Finite Group Databasebeta

← #11713636

Subgroups — #11713636

Order 768.

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

normal subgroups only

This group has 915 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#5421C214yesmaximal, complemented
3C2×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C4×#859C211yesmaximal, characteristic, complemented
5#25756—31nomaximal, Sylow
6C23×SL(2,𝔽3)C2212yes—
7C22×GL(2,𝔽3)C2212yes—
8C2×#940—41nomaximal
9D4×SL(2,𝔽3)—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11#5421—28no—
12C2×#859C2212yes—
13C4⋊GL(2,𝔽3)—22no—
14C2×D4×Q8D311yescharacteristic, complemented
15C2×#492—31no—
16C2×#460—31no—
17C2×#448—34no—
18#2541—31no—
19#2000—34no—
20#1981—31no—
21#1954—31no—
22C4×#103—31no—
23C22×C3×D4—41no—
24C22×#67—44no—
25C22×D12—41no—
26C22×SL(2,𝔽3)—24no—
27C22×SL(2,𝔽3)—22no—
28C22×SL(2,𝔽3)C2311yescharacteristic
29C2×GL(2,𝔽3)—22no—
30C2×GL(2,𝔽3)—24no—
31#940—48no—
32C2×C4×Dic3—41no—
33C4×SL(2,𝔽3)D412yescomplemented
34#859D414yescomplemented
35C23×Q8C2×D312yes—
36C22×SD16—32no—
37C22×SD16—34no—
38D4×Q8—22no—
39D4×Q8—62no—
40C2×C4⋊Q8—31no—
41C2×#127—31no—
42C2×#127—61no—
43C2×C22⋊Q8—32no—
44C2×#121—34no—
45C2×C4×Q8C2×D311yescharacteristic
46C2×C4×D4—31no—
47#492—64no—
48#460—62no—
49#460—62no—
50#448—68no—
51#448—68no—
52C2×C4⋊C8—31no—
53C2×#103—32no—
54C2×#103—34no—
55C2×#102—32no—
56C2×C22⋊C8—34no—
57C2×C4×C8—31no—
58#392—62no—
59#390—61no—
60C4×C4⋊C4—31no—
61#340—32no—
62C24×C3—42no—
63C23×D3—42no—
64C2×C3×D4—44no—
65C2×C3×D4—82no—
66C2×C3×D4—82no—
67C22×C3×C4—41no—
68C2×#67—416no—
69C2×#67—88no—
70C2×#67—88no—
71C22×Dic3—42no—
72C2×D12—44no—
73C2×D12—82no—
74C2×D12—82no—
75C2×SL(2,𝔽3)—42no—
76C2×SL(2,𝔽3)C2×D412yes—
77C2×SL(2,𝔽3)C2×D411yescharacteristic
78GL(2,𝔽3)—42no—
79C4×Dic3—44no—
80C22×Q8—24no—
81C22×Q8—22no—
82C22×Q8—32no—
83C22×Q8—62no—
84C22×Q8—64no—
85C22×Q8C22×D311yescharacteristic
86C22×D4—32no—
87C22×D4—61no—
88C22×D4—64no—
89C22×D4—61no—
90C22×D4S411yescharacteristic
91C23×C4—32no—
92C2×SD16—68no—
93C2×SD16—64no—
94C2×SD16—62no—
95C2×SD16—68no—
96C2×SD16—64no—
97C2×SD16—62no—
98C2×SD16—124no—
99C22×C8—32no—
100C22×C8—32no—
101C4⋊Q8—34no—
102C4⋊Q8—62no—
103#127—62no—
104#127—62no—
105#127—124no—
106C22⋊Q8—64no—
107C22⋊Q8—64no—
108#121—68no—
109#121—124no—
110C4×Q8—32no—
111C4×Q8#6712yescomplemented
112C4×D4—62no—
113C4×D4—62no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—61no—
118C2×C22⋊C4—32no—
119C2×C22⋊C4—64no—
120C2×C42—31no—
121C2×C42—61no—
122C4⋊C8—62no—
123#103—34no—
124#103—34no—
125#103—68no—
126#102—64no—
127C22⋊C8—68no—
128C4×C8—62no—
129#95—61no—
130C23×C3—44no—
131C23×C3—41no—
132C23×C3—84no—
133C23×C3—82no—
134C23×C3—84no—
135C23×C3—82no—
136C22×D3—44no—
137C22×D3—82no—
138C22×D3—84no—
139C22×D3—82no—
140C22×D3—84no—
141C3×D4—88no—
142C2×C3×C4—42no—
143C2×C3×C4—42no—
144C2×C3×C4—42no—
145#67—832no—
146C2×Dic3—44no—
147C2×Dic3—44no—
148C2×Dic3—44no—
149D12—88no—
150SL(2,𝔽3)#12711yescharacteristic, complemented
151C24—62no—
152C24C2×S412yes—
153C2×Q8—32no—
154C2×Q8—31no—
155C2×Q8—34no—
156C2×Q8—42no—
157C2×Q8—64no—
158C2×Q8—64no—
