Finite Group Databasebeta

← #11713102

Subgroups — #11713102

Order 768.

458 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 73 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×#5386C211yesmaximal, characteristic
2C4⋊C4×SL(2,𝔽3)C211yesmaximal, characteristic
3#73063C212yesmaximal
4#73062C212yesmaximal
5C4×#859C211yesmaximal, characteristic
6#22970—31nomaximal, Sylow
7C2×C4×SL(2,𝔽3)C2212yes—
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×#859C2211yescharacteristic
10C2×#859C2212yes—
11C2×#859C2211yescharacteristic
12#5386C414yescomplemented
13#4973—41nomaximal
14C2×#475—31no—
15#2365—31no—
16Q8×C4⋊C4D311yescharacteristic
17#1990—31no—
18#1947—31no—
19#1944—32no—
20#1919—32no—
21#1863—32no—
22#1770—31no—
23#1763—32no—
24C4×#103—31no—
25C22×Dic6—41no—
26C22×SL(2,𝔽3)C2311yescharacteristic
27C2×C3×C4⋊C4—41no—
28C2×C4×Dic3—41no—
29C4×SL(2,𝔽3)—22no—
30C4×SL(2,𝔽3)C2×C412yes—
31#859—21no—
32#859C2×C414yes—
33#859Q812yescomplemented
34C3⋊#95—42no—
35C3⋊#95—42no—
36C2×C4⋊Q8—31no—
37C2×C4×Q8—61no—
38C2×C4×Q8C2×D312yes—
39C2×C4×Q8C2×D311yescharacteristic
40#475—34no—
41#475—34no—
42C2×#107—32no—
43C2×#106—32no—
44C2×C4⋊C8—31no—
45C2×#103—32no—
46C2×#103—31no—
47C2×#103—31no—
48C2×#97—32no—
49C2×C4×C8—31no—
50#391—61no—
51#389—32no—
52#389—32no—
53#387—62no—
54#384—31no—
55C4×C4⋊C4—31no—
56C4×C4⋊C4—32no—
57C4×C4⋊C4—31no—
58#340—32no—
59#340—31no—
60#340—32no—
61#340—31no—
62#340—32no—
63#336—32no—
64C22×C3×C4—41no—
65C22×C3×C4—42no—
66C22×Dic3—41no—
67C22×Dic3—41no—
68C22×Dic3—42no—
69C2×Dic6—44no—
70C2×Dic6—82no—
71C2×Dic6—82no—
72C2×SL(2,𝔽3)C22×C411yescharacteristic
73C2×SL(2,𝔽3)C2×Q811yescharacteristic
74C2×SL(2,𝔽3)#4111yescharacteristic
75C3×C4⋊C4—82no—
76C4×Dic3—82no—
77C22×Q8—61no—
78C22×Q8C22×D311yescharacteristic
79C22×C8—32no—
80C22×C8—31no—
81C22×C8—31no—
82C4⋊Q8—34no—
83C4⋊Q8—62no—
84C4×Q8—22no—
85C4×Q8—32no—
86C4×Q8—62no—
87C4×Q8—124no—
88C4×Q8C4×D312yes—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—61no—
96C2×C4⋊C4—62no—
97C2×C4⋊C4—61no—
98C2×C4⋊C4S411yescharacteristic
99C2×C42—31no—
100C2×C42—32no—
101C2×C42—62no—
102C2×C42—61no—
103#107—122no—
104#106—38no—
105C4⋊C8—34no—
106#103—32no—
107#103—34no—
108#103—34no—
109#103—32no—
110#103—61no—
111#103—61no—
112#97—122no—
113C4×C8—62no—
114#95—32no—
115#95—61no—
116#95—62no—
117#95—62no—
118C23×C3—41no—
119C2×C3×C4—42no—
120C2×C3×C4—42no—
121C2×C3×C4—42no—
122C2×C3×C4—82no—
123C2×C3×C4—82no—
124C2×C3×C4—82no—
125C2×Dic3—44no—
126C2×Dic3—42no—
127C2×Dic3—42no—
128C2×Dic3—81no—
129C2×Dic3—81no—
130C2×Dic3—81no—
131C2×Dic3—82no—
132C2×Dic3—82no—
133C2×Dic3—81no—
134Dic6—88no—
135SL(2,𝔽3)C4×Q811yescharacteristic, complemented
