Finite Group Databasebeta

← #10706115

Subgroups — #10706115

Order 768.

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

normal subgroups only

This group has 368 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
1C3×#2611C211yesmaximal, characteristic
2#71589C211yesmaximal, characteristic
3#71571C211yesmaximal, characteristic
4#67652C212yesmaximal, complemented
5#67650C212yesmaximal, complemented
6#11853—31nomaximal, Sylow
7C2×C3×#125C411yescharacteristic
8C2×C3×#117C2211yescharacteristic
9C3×#400C2211yescharacteristic
10C3×#399C411yescharacteristic
11C3×#398C2211yescharacteristic
12C3×#397C411yescharacteristic
13C3×#389—21no—
14C3×#388—21no—
15C3×#387—21no—
16C3×#385—21no—
17C3×#376C411yescharacteristic
18#5207—21no—
19#4925—21no—
20#4530—21no—
21#4530C2212yes—
22#4528C2212yes—
23#4525—21no—
24#2611D311yescharacteristic
25#1822—31no—
26#1458—31no—
27#1353—32no—
28#1346—32no—
29C23×C3×C4D411yescharacteristic
30C3×#125—41no—
31C3×#125—41no—
32C3×#117—41no—
33C3×#117—41no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C4⋊C4D411yescharacteristic
41C2×C3×C4⋊C4C2311yescharacteristic
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C42—21no—
47C2×C3×C42C2×C411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×#208—22no—
50C3×#95—21no—
51C3×#95—21no—
52C3×#95—21no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95C2×C411yescharacteristic
56C3×#95C2×C411yescharacteristic
57#830—22no—
58#830—41no—
59#830—41no—
60C4⋊(C3⋊C8)—41no—
61C4×C3⋊C8—41no—
62C2×#125Dic311yescharacteristic
63C2×#117C2×D311yescharacteristic
64#431—61no—
65#407—61no—
66#400C2×D311yescharacteristic
67#399Dic311yescharacteristic
68#398C2×D311yescharacteristic
69#397Dic311yescharacteristic
70#389—21no—
71#388—21no—
72#387—21no—
73#385—21no—
74#376Dic311yescharacteristic
75#343—32no—
76#343—61no—
77#324—32no—
78#323—61no—
79C24×C3C22⋊C411yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—22no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—41no—
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C3×C4⋊C4—41no—
95C3×C4⋊C4—41no—
96C3×C4⋊C4—41no—
97C3×C4⋊C4—41no—
98C3×C4⋊C4—42no—
99C3×C4⋊C4—42no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—42no—
103C3×C22⋊C4—41no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—42no—
106C3×C22⋊C4—42no—
107C3×C42—21no—
108C3×C42—21no—
109C3×C42—21no—
110C3×C42—21no—
111C3×C42—42no—
112#208—42no—
113#208—42no—
114C2×C3⋊C8—42no—
115C2×C3⋊C8—42no—
116C23×C4#6711yescharacteristic
117C2×M16—62no—
118#125—41no—
119#125—41no—
120#117—41no—
121#117—41no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4C2×Dic311yescharacteristic
127C2×C4⋊C4C2×Dic311yescharacteristic
128C2×C4⋊C4#6711yescharacteristic
129C2×C4⋊C4C22×D311yescharacteristic
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4C2×Dic311yescharacteristic
133C2×C22⋊C4#6711yescharacteristic
134C2×C42—21no—
135C2×C42C2×Dic311yescharacteristic
136C2×C42#6711yescharacteristic
137C4⋊C8—121no—
138C22⋊C8—62no—
139C22⋊C8—121no—
140C22⋊C8—121no—
141C4×C8—121no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95C2×Dic311yescharacteristic
148#95C2×Dic311yescharacteristic
149C23×C3—21no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—41no—
153C23×C3—41no—
154C23×C3C2×C22⋊C411yescharacteristic
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—22no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—41no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4#9911yescharacteristic
182C2×C3×C4#9911yescharacteristic
183C2×C3×C4#13512yes—
184C3⋊C8—82no—
185C3⋊C8—82no—
186C24#21711yescharacteristic
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—41no—
195C22×C4#21711yescharacteristic
196C22×C4#21711yescharacteristic
197C22×C4#21711yescharacteristic
198C22×C4C22×Dic311yescharacteristic
199C22×C4C2×#6711yescharacteristic
200C22×C4C2×#6711yescharacteristic
201M16—122no—
202M16—122no—
203C2×C8—122no—
204C2×C8—122no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—42no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—42no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—81no—
230C22×C3C2×#9911yescharacteristic
231C22×C3#41611yescharacteristic
232C22×C3#42111yescharacteristic
233C3×C4—22no—
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—42no—
237C3×C4—41no—
238C3×C4—41no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—81no—
242C3×C4—81no—
243C3×C4—81no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23C2×#21711yescharacteristic
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4#83411yescharacteristic
277C2×C4#83411yescharacteristic
278C2×C4#95012yes—
279C8—242no—
280C8—242no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—81no—
284C2×C3#151611yescharacteristic
285C2×C3#212311yescharacteristic
286C2×C3#213211yescharacteristic
287C22—21no—
288C22—21no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22C2×#83411yescharacteristic
295C22#523311yescharacteristic
296C22#523611yescharacteristic
297C4—22no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—81no—
306C4—81no—
307C4—81no—
308C3#1185311yescharacteristic, complemented, Sylow
309C2—21no—
310C2—21no—
311C2—81no—
312C2#7171011yescharacteristic
313C2#7182011yescharacteristic
314C2#7187011yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2611C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#2611D3
hypocenterC3#11853
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
4. #67652C2semidirect
5. #67650C2semidirect
308. C3#11853semidirect

Maximal subgroups

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

subgroupindex
C3×#26112
#715892
#715712
#676522
#676502
#118533