Finite Group Databasebeta

← #10696044

Subgroups — #10696044

Order 768.

696 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 103 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#83351C212yesmaximal, complemented
2#73418C211yesmaximal, characteristic
3C3×#1995C211yesmaximal, characteristic, complemented
4#70496C212yesmaximal, complemented
5#67614C211yesmaximal, characteristic, complemented
6#10015—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×C4×D12—22no—
9C2×C4×D12C2212yes—
10C3×#423C2211yescharacteristic
11C3×#376C2211yescharacteristic
12#5084—22no—
13#5083—22no—
14#5046—22no—
15#5043—22no—
16C4×#215—24no—
17#4925C2212yes—
18#4677—22no—
19#4671—22no—
20C3×#339—22no—
21#4483—24no—
22#3101—32no—
23#2397—31no—
24#1995D311yescharacteristic, complemented
25#1486—32no—
26#1274—31no—
27C22×D12—22no—
28C22×D12D412yes—
29C22×C4×D3—22no—
30C2×C3×M16—21no—
31C3×C4⋊Q8D412yescomplemented
32C3×#125—21no—
33C3×C22⋊Q8—22no—
34C3×C4×Q8—22no—
35C3×#117—21no—
36C2×C3×C42D412yes—
37C2×C3×C42C2311yescharacteristic
38C2×#215—24no—
39C2×(C2×D3)⋊C4—24no—
40C2×(C2×D3)⋊C4—22no—
41C2×C4⋊Dic3—22no—
42C2×C4⋊Dic3D412yes—
43C4×D12—22no—
44C4×D12—22no—
45C4×D12—22no—
46C4×D12—22no—
47C4×D12—44no—
48C4×D12—44no—
49C3×C4⋊C8—22no—
50C3×#95—22no—
51#835—24no—
52#830—22no—
53#809—24no—
54#808—24no—
55C4⋊(C3⋊C8)—22no—
56C4×C3⋊C8D414yescomplemented
57#533C2×D311yescharacteristic
58C2×C4×D4—32no—
59C2×C4×D4—62no—
60#483—62no—
61#476—62no—
62#461—62no—
63#450—62no—
64C4×SD16—64no—
65#431—32no—
66#423C2×D311yescharacteristic
67#399—62no—
68#392—62no—
69#376C2×D311yescharacteristic
70#339—22no—
71D4⋊C8—64no—
72C23×D3—22no—
73C2×C3×Q8—22no—
74C22×C3×C4—22no—
75C22×C3×C4C2×D412yes—
76C22×C3×C4C2×D411yescharacteristic
77C22×Dic3—22no—
78C2×D12—22no—
79C2×D12—22no—
80C2×D12—44no—
81C2×D12—44no—
82C2×D12—48no—
83C2×C4×D3—42no—
84C2×C4×D3—42no—
85C2×C4×D3—44no—
86C2×C4×D3—44no—
87C3×M16—42no—
88C2×C3×C8—41no—
89C3×C4⋊C4—22no—
90C3×C4⋊C4—22no—
91C3×C4⋊C4—42no—
92C3×C22⋊C4—41no—
93C3×C42—22no—
94C3×C42—41no—
95C3×C42SD1614yescomplemented
96C3×C42C2×D411yescharacteristic
97C3×C42C2×D412yes—
98C3×C42#4111yescharacteristic
99#215—48no—
100(C2×D3)⋊C4—48no—
101(C2×D3)⋊C4—42no—
102(C2×D3)⋊C4—42no—
103C4⋊Dic3—22no—
104C4⋊Dic3—22no—
105C4⋊Dic3—44no—
106C2×C3⋊C8—24no—
107C22×D4—32no—
108C22×D4—62no—
109C23×C4—62no—
110C2×SD16—64no—
111C2×M16—21no—
112C4⋊Q8D1212yescomplemented
113#125—21no—
114C22⋊Q8—22no—
115C4×Q8—22no—
116C4×D4—62no—
117C4×D4—62no—
118C4×D4—62no—
119C4×D4—62no—
120C4×D4—124no—
121C4×D4—124no—
122#117—21no—
123C2×C4⋊C4—32no—
124C2×C4⋊C4—62no—
125C2×C22⋊C4—64no—
126C2×C22⋊C4—62no—
127C2×C42#6712yes—
128C2×C42C22×D311yescharacteristic
129#106—64no—
130C4⋊C8—22no—
131C4⋊C8—62no—
132#103—64no—
133#102—64no—
134C22⋊C8—62no—
135C4×C8—34no—
136#95—22no—
137C23×C3#12011yescharacteristic
138C22×D3—24no—
139C22×D3—44no—
140C22×D3—42no—
141C22×D3—42no—
142C22×D3—44no—
143C3×Q8—44no—
144C2×C3×C4—24no—