159C2×Q8—62no—
160C2×Q8—124no—
161C2×Q8—124no—
162C2×Q8—122no—
163C2×Q8D3×D412yes—
164C2×Q8C2×#6711yescharacteristic
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—62no—
168C2×D4—64no—
169C2×D4—64no—
170C2×D4—62no—
171C2×D4—122no—
172C2×D4—128no—
173C2×D4—128no—
174C2×D4—122no—
175C2×D4—122no—
176C2×D4—122no—
177C2×D4—242no—
178C2×D4—244no—
179C2×D4GL(2,𝔽3)14yescomplemented
180C22×C4—31no—
181C22×C4—31no—
182C22×C4—32no—
183C22×C4—64no—
184C22×C4—62no—
185C22×C4—62no—
186C22×C4—64no—
187C22×C4—62no—
188C22×C4C2×S411yescharacteristic
189SD16—1216no—
190SD16—124no—
191SD16—122no—
192SD16—122no—
193C2×C8—34no—
194C2×C8—62no—
195C2×C8—62no—
196C2×C8—64no—
197C2×C8—62no—
198C4⋊C4—32no—
199C4⋊C4—34no—
200C4⋊C4—61no—
201C4⋊C4—62no—
202C4⋊C4—64no—
203C4⋊C4—62no—
204C22⋊C4—64no—
205C22⋊C4—128no—
206C42—32no—
207C42—32no—
208C42—122no—
209C22×C3—42no—
210C22×C3—41no—
211C22×C3—41no—
212C22×C3—42no—
213C22×C3—41no—
214C22×C3—88no—
215C22×C3—84no—
216C22×C3—162no—
217C22×C3—164no—
218C22×C3—162no—
219C2×D3—84no—
220C2×D3—88no—
221C2×D3—162no—
222C2×D3—164no—
223C2×D3—162no—
224C3×C4—44no—
225Dic3—48no—
226C23—24no—
227C23—22no—
228C23—24no—
229C23—22no—
230C23—124no—
231C23—122no—
232C23—122no—
233C23—124no—
234C23—122no—
235C23C2×GL(2,𝔽3)14yes—
236C23C22×S411yescharacteristic
237Q8—32no—
238Q8—31no—
239Q8—61no—
240Q8—68no—
241Q8—122no—
242Q8—122no—
243Q8#94011yescharacteristic, complemented
244D4—28no—
245D4—124no—
246D4—122no—
247D4—122no—
248D4—244no—
249D4—2416no—
250D4—244no—
251C2×C4—32no—
252C2×C4—32no—
253C2×C4—31no—
254C2×C4—31no—
255C2×C4—32no—
256C2×C4—62no—
257C2×C4—64no—
258C2×C4—64no—
259C2×C4—62no—
260C2×C4—122no—
261C2×C4—124no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4C2×GL(2,𝔽3)12yes—
266C2×C4#98312yes—
267C2×C4GL(2,ℤ/4)12yes—
268C8—64no—
269C8—122no—
270C2×C3—41no—
271C2×C3—41no—
272C2×C3—42no—
273C2×C3—42no—
274C2×C3—41no—
275C2×C3—164no—
276D3—164no—
277C22—24no—
278C22—28no—
279C22—42no—
280C22—44no—
281C22—42no—
282C22—242no—
283C22—242no—
284C22—244no—
285C22—244no—
286C22—242no—
287C22D4×S412yes—
288C22C22×GL(2,𝔽3)11yescharacteristic
289C22C2×#98311yescharacteristic
290C22#592612yes—
291C22C2×GL(2,ℤ/4)11yescharacteristic
292C4—32no—
293C4—31no—
294C4—31no—
295C4—62no—
296C4—124no—
297C4—122no—
298C4#542114yescomplemented
299C3—41noSylow
300C2—44no—
301C2—482no—
302C2#8539212yes—
303C2D4×GL(2,𝔽3)12yes—
304C2#8549611yescharacteristic
305C2C2×#542111yescharacteristic
306C2#8554011yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×D4×Q8D3
hypercenterC22×D4S4
hypocenterSL(2,𝔽3)#127
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×SL(2,𝔽3)C2semidirect
2. C2×#5421C2semidirect
3. C2×C4⋊GL(2,𝔽3)C2semidirect
4. C4×#859C2semidirect
14. C2×D4×Q8D3semidirect
33. C4×SL(2,𝔽3)D4semidirect
34. #859D4semidirect
111. C4×Q8#67semidirect
150. SL(2,𝔽3)#127semidirect
179. C2×D4GL(2,𝔽3)semidirect
243. Q8#940semidirect
298. C4#5421semidirect

Maximal subgroups

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

subgroupindex
C2×D4×SL(2,𝔽3)2
C2×#54212
C2×C4⋊GL(2,𝔽3)2
C4×#8592
#257563
C2×#9404