Finite Group Databasebeta

← #11714093

Subgroups — #11714093

Order 768.

655 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 655 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#5395C214yesmaximal, complemented
2C2×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
3C2×C4⋊#226C211yesmaximal, characteristic, complemented
4C42×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
5#33165—31nomaximal, Sylow
6C22×GL(2,𝔽3)C2212yes—
7C22×#226C2212yes—
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×C4×SL(2,𝔽3)C2212yes—
10#5395—28no—
11C4⋊GL(2,𝔽3)—22no—
12C4⋊#226—22no—
13#4939—41nomaximal
14C2×#464—34no—
15C2×#462—31no—
16C2×#460—31no—
17#2858—31no—
18C42×Q8D311yescharacteristic, complemented
19#1983—34no—
20#1982—31no—
21#1981—31no—
22#1842—31no—
23C22×D12—41no—
24C22×SL(2,𝔽3)C2311yescharacteristic
25C2×GL(2,𝔽3)—22no—
26C2×GL(2,𝔽3)—24no—
27C2×#226—22no—
28C2×#226—24no—
29C2×C3×C42—41no—
30C2×(C2×D3)⋊C4—44no—
31C2×C4⋊Dic3—41no—
32C4×SL(2,𝔽3)D412yes—
33C4×SL(2,𝔽3)D414yescomplemented
34C22×Q16—32no—
35C22×SD16—32no—
36C2×C4⋊Q8—31no—
37C2×#127—31no—
38C2×#124—34no—
39C2×C4×Q8C2×D311yescharacteristic
40C2×C4×Q8C2×D312yes—
41#464—68no—
42#464—68no—
43#462—62no—
44#462—62no—
45#460—62no—
46#460—62no—
47C2×C4⋊C8—31no—
48C2×C4⋊C8—31no—
49C2×C4⋊C8—34no—
50C2×#103—34no—
51C2×#103—32no—
52C2×#102—34no—
53C2×#102—32no—
54#390—61no—
55#390—62no—
56#390—61no—
57C4×C4⋊C4—31no—
58C4×C4⋊C4—32no—
59C43—31no—
60C23×D3—42no—
61C22×C3×C4—41no—
62C22×C3×C4—42no—
63C22×Dic3—42no—
64C2×D12—44no—
65C2×D12—82no—
66C2×D12—82no—
67C2×SL(2,𝔽3)C2×D411yescharacteristic
68C2×SL(2,𝔽3)C2×D412yes—
69GL(2,𝔽3)—42no—
70#226—42no—
71C3×C42—82no—
72(C2×D3)⋊C4—88no—
73C4⋊Dic3—82no—
74C22×Q8—32no—
75C22×Q8C22×D311yescharacteristic
76C22×D4—32no—
77C22×D4—61no—
78C22×D4—61no—
79C2×Q16—62no—
80C2×Q16—64no—
81C2×Q16—62no—
82C2×Q16—64no—
83C2×SD16—62no—
84C2×SD16—64no—
85C2×SD16—62no—
86C2×SD16—64no—
87C22×C8—32no—
88C22×C8—32no—
89C4⋊Q8—62no—
90C4⋊Q8—62no—
91#127—62no—
92#127—62no—
93#124—68no—
94#124—68no—
95C4×Q8—32no—
96C4×Q8—34no—
97C4×Q8D1212yes—
98C4×Q8D1214yescomplemented
99C2×C4⋊C4—31no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—61no—
102C2×C4⋊C4—61no—
103C2×C22⋊C4—64no—
104C2×C22⋊C4—64no—
105C2×C42—32no—
106C2×C42—31no—
107C2×C42—31no—
108C2×C42—32no—
109C2×C42S411yescharacteristic
110C4⋊C8—62no—
111C4⋊C8—62no—
112C4⋊C8—68no—
113#103—64no—
114#103—68no—
115#102—64no—
116#102—68no—
117#95—61no—
118C23×C3—41no—
119C22×D3—44no—
120C22×D3—84no—
121C22×D3—82no—
122C22×D3—82no—
123C22×D3—84no—
124C2×C3×C4—42no—
125C2×C3×C4—44no—
126C2×C3×C4—42no—
127C2×C3×C4—42no—
128C2×C3×C4—82no—
129C2×C3×C4—82no—
130C2×Dic3—82no—
131C2×Dic3—84no—
132D12—88no—
