Finite Group Databasebeta

← #10705479

Subgroups — #10705479

Order 768.

448 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 107 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#82264C211yesmaximal, characteristic
2C3×#1740C211yesmaximal, characteristic
3#71571C211yesmaximal, characteristic
4#67492C214yesmaximal
5#11718—31nomaximal, Sylow
6C2×C3×#117—21no—
7C2×C3×#117C2211yescharacteristic
8C2×#869C411yescharacteristic, complemented
9#5212—21no—
10#4986C412yescomplemented
11#4979—21no—
12#4949C2212yes—
13#4947—21no—
14#4936C411yescharacteristic, complemented
15#4928—21no—
16#4925—21no—
17C3×#342—21no—
18C3×#342C2212yes—
19#4528—21no—
20#4525C2212yes—
21#2628—31no—
22#1740D311yescharacteristic
23#1458—31no—
24#1352—34no—
25C23×C3×C4D411yescharacteristic
26C22×Dic6—21no—
27C3×#117—24no—
28C3×#117—41no—
29C3×#117—41no—
30C3×#117—42no—
31C2×C3×C4⋊C4—41no—
32C2×C3×C4⋊C4C2×C411yescharacteristic
33C2×C3×C4⋊C4D411yescharacteristic
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4D411yescharacteristic
36C2×C3×C22⋊C4C2311yescharacteristic
37C2×C3×C42—41no—
38C2×C3×C42C2×C411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×#217—22no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3C2×C412yes—
44C2×Dic3⋊C4—22no—
45#869—41no—
46#869—41no—
47C3⋊#95—22no—
48C3⋊#95—21no—
49C3⋊#95—22no—
50C3⋊#95C2×C412yes—
51#830—22no—
52C4⋊(C3⋊C8)—41no—
53C4×C3⋊C8—41no—
54C2×C4⋊Q8—31no—
55C2×#117—21no—
56C2×#117C2×D311yescharacteristic
57#431—61no—
58#400—32no—
59#399—32no—
60#395—61no—
61#393—61no—
62#389—61no—
63#382—31no—
64#380—61no—
65#342—21no—
66#342C2×D312yes—
67#324—61no—
68#323—32no—
69C24×C3C2×D411yescharacteristic
70C22×C3×C4—21no—
71C22×C3×C4—21no—
72C22×C3×C4—22no—
73C22×C3×C4—42no—
74C22×C3×C4C22⋊C411yescharacteristic
75C22×C3×C4C22⋊C411yescharacteristic
76C22×C3×C4C22⋊C411yescharacteristic
77C22×C3×C4C22⋊C411yescharacteristic
78C22×C3×C4SD1612yes—
79C22×C3×C4Q1612yes—
80C22×C3×C4C22×C411yescharacteristic
81C22×C3×C4C2×D411yescharacteristic
82C22×Dic3—22no—
83C22×Dic3—22no—
84C2×Dic6—42no—
85C2×Dic6—82no—
86C3×C4⋊C4—41no—
87C3×C4⋊C4—41no—
88C3×C4⋊C4—44no—
89C3×C22⋊C4—24no—
90C3×C22⋊C4—41no—
91C3×C22⋊C4—41no—
92C3×C22⋊C4—42no—
93C3×C42—21no—
94C3×C42—21no—
95C3×C42—21no—
96C3×C42—21no—
97C3×C42—44no—
98#217—82no—
99C4⋊Dic3—41no—
100C4⋊Dic3—41no—
101C4⋊Dic3—42no—
102C4⋊Dic3—42no—
103Dic3⋊C4—82no—
104C2×C3⋊C8—42no—
105C22×Q8—61no—
106C23×C4#6711yescharacteristic
107C4⋊Q8—121no—
108C4⋊Q8—121no—
109#117—24no—
110#117—41no—
111#117—41no—
112#117—42no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4—61no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4C4×D311yescharacteristic
119C2×C4⋊C4D1211yescharacteristic
120C2×C22⋊C4—22no—
121C2×C22⋊C4—62no—
122C2×C22⋊C4#6711yescharacteristic
123C2×C22⋊C4C22×D311yescharacteristic
124C2×C42—41no—
125C2×C42C4×D311yescharacteristic
126C2×C42D1211yescharacteristic
127C4⋊C8—121no—
128C22⋊C8—62no—
129C4×C8—121no—
130#95—32no—
131#95—62no—
132#95—61no—
133#95—62no—
134C23×C3—21no—
135C23×C3—21no—
136C23×C3—41no—
137C23×C3—41no—
138C23×C3C2×C22⋊C411yescharacteristic
139C23×C3C2×SD1611yescharacteristic
140C23×C3C2×Q1611yescharacteristic
