Finite Group Databasebeta

← #10704591

Subgroups — #10704591

Order 768.

446 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2285C211yesmaximal, characteristic
2#71572C212yesmaximal
3#67635C212yesmaximal, complemented
4#67632C212yesmaximal, complemented
5#11513—31nomaximal, Sylow
6C2×C3×#117C411yescharacteristic
7C2×C3×#117C2211yescharacteristic
8C22×C3×C42C411yescharacteristic
9C3×#384C411yescharacteristic
10C3×#384C2211yescharacteristic
11C3×#382C411yescharacteristic
12C3×#382C2211yescharacteristic
13C3×#380—22no—
14C3×C4×C4⋊C4—22no—
15#4926—22no—
16#4528C2214yes—
17#4525—22no—
18#2285D311yescharacteristic
19#1464—32no—
20#1325—32no—
21#1324—32no—
22C23×C3×C4—21no—
23C23×C3×C4C2×C411yescharacteristic
24C3×#117—41no—
25C3×#117—41no—
26C3×#117—41no—
27C3×#117—41no—
28C2×C3×C4⋊C4—22no—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4C2×C411yescharacteristic
31C2×C3×C4⋊C4C2×C411yescharacteristic
32C2×C3×C4⋊C4C2×C411yescharacteristic
33C2×C3×C4⋊C4C2311yescharacteristic
34C2×C3×C22⋊C4—21no—
35C2×C3×C22⋊C4D412yes—
36C2×C3×C42—21no—
37C2×C3×C42—22no—
38C2×C3×C42—22no—
39C2×C3×C42—21no—
40C2×C3×C42C2×C411yescharacteristic
41C2×C3×C42C2×C411yescharacteristic
42C2×C3×C42D412yes—
43C3×#95—22no—
44C3×#95—22no—
45#830—24no—
46C4⋊(C3⋊C8)—42no—
47#803—42no—
48C2×#117Dic311yescharacteristic
49C2×#117C2×D311yescharacteristic
50C22×C42Dic311yescharacteristic
51#432—62no—
52#384Dic311yescharacteristic
53#384C2×D311yescharacteristic
54#382Dic311yescharacteristic
55#382C2×D311yescharacteristic
56#380—22no—
57C4×C4⋊C4—22no—
58#324—34no—
59#323—62no—
60C24×C3C22⋊C411yescharacteristic
61C22×C3×C4—22no—
62C22×C3×C4—21no—
63C22×C3×C4—21no—
64C22×C3×C4—21no—
65C22×C3×C4—21no—
66C22×C3×C4—21no—
67C22×C3×C4—24no—
68C22×C3×C4—21no—
69C22×C3×C4—21no—
70C22×C3×C4—21no—
71C22×C3×C4—42no—
72C22×C3×C4—42no—
73C22×C3×C4C22⋊C412yes—
74C22×C3×C4C22⋊C411yescharacteristic
75C22×C3×C4C22×C411yescharacteristic
76C22×C3×C4C2×D412yes—
77C3×C4⋊C4—41no—
78C3×C4⋊C4—41no—
79C3×C4⋊C4—41no—
80C3×C4⋊C4—41no—
81C3×C4⋊C4—44no—
82C3×C4⋊C4—44no—
83C3×C22⋊C4—42no—
84C3×C22⋊C4—42no—
85C3×C42—24no—
86C3×C42—22no—
87C3×C42—22no—
88C3×C42—21no—
89C3×C42—21no—
90C3×C42—41no—
91C3×C42—41no—
92C3×C42M1612yes—
93C3×C42M1612yes—
94C2×C3⋊C8—44no—
95C23×C4—21no—
96C23×C4C2×Dic311yescharacteristic
97#117—41no—
98#117—41no—
99#117—41no—
100#117—41no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4C2×Dic311yescharacteristic
104C2×C4⋊C4C2×Dic311yescharacteristic
105C2×C4⋊C4C2×Dic311yescharacteristic
106C2×C4⋊C4C22×D311yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4#6712yes—
109C2×C42—21no—
110C2×C42—22no—
111C2×C42—22no—
112C2×C42—21no—
113C2×C42C2×Dic311yescharacteristic
114C2×C42C2×Dic311yescharacteristic
115C2×C42#6712yes—
116C4⋊C8—122no—
117C22⋊C8—64no—
118#97—122no—
119#95—22no—
120#95—22no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3—41no—
124C23×C3—41no—
125C23×C3#10112yes—
126C23×C3C2×C22⋊C411yescharacteristic
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—48no—
142C2×C3×C4—41no—
143C2×C3×C4—41no—
144C2×C3×C4—42no—
145C2×C3×C4—42no—
146C2×C3×C4—42no—
147C2×C3×C4—42no—
148C2×C3×C4—41no—
149C2×C3×C4—41no—
150C2×C3×C4#9911yescharacteristic
151C2×C3×C4#9911yescharacteristic
152C2×C3×C4C2×M1611yescharacteristic
153C2×C3×C4C2×M1611yescharacteristic
154C3⋊C8—84no—
155C24#21711yescharacteristic
156C22×C4—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—24no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4#21712yes—
169C22×C4#21711yescharacteristic
170C22×C4C22×Dic311yescharacteristic
171C22×C4C2×#6712yes—
172C2×C8—124no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—44no—
178C4⋊C4—44no—
179C22⋊C4—42no—
180C22⋊C4—42no—
181C42—24no—
182C42—22no—
183C42—22no—
184C42—21no—
185C42—21no—
186C42—41no—
187C42—41no—
188C42#20812yes—
189C42#20812yes—
190C22×C3—21no—
191C22×C3—21no—
192C22×C3—22no—
193C22×C3—42no—
194C22×C3—42no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3#40711yescharacteristic
198C22×C3C2×#9911yescharacteristic
199C22×C3C2×#10111yescharacteristic
200C3×C4—24no—
201C3×C4—21no—
202C3×C4—21no—
203C3×C4—21no—
204C3×C4—21no—
205C3×C4—42no—
206C3×C4—42no—
207C3×C4—42no—
208C3×C4—81no—
209C3×C4—81no—
210C23—21no—
211C23—21no—
212C23—41no—
213C23—41no—
214C23#83612yes—
215C23C2×#21711yescharacteristic
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—48no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4#83411yescharacteristic
240C2×C4#83411yescharacteristic
241C2×C4C2×#20811yescharacteristic
242C2×C4C2×#20811yescharacteristic
243C8—244no—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3—42no—
247C2×C3#146311yescharacteristic
248C2×C3#212611yescharacteristic
249C2×C3#213211yescharacteristic
250C22—21no—
251C22—21no—
252C22—22no—
253C22—42no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22#520711yescharacteristic
258C22C2×#83411yescharacteristic
259C22C2×#83611yescharacteristic
260C4—24no—
261C4—21no—
262C4—21no—
263C4—21no—
264C4—21no—
265C4—42no—
266C4—42no—
267C4—42no—
268C4—81no—
269C4—81no—
270C3#1151311yescharacteristic, complemented, Sylow
271C2—21no—
272C2—21no—
273C2—42no—
274C2#7155211yescharacteristic
275C2#7181911yescharacteristic
276C2#7182011yescharacteristic
subgroupG/H
centerC22C2×#834
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2285C2
socleC22×C3C2×#99
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#2285D3
hypocenterC3#11513
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2285

Complements

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

NH ≅ G/Ntype
3. #67635C2semidirect
4. #67632C2semidirect
270. C3#11513semidirect

Maximal subgroups

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

subgroupindex
C3×#22852
#715722
#676352
#676322
#115133