133SL(2,𝔽3)#12711yescharacteristic
134C24—62no—
135C2×Q8—31no—
136C2×Q8—32no—
137C2×Q8—64no—
138C2×Q8—62no—
139C2×Q8—64no—
140C2×Q8—62no—
141C2×Q8C2×D1211yescharacteristic
142C2×Q8C2×D1212yes—
143C2×D4—64no—
144C2×D4—62no—
145C2×D4—64no—
146C2×D4—62no—
147C2×D4—122no—
148C2×D4—124no—
149C2×D4—122no—
150C2×D4—122no—
151C2×D4—122no—
152C22×C4—31no—
153C22×C4—32no—
154C22×C4—31no—
155C22×C4—62no—
156C22×C4C2×S412yes—
157C22×C4C2×S411yescharacteristic
158Q16—124no—
159Q16—122no—
160Q16—122no—
161SD16—124no—
162SD16—122no—
163SD16—122no—
164C2×C8—62no—
165C2×C8—64no—
166C2×C8—64no—
167C2×C8—62no—
168C4⋊C4—32no—
169C4⋊C4—34no—
170C4⋊C4—62no—
171C4⋊C4—61no—
172C4⋊C4—122no—
173C4⋊C4—122no—
174C22⋊C4—128no—
175C22⋊C4—128no—
176C42—22no—
177C42—32no—
178C42—34no—
179C42—34no—
180C42—32no—
181C42—62no—
182C42—122no—
183C22×C3—41no—
184C22×C3—42no—
185C22×C3—41no—
186C22×C3—41no—
187C22×C3—42no—
188C2×D3—84no—
189C2×D3—88no—
190C2×D3—164no—
191C2×D3—162no—
192C2×D3—162no—
193C3×C4—44no—
194C3×C4—84no—
195Dic3—162no—
196C23—124no—
197C23—122no—
198C23—122no—
199C23—122no—
200C23—124no—
201C23C22×S411yescharacteristic
202Q8—31no—
203Q8—32no—
204Q8—61no—
205Q8—122no—
206Q8—124no—
207Q8—122no—
208Q8#87411yescharacteristic
209D4—122no—
210D4—124no—
211D4—122no—
212D4—244no—
213D4—244no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—31no—
217C2×C4—32no—
218C2×C4—31no—
219C2×C4—32no—
220C2×C4—34no—
221C2×C4—32no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—124no—
226C2×C4—122no—
227C2×C4C4⋊S414yes—
228C2×C4C4⋊S412yes—
229C2×C4#98312yes—
230C2×C4#98512yes—
231C8—122no—
232C8—122no—
233C2×C3—41no—
234C2×C3—41no—
235C2×C3—42no—
236C2×C3—42no—
237C2×C3—41no—
238D3—164no—
239C22—242no—
240C22—242no—
241C22—244no—
242C22—244no—
243C22—242no—
244C22C2×C4⋊S411yescharacteristic
245C22C2×C4⋊S412yes—
246C22C2×#98311yescharacteristic
247C22C2×#98511yescharacteristic
248C22#592312yes—
249C4—24no—
250C4—31no—
251C4—31no—
252C4—32no—
253C4—64no—
254C4—62no—
255C4—242no—
256C4#539514yescomplemented
257C3—41noSylow
258C2—482no—
259C2#8528911yescharacteristic
260C2C2×#539511yescharacteristic
261C2#8533112yes—
262C2#8534412yes—
263C2#8534711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC42×Q8D3
prime radicalC2×C4⋊GL(2,𝔽3)C2
hypercenterC2×C42S4
hypocenterSL(2,𝔽3)#127
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×#5395C2semidirect
3. C2×C4⋊#226C2semidirect
4. C42×SL(2,𝔽3)C2semidirect
18. C42×Q8D3semidirect
33. C4×SL(2,𝔽3)D4semidirect
98. C4×Q8D12semidirect
256. C4#5395semidirect

Maximal subgroups

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

subgroupindex
C2×#53952
C2×C4⋊GL(2,𝔽3)2
C2×C4⋊#2262
C42×SL(2,𝔽3)2
#331653
#49394