—
164D12—84no—
165D12—82no—
166D12—84no—
167D12—82no—
168C3×C8—82no—
169C3⋊C8—82no—
170C24—62no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8—41no—
174C2×D4—64no—
175C2×D4—122no—
176C2×D4—122no—
177C2×D4—122no—
178C2×D4—122no—
179C2×D4—122no—
180C2×D4—121no—
181C2×D4—121no—
182C2×D4—242no—
183C22×C4—62no—
184C22×C4(C2×D3)⋊C411yescharacteristic
185C22×C4(C2×D3)⋊C411yescharacteristic
186C22×C4(C2×D3)⋊C411yescharacteristic
187C22×C4(C2×D3)⋊C411yescharacteristic
188C22×C4C2×C4×D311yescharacteristic
189C22×C4C2×D1211yescharacteristic
190C22×C4C2×#6711yescharacteristic
191C2×C8—42no—
192C2×C8—122no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—122no—
197C22⋊C4—124no—
198C22⋊C4—124no—
199C22⋊C4—242no—
200C22⋊C4—242no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—122no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3#40811yescharacteristic
211C22×C3C2×#9911yescharacteristic
212C22×C3C2×#10011yescharacteristic
213C2×D3—82no—
214C2×D3—82no—
215C2×D3—82no—
216C2×D3—82no—
217C2×D3—82no—
218C2×D3—82no—
219C2×D3—82no—
220C3×C4—21no—
221C3×C4—22no—
222C3×C4—22no—
223C3×C4—41no—
224C3×C4—41no—
225C3×C4—41no—
226C3×C4—41no—
227C3×C4#35512yes—
228Dic3—82no—
229C23—122no—
230C23—122no—
231C23—122no—
232C23—122no—
233C23—122no—
234C23—122no—
235C23—122no—
236C23C2×(C2×D3)⋊C411yescharacteristic
237Q8—44no—
238D4—244no—
239D4—242no—
240D4—244no—
241D4—242no—
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—122no—
257C2×C4—122no—
258C2×C4—122no—
259C2×C4#80611yescharacteristic
260C2×C4#80611yescharacteristic
261C2×C4#82312yes—
262C2×C4#90712yes—
263C8—82no—
264C8—242no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3#147111yescharacteristic
268C2×C3#212811yescharacteristic
269C2×C3C2×#35511yescharacteristic
270D3—162no—
271C22—21no—
272C22—21no—
273C22—242no—
274C22—242no—
275C22—242no—
276C22—242no—
277C22—242no—
278C22—242no—
279C22—242no—
280C22C2×#80611yescharacteristic
281C22#512911yescharacteristic
282C22C2×#82311yescharacteristic
283C4—22no—
284C4—22no—
285C4—21no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—242no—
291C4#447712yes—
292C3#1156711yescharacteristic, complemented, Sylow
293C2—21no—
294C2—21no—
295C2—482no—
296C2#6997011yescharacteristic
297C2#7006111yescharacteristic
298C2C2×#447711yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1467C2
socleC22×C3C2×#99
prime radicalC2×#874C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1467D3
hypocenterC3#11567
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. C3×#1467C2semidirect
3. #71624C2semidirect
4. #67454C2semidirect
22. #1467D3semidirect
292. C3#11567semidirect

Maximal subgroups

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

subgroupindex
#824762
C3×#14672
#716242
#674542
#115673