Finite Group Databasebeta

← #10704759

Subgroups — #10704759

Order 768.

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

normal subgroups only

This group has 366 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#82256C211yesmaximal, characteristic
2C3×#1464C211yesmaximal, characteristic
3#71571C211yesmaximal, characteristic
4#67455C212yesmaximal, complemented
5#67453C212yesmaximal, complemented
6#11552—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#924C411yescharacteristic
9C3×#432—21no—
10C4×#217C411yescharacteristic
11#4977C411yescharacteristic
12#4975C2211yescharacteristic
13#4973C411yescharacteristic
14#4973C2211yescharacteristic
15#4952—21no—
16#4952—21no—
17#4933—21no—
18#4933—21no—
19#4925—21no—
20C3×#324C2212yes—
21C3×#323—21no—
22#4528C2212yes—
23#4525—21no—
24#2499—31no—
25#1464D311yescharacteristic
26#1458—31no—
27#1325—32no—
28#1319—32no—
29C23×C3×C4C2×C411yescharacteristic
30C22×Dic6C2×C411yescharacteristic
31C3×#117—41no—
32C3×#117—41no—
33C2×C3×C4⋊C4C2×C411yescharacteristic
34C2×C3×C4⋊C4C2311yescharacteristic
35C2×C3×C22⋊C4D411yescharacteristic
36C2×C3×C22⋊C4D411yescharacteristic
37C2×C3×C42D411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×#217—21no—
40C2×#217—21no—
41C2×C4⋊Dic3C2×C411yescharacteristic
42#924—41no—
43#924—41no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×C4×Dic3C2×C411yescharacteristic
48C2×C4×Dic3C2×C411yescharacteristic
49C3×C4⋊C8—41no—
50C3×C22⋊C8—22no—
51C3×#97—41no—
52C3⋊#95—21no—
53C3⋊#95—21no—
54C3⋊#95—21no—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59#830—22no—
60C4⋊(C3⋊C8)—41no—
61C4×C3⋊C8—41no—
62C2×C22⋊Q8—31no—
63C2×#117C2×D311yescharacteristic
64#432—21no—
65#431—61no—
66#397—61no—
67#397—61no—
68#391—31no—
69#391—31no—
70#387—61no—
71#387—61no—
72#384—31no—
73#382—31no—
74C4×C22⋊C4—31no—
75#324—32no—
76#324C2×D312yes—
77#323—21no—
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—
94C2×Dic6—22no—
95C2×Dic6—41no—
96C2×Dic6—41no—
97C2×C3×C8—42no—
98C3×C4⋊C4—41no—
99C3×C4⋊C4—41no—
100C3×C22⋊C4—41no—
101C3×C22⋊C4—41no—
102C3×C42—21no—
103C3×C42—21no—
104C3×C42—21no—
105C3×C42—21no—
106#217—42no—
107#217—42no—
108C4⋊Dic3—41no—
109Dic3⋊C4—42no—
110Dic3⋊C4—42no—
111Dic3⋊C4—41no—
112Dic3⋊C4—41no—
113C4×Dic3—22no—
114C4×Dic3—41no—
115C2×C3⋊C8—42no—
116C22×Q8—31no—
117C23×C4C4×D311yescharacteristic
118C22⋊Q8—121no—
119C22⋊Q8—121no—
120#117—41no—
121#117—41no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—61no—
124C2×C4⋊C4—61no—
125C2×C4⋊C4—61no—
126C2×C4⋊C4C4×D311yescharacteristic
127C2×C4⋊C4C22×D311yescharacteristic
128C2×C22⋊C4—61no—
129C2×C22⋊C4—61no—
130C2×C22⋊C4D1211yescharacteristic
131C2×C22⋊C4#6711yescharacteristic
132C2×C42—31no—
133C2×C42—31no—
134C2×C42D1211yescharacteristic
135C2×C42#6711yescharacteristic
136C4⋊C8—41no—
137C4⋊C8—121no—
138C22⋊C8—22no—
139C22⋊C8—62no—
140#97—41no—
141C4×C8—121no—
142#95—61no—
143#95—61no—
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#13112yes—
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—
184Dic6—82no—
185C3×C8—82no—
186C3⋊C8—82no—
187C24(C2×D3)⋊C411yescharacteristic
188C2×Q8—62no—
189C2×Q8—121no—
190C2×Q8—121no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—41no—
195C22×C4—61no—
196C22×C4—61no—
197C22×C4—61no—
198C22×C4—61no—
199C22×C4(C2×D3)⋊C411yescharacteristic
200C22×C4(C2×D3)⋊C411yescharacteristic
201C22×C4(C2×D3)⋊C411yescharacteristic
202C22×C4C2×C4×D311yescharacteristic
203C22×C4C2×D1211yescharacteristic
204C22×C4C2×#6711yescharacteristic
205C2×C8—42no—
206C2×C8—122no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—121no—
210C4⋊C4—122no—
211C4⋊C4—122no—
212C4⋊C4—121no—
213C4⋊C4—121no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C22⋊C4—122no—
217C22⋊C4—122no—
218C42—21no—
219C42—21no—
220C42—21no—
221C42—21no—
222C42—62no—
223C42—121no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—81no—
230C22×C3—81no—
231C22×C3#40811yescharacteristic
232C22×C3C2×#9911yescharacteristic
233C22×C3#41311yescharacteristic
234C3×C4—22no—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—41no—
238C3×C4—41no—
239C3×C4—41no—
240C3×C4—41no—
241Dic3—42no—
242Dic3—81no—
243Dic3—81no—
244Dic3—81no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23C2×(C2×D3)⋊C411yescharacteristic
251Q8—242no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—62no—
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—121no—
277C2×C4—121no—
278C2×C4#80611yescharacteristic
279C2×C4#80611yescharacteristic
280C2×C4#90712yes—
281C8—82no—
282C8—242no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—81no—
286C2×C3#146811yescharacteristic
287C2×C3#212611yescharacteristic
288C2×C3#213511yescharacteristic
289C22—21no—
290C22—21no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22—81no—
296C22C2×#80611yescharacteristic
297C22#512911yescharacteristic
298C22#513211yescharacteristic
299C4—21no—
300C4—21no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—122no—
307C4—241no—
308C4—241no—
309C4—241no—
310C3#1155211yescharacteristic, complemented, Sylow
311C2—21no—
312C2—21no—
313C2—81no—
314C2#6996911yescharacteristic
315C2#7006011yescharacteristic
316C2#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#11552
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
5. #67453C2semidirect
310. C3#11552semidirect

Maximal subgroups

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

subgroupindex
#822562
C3×#14642
#715712
#674552
#674532
#115523