Finite Group Databasebeta

← #10695839

Subgroups — #10695839

Order 768.

602 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 81 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#78624C212yesmaximal, complemented
2#74144C211yesmaximal, characteristic
3#69978C212yesmaximal, complemented
4#69534C211yesmaximal, characteristic, complemented
5C3×#1306C211yesmaximal, characteristic, complemented
6#9983—31nomaximal, Sylow
7C2×#930C2212yes—
8C2×#926C2211yescharacteristic
9C3×#407C2212yes—
10C3×#381C2211yescharacteristic
11#5211C2211yescharacteristic
12#5131—22no—
13#5121—22no—
14#4960—22no—
15#4737—24no—
16#4736—22no—
17#4735—22no—
18#4674—22no—
19#4671—22no—
20#4527—22no—
21#4470—24no—
22#3094—32no—
23#2815—31no—
24#1820—31no—
25#1510—32no—
26#1306D311yescharacteristic, complemented
27C23×C3×C4C2311yescharacteristic
28C22×#67—22no—
29C22×D12—22no—
30C22×D12D412yes—
31C2×C3×M16—22no—
32C2×C3×C22⋊C4D412yes—
33C2×#217—21no—
34C2×#217—21no—
35#930—22no—
36#930—22no—
37#930—22no—
38#930—22no—
39C2×(C2×D3)⋊C4—22no—
40C2×(C2×D3)⋊C4—24no—
41#926—22no—
42#926—22no—
43C2×C4⋊Dic3D412yes—
44C2×C4⋊Dic3D412yescomplemented
45C2×C4×Dic3—21no—
46C3×C22⋊C8—22no—
47C3×C22⋊C8D414yescomplemented
48C3×#95—22no—
49C3⋊#95—22no—
50C3⋊#95—22no—
51#821—24no—
52#821—24no—
53#818—24no—
54#817—24no—
55C2×#121—32no—
56C2×#117—31no—
57#482—64no—
58#481—62no—
59#480—62no—
60#428—62no—
61#418—62no—
62#407C2×D312yes—
63#397—62no—
64#392—62no—
65#392—62no—
66#381—31no—
67#381C2×D311yescharacteristic
68#342—62no—
69#327—64no—
70C24×C3C2×D411yescharacteristic
71C23×D3—22no—
72C22×C3×C4—22no—
73C22×C3×C4—42no—
74C22×C3×C4C2×D412yes—
75C22×C3×C4C2×D412yes—
76C22×C3×C4C2×D411yescharacteristic
77C22×C3×C4#4111yescharacteristic
78C2×#67—42no—
79C2×#67—42no—
80C2×#67—84no—
81C22×Dic3—22no—
82C22×Dic3—22no—
83C22×Dic3—41no—
84C2×D12—22no—
85C2×D12—22no—
86C2×D12—44no—
87C2×D12—84no—
88C2×D12—82no—
89C3×M16—44no—
90C2×C3×C8—24no—
91C3×C22⋊C4—42no—
92#217—41no—
93#217—41no—
94#217—42no—
95(C2×D3)⋊C4—42no—
96(C2×D3)⋊C4—42no—
97(C2×D3)⋊C4—84no—
98C4⋊Dic3—22no—
99C4⋊Dic3—22no—
100C4⋊Dic3—22no—
101C4⋊Dic3—22no—
102C4×Dic3—42no—
103C22×D4—32no—
104C22×D4—62no—
105C22×D4—62no—
106C23×C4C22×D311yescharacteristic
107C2×M16—22no—
108#121—62no—
109#121—62no—
110#121—62no—
111#121—62no—
112#117—62no—
113#117—62no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—32no—
116C2×C22⋊C4—61no—
117C2×C22⋊C4—62no—
118C2×C22⋊C4—61no—
119C2×C22⋊C4—64no—
120C2×C22⋊C4D1212yes—
121C2×C42—61no—
122#107—64no—
123#106—64no—
124#102—64no—
125#102—64no—
126C22⋊C8—22no—
127C22⋊C8D1214yescomplemented
128#95—22no—
129#95—62no—
130#95—62no—
131C23×C3—22no—
132C23×C3—42no—
133C23×C3#12011yescharacteristic
134C23×C3#12111yescharacteristic
135C23×C3#12711yescharacteristic
136C22×D3—42no—
137C22×D3—42no—
138C22×D3—44no—
