Finite Group Databasebeta

← #10706113

Subgroups — #10706113

Order 768.

368 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 79 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#82261C211yesmaximal, characteristic
2C3×#1822C211yesmaximal, characteristic
3#71571C211yesmaximal, characteristic
4#67496C212yesmaximal, complemented
5#67495C212yesmaximal, complemented
6#11853—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#870C411yescharacteristic
9C3×#407—21no—
10#5210—21no—
11#4986C2211yescharacteristic
12#4981C2211yescharacteristic
13#4976—21no—
14#4953C411yescharacteristic
15#4949C411yescharacteristic
16#4945—21no—
17#4933—21no—
18#4932C411yescharacteristic
19#4925—21no—
20C3×#343—21no—
21C3×#343C2212yes—
22#4528C2212yes—
23#4525—21no—
24#2611—31no—
25#1822D311yescharacteristic
26#1458—31no—
27#1353—32no—
28#1346—32no—
29C23×C3×C4D411yescharacteristic
30C2×C3×M16—22no—
31C3×#117—41no—
32C3×#117—41no—
33C2×C3×C4⋊C4D411yescharacteristic
34C2×C3×C4⋊C4C2311yescharacteristic
35C2×C3×C22⋊C4C2×C411yescharacteristic
36C2×C3×C22⋊C4D411yescharacteristic
37C2×C3×C42C2×C411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×#217—21no—
40C2×#217—21no—
41C2×C4⋊Dic3C2×C411yescharacteristic
42C2×C4⋊Dic3C2×C411yescharacteristic
43C2×Dic3⋊C4—21no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×C4×Dic3—21no—
48#870—41no—
49#870—41no—
50C3×C22⋊C8—41no—
51C3×C22⋊C8—41no—
52C3⋊#95—21no—
53C3⋊#95—21no—
54C3⋊#95—21no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95C2×C411yescharacteristic
58C3⋊#95C2×C411yescharacteristic
59#830—22no—
60C4⋊(C3⋊C8)—41no—
61C4×C3⋊C8—41no—
62C2×#125—31no—
63C2×#117C2×D311yescharacteristic
64#431—61no—
65#407—21no—
66#400—31no—
67#399—31no—
68#398—31no—
69#397—31no—
70#389—61no—
71#388—61no—
72#387—61no—
73#385—61no—
74#376—31no—
75#343—21no—
76#343C2×D312yes—
77#324—32no—
78#323—61no—
79C24×C3C22⋊C411yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—22no—
83C22×C3×C4—41no—
84C22×C3×C4C22⋊C411yescharacteristic
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22×C411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×Dic3—21no—
91C22×Dic3—21no—
92C22×Dic3—21no—
93C22×Dic3—21no—
94C3×M16—42no—
95C3×M16—42no—
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—
106#217—42no—
107C4⋊Dic3—41no—
108C4⋊Dic3—41no—
109Dic3⋊C4—42no—
110Dic3⋊C4—42no—
111Dic3⋊C4—41no—
112Dic3⋊C4—41no—
113Dic3⋊C4—42no—
114C4×Dic3—42no—
115C2×C3⋊C8—42no—
116C23×C4#6711yescharacteristic
117C2×M16—22no—
118#125—121no—
119#125—121no—
120#117—41no—
121#117—41no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—61no—
125C2×C4⋊C4—61no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4#6711yescharacteristic
129C2×C4⋊C4C22×D311yescharacteristic
130C2×C22⋊C4—61no—
131C2×C22⋊C4—61no—
132C2×C22⋊C4C4×D311yescharacteristic
133C2×C22⋊C4D1211yescharacteristic
134C2×C42—61no—
135C2×C42C4×D311yescharacteristic
136C2×C42D1211yescharacteristic
137C4⋊C8—121no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—62no—
141C4×C8—121no—
142#95—31no—
143#95—31no—
144#95—61no—
145#95—61no—
146#95—61no—
147#95—61no—
148#95—61no—
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—21no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—41no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4#9911yescharacteristic
170C2×C3×C4#9911yescharacteristic
171C2×C3×C4#13512yes—
172C2×Dic3—22no—
173C2×Dic3—41no—
174C2×Dic3—41no—
175C2×Dic3—41no—
176C2×Dic3—41no—
177C2×Dic3—41no—
178C2×Dic3—41no—
179C2×Dic3—41no—
180C2×Dic3—41no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C3×C8—82no—
185C3⋊C8—82no—
186C24(C2×D3)⋊C411yescharacteristic
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—41no—
191C22×C4—61no—
192C22×C4—61no—
193C22×C4—61no—
194C22×C4—61no—
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
201M16—42no—
202M16—42no—
203C2×C8—42no—
204C2×C8—122no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—121no—
208C4⋊C4—121no—
209C4⋊C4—122no—
210C4⋊C4—122no—
211C4⋊C4—121no—
212C4⋊C4—121no—
213C4⋊C4—122no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—122no—
217C22⋊C4—122no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—122no—
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—41no—
237C3×C4—41no—
238C3×C4—41no—
239C3×C4—41no—
240Dic3—42no—
241Dic3—81no—
242Dic3—81no—
243Dic3—81no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—41no—
248C23—41no—
249C23C2×(C2×D3)⋊C411yescharacteristic
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—62no—
265C2×C4—121no—
266C2×C4—121no—
267C2×C4—121no—
268C2×C4—121no—
269C2×C4—121no—
270C2×C4—121no—
271C2×C4—121no—
272C2×C4—121no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4#80611yescharacteristic
277C2×C4#80611yescharacteristic
278C2×C4#95012yes—
279C8—82no—
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×#80611yescharacteristic
295C22#500211yescharacteristic
296C22#500511yescharacteristic
297C4—22no—
298C4—21no—
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—122no—
305C4—241no—
306C4—241no—
307C4—241no—
308C3#1185311yescharacteristic, complemented, Sylow
309C2—21no—
310C2—21no—
311C2—81no—
312C2#6953211yescharacteristic
313C2#7014711yescharacteristic
314C2#7014811yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1822C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1822D3
hypocenterC3#11853
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
4. #67496C2semidirect
5. #67495C2semidirect
308. C3#11853semidirect

Maximal subgroups

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

subgroupindex
#822612
C3×#18222
#715712
#674962
#674952
#118533