Finite Group Databasebeta

← #10695838

Subgroups — #10695838

Order 768.

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

normal subgroups only

This group has 602 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#82217C212yesmaximal, complemented
2#74144C211yesmaximal, characteristic
3C3×#1820C211yesmaximal, characteristic, complemented
4#69531C212yesmaximal, complemented
5#67629C211yesmaximal, characteristic, complemented
6#9983—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#930C2212yes—
9C3×#381C2211yescharacteristic
10C3×#381C2211yescharacteristic
11#5207C2212yes—
12#5003—24no—
13#5001—22no—
14#4995—22no—
15#4968—22no—
16#4967—22no—
17#4960—22no—
18#4674—22no—
19#4671—22no—
20C3×#342—22no—
21#4451—24no—
22#3094—32no—
23#2815—31no—
24#1820D311yescharacteristic, complemented
25#1510—32no—
26#1306—31no—
27C23×C3×C4C2311yescharacteristic
28C22×#67—22no—
29C22×D12—22no—
30C22×D12D412yes—
31C3×#117—22no—
32C3×#117—22no—
33C2×C3×C4⋊C4D412yescomplemented
34C2×C3×C22⋊C4—21no—
35C2×C3×C22⋊C4—21no—
36C2×C3×C22⋊C4D412yes—
37C2×C3×C42—21no—
38#930—22no—
39#930—22no—
40#930—22no—
41#930—22no—
42C2×(C2×D3)⋊C4—22no—
43C2×(C2×D3)⋊C4—24no—
44C2×C4⋊Dic3D412yes—
45C2×#208—22no—
46C3×#95—22no—
47C3×#95—22no—
48C3⋊#95—22no—
49#830—22no—
50#830D414yescomplemented
51#809—24no—
52#809—24no—
53#808—24no—
54#807—24no—
55C2×#121—32no—
56C2×#117C2×D311yescharacteristic
57#482—64no—
58#481—62no—
59#480—62no—
60#428—62no—
61#418—62no—
62#407—32no—
63#397—62no—
64#392—62no—
65#392—62no—
66#381C2×D311yescharacteristic
67#381C2×D311yescharacteristic
68#342—22no—
69#327—64no—
70C24×C3C2×D411yescharacteristic
71C23×D3—22no—
72C22×C3×C4—22no—
73C22×C3×C4—22no—
74C22×C3×C4—42no—
75C22×C3×C4—41no—
76C22×C3×C4C2×D411yescharacteristic
77C22×C3×C4C2×D412yes—
78C22×C3×C4C2×D412yes—
79C22×C3×C4#4111yescharacteristic
80C2×#67—42no—
81C2×#67—42no—
82C2×#67—84no—
83C22×Dic3—22no—
84C2×D12—22no—
85C2×D12—22no—
86C2×D12—44no—
87C2×D12—84no—
88C2×D12—82no—
89C3×C4⋊C4—22no—
90C3×C4⋊C4—22no—
91C3×C22⋊C4—41no—
92C3×C22⋊C4—41no—
93C3×C22⋊C4—42no—
94C3×C22⋊C4—42no—
95C3×C42—42no—
96(C2×D3)⋊C4—42no—
97(C2×D3)⋊C4—42no—
98(C2×D3)⋊C4—84no—
99C4⋊Dic3—22no—
100C4⋊Dic3—22no—
101#208—44no—
102C2×C3⋊C8—24no—
103C22×D4—32no—
104C22×D4—62no—
105C22×D4—62no—
106C23×C4C22×D311yescharacteristic
107C2×M16—62no—
108#121—62no—
109#121—62no—
110#121—62no—
111#121—62no—
112#117—22no—
113#117—22no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4D1212yescomplemented
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—62no—
119C2×C22⋊C4—64no—
120C2×C22⋊C4#6712yes—
121C2×C42—21no—
122#107—64no—
123#106—64no—
124#102—64no—
125#102—64no—
126C22⋊C8—34no—
127C22⋊C8—62no—
128#95—22no—
129#95—22no—
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—41no—
143C2×C3×C4—42no—
144C2×C3×C4—41no—
145C2×C3×C4—44no—
146C2×C3×C4—42no—
147C2×C3×C4—42no—
148C2×C3×C4—82no—
149C2×C3×C4—81no—
150C2×C3×C4—82no—
151C2×C3×C4#12012yes—
152C2×C3×C4#12112yes—
153#67—84no—
154#67—84no—
155C2×Dic3—44no—
156C2×Dic3—42no—
157D12—44no—
158D12—82no—
159D12—88no—
160C3⋊C8—44no—
161C24—62no—
162C24C2×#6711yescharacteristic
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—22no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—62no—
176C22×C4C2×D1211yescharacteristic
177C22×C4#23511yescharacteristic
178C22×C4D3×D412yes—
179C22×C4D3×D412yes—
180M16—124no—
181C2×C8—64no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—62no—
185C4⋊C4—62no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—122no—
191C22⋊C4—122no—
192C22⋊C4—244no—
193C42—42no—
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—
210C3×C4—82no—
211C3×C4—161no—
212Dic3—82no—
213C23—22no—
214C23—42no—
215C23—122no—
216C23—122no—
217C23—124no—
218C23—122no—
219C23—124no—
220C23#93011yescharacteristic
221C23#93711yescharacteristic
222C23#94011yescharacteristic
223D4—124no—
224D4—242no—
225D4—248no—
226D4—244no—
227D4—244no—
228C2×C4—24no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—81no—
237C2×C4—82no—
238C2×C4—124no—
239C2×C4—122no—
240C2×C4#88212yes—
241C2×C4#88412yes—
242C8—124no—
243C2×C3—22no—
244C2×C3—41no—
245C2×C3—41no—
246C2×C3#200812yes—
247C2×C3#202211yescharacteristic
248D3—162no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—21no—
253C22—21no—
254C22—82no—
255C22—244no—
256C22—244no—
257C22—242no—
258C22—242no—
259C22—242no—
260C22#496011yescharacteristic
261C22#505312yes—
262C4—22no—
263C4—22no—
264C4—82no—
265C4—82no—
266C4—161no—
267C4—242no—
268C3#998311yescharacteristic, complemented, Sylow
269C2—22no—
270C2—41no—
271C2—41no—
272C2—482no—
273C2#6953911yescharacteristic
274C2#7005012yes—
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1820C2
socleC22×C3#392
prime radical#74144C2
hypercenter#1820D3
hypocenterC3#9983
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
1. #82217C2semidirect
3. C3×#1820C2semidirect
4. #69531C2semidirect
5. #67629C2semidirect
24. #1820D3semidirect
33. C2×C3×C4⋊C4D4semidirect
50. #830D4semidirect
115. C2×C4⋊C4D12semidirect
268. C3#9983semidirect

Maximal subgroups

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

subgroupindex
#822172
#741442
C3×#18202
#695312
#676292
#99833