139C22×D3—42no—
140C22×D3—44no—
141C2×C3×C4—24no—
142C2×C3×C4—42no—
143C2×C3×C4—41no—
144C2×C3×C4—41no—
145C2×C3×C4—44no—
146C2×C3×C4—42no—
147C2×C3×C4#12012yes—
148C2×C3×C4#12112yes—
149#67—84no—
150#67—84no—
151C2×Dic3—44no—
152C2×Dic3—42no—
153C2×Dic3—42no—
154C2×Dic3—82no—
155C2×Dic3—81no—
156C2×Dic3—82no—
157D12—44no—
158D12—82no—
159D12—88no—
160C3×C8—44no—
161C24—62no—
162C24C2×D1211yescharacteristic
163C2×D4—62no—
164C2×D4—62no—
165C2×D4—124no—
166C2×D4—122no—
167C2×D4—122no—
168C2×D4—244no—
169C2×D4—242no—
170C2×D4—244no—
171C22×C4—22no—
172C22×C4—42no—
173C22×C4—62no—
174C22×C4—62no—
175C22×C4—121no—
176C22×C4C2×D1212yes—
177C22×C4D3×D412yes—
178C22×C4D3×D411yescharacteristic
179C22×C4#23911yescharacteristic
180M16—44no—
181C2×C8—24no—
182C4⋊C4—62no—
183C4⋊C4—62no—
184C4⋊C4—62no—
185C4⋊C4—62no—
186C22⋊C4—42no—
187C22⋊C4—121no—
188C22⋊C4—121no—
189C22⋊C4—122no—
190C22⋊C4—122no—
191C22⋊C4—122no—
192C22⋊C4—244no—
193C42—122no—
194C22×C3—22no—
195C22×C3—22no—
196C22×C3—22no—
197C22×C3—21no—
198C22×C3—21no—
199C22×C3—82no—
200C22×C3#39211yescharacteristic
201C22×C3#45812yes—
202C2×D3—84no—
203C2×D3—84no—
204C2×D3—82no—
205C2×D3—82no—
206C2×D3—82no—
207C3×C4—22no—
208C3×C4—22no—
209C3×C4—82no—
210Dic3—82no—
211Dic3—82no—
212Dic3—161no—
213C23—22no—
214C23—42no—
215C23—122no—
216C23—122no—
217C23—124no—
218C23—122no—
219C23—124no—
220C23#87411yescharacteristic
221C23#88211yescharacteristic
222C23#89511yescharacteristic
223D4—124no—
224D4—242no—
225D4—248no—
226D4—244no—
227D4—244no—
228C2×C4—24no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—124no—
235C2×C4—122no—
236C2×C4—122no—
237C2×C4—242no—
238C2×C4—241no—
239C2×C4—242no—
240C2×C4#88212yes—
241C2×C4#89512yes—
242C8—44no—
243C2×C3—22no—
244C2×C3—41no—
245C2×C3—41no—
246C2×C3#200812yes—
247C2×C3#202211yescharacteristic
248D3—162no—
249C22—22no—
250C22—21no—
251C22—21no—
252C22—22no—
253C22—22no—
254C22—82no—
255C22—244no—
256C22—244no—
257C22—242no—
258C22—242no—
259C22—242no—
260C22#467111yescharacteristic
261C22#474212yes—
262C4—22no—
263C4—22no—
264C4—82no—
265C4—242no—
266C4—242no—
267C4—481no—
268C3#998311yescharacteristic, complemented, Sylow
269C2—22no—
270C2—41no—
271C2—41no—
272C2—482no—
273C2#6896611yescharacteristic
274C2#6902812yes—
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1306C2
socleC22×C3#392
prime radical#74144C2
hypercenter#1306D3
hypocenterC3#9983
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
1. #78624C2semidirect
3. #69978C2semidirect
4. #69534C2semidirect
5. C3×#1306C2semidirect
26. #1306D3semidirect
44. C2×C4⋊Dic3D4semidirect
47. C3×C22⋊C8D4semidirect
127. C22⋊C8D12semidirect
268. C3#9983semidirect

Maximal subgroups

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

subgroupindex
#786242
#741442
#699782
#695342
C3×#13062
#99833