Finite Group Databasebeta

← #10704739

Subgroups — #10704739

Order 768.

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

normal subgroups only

This group has 380 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#82254C211yesmaximal, characteristic
2C3×#1464C211yesmaximal, characteristic
3#71571C211yesmaximal, characteristic
4#67446C212yesmaximal
5#67445C212yesmaximal
6#11548—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C22×Dic3⋊C4C411yescharacteristic
9C3×#432—21no—
10#5211C411yescharacteristic
11#4985—21no—
12#4983—21no—
13#4946C411yescharacteristic
14#4946C2211yescharacteristic
15#4943C411yescharacteristic
16#4943C2211yescharacteristic
17#4933—21no—
18#4933—21no—
19#4925—21no—
20C3×#324—21no—
21C3×#323C2212yes—
22#4528—21no—
23#4525C2212yes—
24#2484—31no—
25#1464D311yescharacteristic
26#1458—31no—
27#1326—32no—
28#1317—32no—
29C23×C3×C4C2×C411yescharacteristic
30C23×Dic3C2×C411yescharacteristic
31C23×Dic3C2×C411yescharacteristic
32C3×#117—41no—
33C3×#117—41no—
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C22⋊C4C2×C411yescharacteristic
37C2×C3×C22⋊C4C2311yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×#217C2×C411yescharacteristic
41C2×#217C2×C411yescharacteristic
42C2×Dic3⋊C4—21no—
43C2×Dic3⋊C4—21no—
44C2×Dic3⋊C4—21no—
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—41no—
47C2×Dic3⋊C4—41no—
48C3×C4⋊C8—41no—
49C3×C22⋊C8—22no—
50C3×#97—41no—
51C3⋊#95—21no—
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×#117C2×D311yescharacteristic
63C22×C4⋊C4—31no—
64#432—21no—
65#431—61no—
66#400—61no—
67#398—61no—
68#387—61no—
69#387—61no—
70#385—31no—
71#385—31no—
72#381—31no—
73#381—31no—
74#381—31no—
75#324—21no—
76#324—61no—
77#323—32no—
78#323C2×D312yes—
79C24×C3C22×C411yescharacteristic
80C22×C3×C4—22no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
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—
94C22×Dic3—22no—
95C22×Dic3—42no—
96C22×Dic3—41no—
97C22×Dic3—41no—
98C22×Dic3—41no—
99C2×C3×C8—42no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—41no—
102C3×C22⋊C4—41no—
103C3×C22⋊C4—41no—
104C3×C42—21no—
105C3×C42—21no—
106C3×C42—21no—
107C3×C42—21no—
108#217—22no—
109#217—41no—
110Dic3⋊C4—42no—
111Dic3⋊C4—42no—
112Dic3⋊C4—42no—
113Dic3⋊C4—41no—
114Dic3⋊C4—41no—
115C2×C3⋊C8—42no—
116C23×C4—31no—
117C23×C4—31no—
118C23×C4C4×D311yescharacteristic
119#117—41no—
120#117—41no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4—61no—
123C2×C4⋊C4—61no—
124C2×C4⋊C4—61no—
125C2×C4⋊C4—121no—
126C2×C4⋊C4—121no—
127C2×C4⋊C4D1211yescharacteristic
128C2×C4⋊C4#6711yescharacteristic
129C2×C22⋊C4—31no—
130C2×C22⋊C4—31no—
131C2×C22⋊C4C4×D311yescharacteristic
132C2×C22⋊C4C22×D311yescharacteristic
133C2×C42D1211yescharacteristic
134C2×C42#6711yescharacteristic
135C4⋊C8—41no—
136C4⋊C8—121no—
137C22⋊C8—22no—
138C22⋊C8—62no—
139#97—41no—
140C4×C8—121no—
141#95—61no—
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—41no—
152C23×C3—41no—
153C23×C3C2×C22⋊C411yescharacteristic
154C23×C3#13112yes—
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—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#9911yescharacteristic
171C2×C3×C4#9911yescharacteristic
172C2×Dic3—22no—
173C2×Dic3—41no—
174C2×Dic3—42no—
175C2×Dic3—41no—
176C2×Dic3—42no—
177C2×Dic3—41no—
178C2×Dic3—41no—
179C2×Dic3—41no—
180C2×Dic3—41no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—82no—
187C3×C8—82no—
188C3⋊C8—82no—
189C24C2×C4×D311yescharacteristic
190C22×C4—22no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4—61no—
195C22×C4—61no—
196C22×C4—61no—
197C22×C4—61no—
198C22×C4—62no—
199C22×C4—122no—
200C22×C4—121no—
201C22×C4—121no—
202C22×C4—121no—
203C22×C4(C2×D3)⋊C411yescharacteristic
204C22×C4(C2×D3)⋊C411yescharacteristic
205C22×C4(C2×D3)⋊C411yescharacteristic
206C22×C4(C2×D3)⋊C411yescharacteristic
207C22×C4C2×D1211yescharacteristic
208C22×C4C2×#6711yescharacteristic
209C2×C8—42no—
210C2×C8—122no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—122no—
214C4⋊C4—122no—
215C4⋊C4—122no—
216C4⋊C4—121no—
217C4⋊C4—121no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—62no—
221C22⋊C4—121no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—22no—
229C22×C3—42no—
230C22×C3—42no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3#40811yescharacteristic
234C22×C3C2×#9911yescharacteristic
235C22×C3#41311yescharacteristic
236C3×C4—21no—
237C3×C4—21no—
238C3×C4—41no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4—41no—
242C3×C4—41no—
243Dic3—42no—
244Dic3—81no—
245Dic3—81no—
246Dic3—81no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23#90712yes—
252C23C2×(C2×D3)⋊C411yescharacteristic
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—62no—
269C2×C4—121no—
270C2×C4—122no—
271C2×C4—121no—
272C2×C4—122no—
273C2×C4—121no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—242no—
283C2×C4#80611yescharacteristic
284C2×C4#80611yescharacteristic
285C8—82no—
286C8—242no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—42no—
290C2×C3#146811yescharacteristic
291C2×C3#212011yescharacteristic
292C2×C3#212311yescharacteristic
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22C2×#80611yescharacteristic
301C22#512911yescharacteristic
302C22#513211yescharacteristic
303C4—21no—
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—122no—
311C4—241no—
312C4—241no—
313C4—241no—
314C3#1154811yescharacteristic, complemented, Sylow
315C2—21no—
316C2—21no—
317C2—42no—
318C2#6996911yescharacteristic
319C2#7005611yescharacteristic
320C2#7014711yescharacteristic
subgroupG/H
centerC22C2×#806
commutator, prime radicalC24×C3C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1464C2
socleC22×C3C2×#99
cosocleC2×C3×C22⋊C4C23
hypercenter#1464D3
hypocenterC3#11548
LCSG ▷ C24×C3 ▷ 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
314. C3#11548semidirect

Maximal subgroups

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

subgroupindex
#822542
C3×#14642
#715712
#674462
#674452
#115483