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—22no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—41no—
152C2×C3×C4—41no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4—44no—
156C2×C3×C4—82no—
157C2×C3×C4—82no—
158C2×C3×C4#9911yescharacteristic
159C2×C3×C4#9911yescharacteristic
160C2×C3×C4#10312yes—
161C2×C3×C4#10312yes—
162C2×C3×C4#10412yes—
163C2×Dic3—42no—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—42no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169Dic6—84no—
170C3⋊C8—82no—
171C24C2×#6711yescharacteristic
172C2×Q8—122no—
173C2×Q8—242no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4(C2×D3)⋊C411yescharacteristic
181C22×C4(C2×D3)⋊C411yescharacteristic
182C22×C4(C2×D3)⋊C411yescharacteristic
183C22×C4(C2×D3)⋊C411yescharacteristic
184C22×C4#21412yes—
185C22×C4#21612yes—
186C22×C4C2×C4×D311yescharacteristic
187C22×C4C2×D1211yescharacteristic
188C2×C8—122no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—44no—
192C4⋊C4—122no—
193C4⋊C4—122no—
194C4⋊C4—121no—
195C4⋊C4—121no—
196C4⋊C4—242no—
197C22⋊C4—24no—
198C22⋊C4—41no—
199C22⋊C4—41no—
200C22⋊C4—42no—
201C22⋊C4—242no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—44no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—41no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—81no—
216C22×C3C2×#9911yescharacteristic
217C22×C3C2×#10311yescharacteristic
218C22×C3C2×#10411yescharacteristic
219C3×C4—22no—
220C3×C4—41no—
221C3×C4—41no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4—84no—
225C3×C4#32812yes—
226C3×C4#32812yes—
227Dic3—82no—
228Dic3—82no—
229C23—21no—
230C23—21no—
231C23—41no—
232C23—41no—
233C23C2×(C2×D3)⋊C411yescharacteristic
234C23C2×#21411yescharacteristic
235C23C2×#21611yescharacteristic
236Q8—244no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—44no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—122no—
255C2×C4—122no—
256C2×C4—122no—
257C2×C4—122no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4#80512yes—
261C2×C4#80611yescharacteristic
262C2×C4#80611yescharacteristic
263C2×C4#81012yes—
264C2×C4#81012yes—
265C8—242no—
266C2×C3—21no—
267C2×C3—21no—
268C2×C3—41no—
269C2×C3—41no—
270C2×C3C2×#32811yescharacteristic
271C2×C3#212311yescharacteristic
272C2×C3#212811yescharacteristic
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22C2×#80511yescharacteristic
283C22C2×#80611yescharacteristic
284C22C2×#81011yescharacteristic
285C4—22no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—84no—
291C4—242no—
292C4—242no—
293C4#445212yes—
294C4#445212yes—
295C3#1171811yescharacteristic, complemented, Sylow
296C2—21no—
297C2—21no—
298C2—41no—
299C2—41no—
300C2C2×#445211yescharacteristic
301C2#7005811yescharacteristic
302C2#7006111yescharacteristic
subgroupG/H
centerC22C2×#806
commutatorC22×C3×C4C22×C4
FrattiniC2×C22⋊C4C22×D3
FittingC3×#1740C2
socleC22×C3C2×#99
prime radicalC24×C3C2×D4
cosocleC2×C3×C22⋊C4C23
hypercenter#1740D3
hypocenterC3#11718
LCSG ▷ C22×C3×C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
8. C2×#869C4semidirect
10. #4986C4semidirect
14. #4936C4semidirect
295. C3#11718semidirect

Maximal subgroups

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

subgroupindex
#822642
C3×#17402
#715712
#674922
#117183