145C2×C3×C4—42no—
146C2×C3×C4—44no—
147C2×C3×C4—42no—
148C2×C3×C4#12012yes—
149C2×C3×C4#12112yes—
150C2×C3×C4#12111yescharacteristic
151C2×C3×C4#12711yescharacteristic
152C2×C3×C4C2×SD1612yes—
153C2×C3×C4AGL(1,ℤ/8)12yes—
154C2×Dic3—42no—
155C2×Dic3—44no—
156D12—24no—
157D12—24no—
158D12—42no—
159D12—42no—
160D12—416no—
161C4×D3—84no—
162C4×D3—82no—
163C4×D3—82no—
164C3×C8—81no—
165C3⋊C8—44no—
166C24—62no—
167C2×Q8—22no—
168C2×D4—62no—
169C2×D4—62no—
170C2×D4—124no—
171C2×D4—124no—
172C2×D4—128no—
173C22×C4—22no—
174C22×C4—62no—
175C22×C4—122no—
176C22×C4—122no—
177C22×C4—124no—
178C22×C4—124no—
179C22×C4D3×D412yes—
180C22×C4C2×#6711yescharacteristic
181SD16—128no—
182M16—42no—
183C2×C8—41no—
184C2×C8—64no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4—62no—
189C4⋊C4—62no—
190C4⋊C4—124no—
191C22⋊C4—41no—
192C22⋊C4—128no—
193C22⋊C4—122no—
194C22⋊C4—122no—
195C42—22no—
196C42—41no—
197C42#21514yescomplemented
198C42C2×D1211yescharacteristic
199C42#23511yescharacteristic
200C42D3×D412yes—
201C22×C3—22no—
202C22×C3#39211yescharacteristic
203C22×C3#45012yes—
204C2×D3—48no—
205C2×D3—44no—
206C2×D3—84no—
207C2×D3—82no—
208C2×D3—82no—
209C3×C4—22no—
210C3×C4—22no—
211C3×C4—44no—
212C3×C4—82no—
213C3×C4#46012yes—
214C3×C4#49211yescharacteristic
215C3×C4#49611yescharacteristic
216Dic3—82no—
217C23—64no—
218C23—124no—
219C23—122no—
220C23—122no—
221C23—124no—
222C23#93711yescharacteristic
223Q8—44no—
224D4—64no—
225D4—64no—
226D4—122no—
227D4—122no—
228D4—1216no—
229C2×C4—24no—
230C2×C4—42no—
231C2×C4—44no—
232C2×C4—42no—
233C2×C4—122no—
234C2×C4—124no—
235C2×C4—244no—
236C2×C4—242no—
237C2×C4—242no—
238C2×C4#88212yes—
239C2×C4#88412yes—
240C2×C4#93011yescharacteristic
241C2×C4#94011yescharacteristic
242C2×C4C2×#21512yes—
243C2×C4#94912yes—
244C8—81no—
245C8—124no—
246C2×C3—22no—
247C2×C3#201012yes—
248C2×C3#201411yescharacteristic
249D3—84no—
250C22—22no—
251C22—128no—
252C22—124no—
253C22—244no—
254C22—242no—
255C22—242no—
256C22#496011yescharacteristic
257C22#504612yes—
258C4—22no—
259C4—22no—
260C4—44no—
261C4—82no—
262C4—242no—
263C4#502712yes—
264C4#505811yescharacteristic
265C4#508511yescharacteristic
266C3#1001511yescharacteristic, complemented, Sylow
267C2—22no—
268C2—244no—
269C2#6968811yescharacteristic
270C2#7040612yes—
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1995C2
socleC22×C3#392
prime radical#73418C2
hypercenter#1995D3
hypocenterC3#10015
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1995

Complements

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

NH ≅ G/Ntype
1. #83351C2semidirect
3. C3×#1995C2semidirect
4. #70496C2semidirect
5. #67614C2semidirect
24. #1995D3semidirect
31. C3×C4⋊Q8D4semidirect
56. C4×C3⋊C8D4semidirect
95. C3×C42SD16semidirect
112. C4⋊Q8D12semidirect
197. C42#215semidirect
266. C3#10015semidirect

Maximal subgroups

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

subgroupindex
#833512
#734182
C3×#19952
#704962
#676142
#100153