136C2×Q8—31no—
137C2×Q8—31no—
138C2×Q8—31no—
139C2×Q8—64no—
140C2×Q8—122no—
141C2×Q8—122no—
142C2×Q8C2×C4×D311yescharacteristic
143C2×Q8#23711yescharacteristic
144C2×Q8D3×Q811yescharacteristic
145C22×C4—31no—
146C22×C4—31no—
147C22×C4—32no—
148C22×C4—61no—
149C22×C4—62no—
150C22×C4—61no—
151C22×C4C2×S412yes—
152C22×C4C2×S411yescharacteristic
153C2×C8—34no—
154C2×C8—32no—
155C2×C8—61no—
156C2×C8—61no—
157C2×C8—62no—
158C2×C8—62no—
159C2×C8—61no—
160C2×C8—61no—
161C2×C8—61no—
162C4⋊C4—22no—
163C4⋊C4—32no—
164C4⋊C4—34no—
165C4⋊C4—32no—
166C4⋊C4—61no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—61no—
170C4⋊C4—62no—
171C4⋊C4—64no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4—62no—
175C4⋊C4—124no—
176C4⋊C4—241no—
177C42—32no—
178C42—32no—
179C42—62no—
180C42—62no—
181C42—124no—
182C42—122no—
183C22×C3—41no—
184C22×C3—41no—
185C22×C3—41no—
186C22×C3—41no—
187C22×C3—41no—
188C22×C3—41no—
189C22×C3—41no—
190C3×C4—44no—
191C3×C4—162no—
192Dic3—84no—
193Dic3—82no—
194Dic3—82no—
195C23C22×S411yescharacteristic
196Q8—31no—
197Q8—31no—
198Q8—31no—
199Q8—61no—
200Q8—128no—
201Q8#88711yescharacteristic
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—31no—
209C2×C4—31no—
210C2×C4—31no—
211C2×C4—32no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—64no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—122no—
220C2×C4—121no—
221C2×C4—122no—
222C2×C4—121no—
223C2×C4—121no—
224C2×C4—121no—
225C2×C4—121no—
226C2×C4C4×S412yes—
227C2×C4#98312yes—
228C2×C4#98612yes—
229C8—64no—
230C8—62no—
231C8—121no—
232C2×C3—41no—
233C2×C3—41no—
234C2×C3—41no—
235C2×C3—41no—
236C2×C3—41no—
237C2×C3—41no—
238C2×C3—41no—
239C22C2×C4×S411yescharacteristic
240C22#591411yescharacteristic
241C22C2×#98311yescharacteristic
242C22Q8×S411yescharacteristic
243C22C2×#98611yescharacteristic
244C22#592411yescharacteristic
245C22#592711yescharacteristic
246C4—31no—
247C4—31no—
248C4—31no—
249C4—31no—
250C4—42no—
251C4—62no—
252C4—124no—
253C4—122no—
254C4—122no—
255C4—241no—
256C4#539414yes—
257C3—41noSylow
258C2#8530211yescharacteristic
259C2C2×#539411yescharacteristic
260C2#8536011yescharacteristic
261C2#8539511yescharacteristic
262C2#8541211yescharacteristic
263C2#8542411yescharacteristic
264C2#8542611yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
FittingQ8×C4⋊C4D3
prime radical, cosocleC22×SL(2,𝔽3)C23
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)C4×Q8
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
12. #5386C4semidirect
33. #859Q8semidirect
135. SL(2,𝔽3)C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#53862
C4⋊C4×SL(2,𝔽3)2
#730632
#730622
C4×#8592
#229703
#49734