Finite Group Databasebeta

← #11713103

Subgroups — #11713103

Order 768.

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

normal subgroups only

This group has 436 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×#5391C211yesmaximal, characteristic
2C4⋊C4×SL(2,𝔽3)C211yesmaximal, characteristic
3#73063C212yesmaximal
4#73059C212yesmaximal
5C4×#859C211yesmaximal, characteristic
6#22971—31nomaximal, Sylow
7C2×C4×SL(2,𝔽3)C2212yes—
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×#859C2211yescharacteristic
10C2×#859C2212yes—
11C2×#859C2211yescharacteristic
12#5391C414yescomplemented
13#4972—41nomaximal
14C2×#477—31no—
15#2369—31no—
16Q8×C4⋊C4D311yescharacteristic
17#1990—31no—
18#1948—31no—
19#1943—32no—
20#1918—32no—
21#1865—32no—
22#1770—31no—
23#1763—32no—
24C4×#103—31no—
25C22×SL(2,𝔽3)C2311yescharacteristic
26C2×C3×C4⋊C4—41no—
27C2×C4⋊Dic3—41no—
28C2×Dic3⋊C4—42no—
29C2×C4×Dic3—41no—
30C4×SL(2,𝔽3)—22no—
31C4×SL(2,𝔽3)C2×C412yes—
32#859—21no—
33#859C2×C414yes—
34#859Q812yescomplemented
35C3⋊#95—42no—
36C2×C4⋊Q8—31no—
37C2×C4×Q8C2×D312yes—
38C2×C4×Q8C2×D311yescharacteristic
39#477—34no—
40#477—34no—
41C2×#107—32no—
42C2×#106—32no—
43C2×C4⋊C8—31no—
44C2×#103—32no—
45C2×#103—31no—
46C2×#103—31no—
47C2×#97—32no—
48C2×C4×C8—31no—
49#391—62no—
50#389—32no—
51#389—32no—
52#389—61no—
53#384—31no—
54C4×C4⋊C4—31no—
55C4×C4⋊C4—32no—
56C4×C4⋊C4—31no—
57C4×C4⋊C4—61no—
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—42no—
68C22×Dic3—41no—
69C2×SL(2,𝔽3)C22×C411yescharacteristic
70C2×SL(2,𝔽3)C2×Q811yescharacteristic
71C2×SL(2,𝔽3)#4111yescharacteristic
72C3×C4⋊C4—82no—
73C4⋊Dic3—44no—
74Dic3⋊C4—84no—
75C4×Dic3—82no—
76C22×Q8—61no—
77C22×Q8C22×D311yescharacteristic
78C22×C8—32no—
79C22×C8—31no—
80C22×C8—31no—
81C4⋊Q8—34no—
82C4⋊Q8—62no—
83C4×Q8—22no—
84C4×Q8—32no—
85C4×Q8—62no—
86C4×Q8C4×D312yes—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—32no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—61no—
94C2×C4⋊C4—61no—
95C2×C4⋊C4—62no—
96C2×C4⋊C4S411yescharacteristic
97C2×C42—31no—
98C2×C42—32no—
99C2×C42—61no—
100C2×C42—62no—
101#107—38no—
102#106—122no—
103C4⋊C8—34no—
104#103—32no—
105#103—34no—
106#103—34no—
107#103—32no—
108#103—61no—
109#103—61no—
110#97—122no—
111C4×C8—62no—
112#95—32no—
113#95—61no—
114#95—62no—
115#95—62no—
116C23×C3—41no—
117C2×C3×C4—42no—
118C2×C3×C4—42no—
119C2×C3×C4—42no—
120C2×C3×C4—82no—
121C2×C3×C4—82no—
122C2×C3×C4—82no—
123C2×Dic3—44no—
124C2×Dic3—42no—
125C2×Dic3—42no—
126C2×Dic3—81no—
127C2×Dic3—81no—
128C2×Dic3—81no—
129C2×Dic3—82no—
130C2×Dic3—81no—
131C2×Dic3—82no—
132SL(2,𝔽3)C4×Q811yescharacteristic
133C2×Q8—31no—
134C2×Q8—31no—
135C2×Q8—31no—
136C2×Q8—64no—
137C2×Q8—241no—
138C2×Q8—241no—
139C2×Q8C2×C4×D311yescharacteristic
140C2×Q8#23711yescharacteristic
141C2×Q8D3×Q811yescharacteristic
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—32no—
145C22×C4—61no—
146C22×C4—62no—
147C22×C4—61no—
148C22×C4C2×S412yes—
149C22×C4C2×S411yescharacteristic
150C2×C8—34no—
151C2×C8—32no—
152C2×C8—61no—
153C2×C8—61no—
154C2×C8—62no—
155C2×C8—62no—
156C2×C8—61no—
157C2×C8—61no—
158C2×C8—61no—
159C4⋊C4—22no—
160C4⋊C4—32no—
161C4⋊C4—34no—
162C4⋊C4—32no—
163C4⋊C4—61no—
164C4⋊C4—62no—
165C4⋊C4—62no—
166C4⋊C4—61no—
167C4⋊C4—62no—
168C4⋊C4—61no—
169C4⋊C4—61no—
170C4⋊C4—64no—
171C4⋊C4—62no—
172C4⋊C4—242no—
173C4⋊C4—241no—
174C42—32no—
175C42—32no—
176C42—62no—
177C42—62no—
178C42—122no—
179C42—242no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3—41no—
183C22×C3—41no—
184C22×C3—41no—
185C22×C3—41no—
186C22×C3—41no—
187C3×C4—44no—
188C3×C4—162no—
189Dic3—84no—
190Dic3—82no—
191Dic3—82no—
192C23C22×S411yescharacteristic
193Q8—31no—
194Q8—31no—
195Q8—31no—
196Q8—61no—
197Q8—244no—
198Q8#88711yescharacteristic
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—31no—
203C2×C4—31no—
204C2×C4—31no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—32no—
209C2×C4—61no—
210C2×C4—61no—
211C2×C4—64no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—121no—
217C2×C4—121no—
218C2×C4—122no—
219C2×C4—122no—
220C2×C4—121no—
221C2×C4—121no—
222C2×C4—121no—
223C2×C4C4×S412yes—
224C2×C4#98312yes—
225C2×C4#98412yes—
226C8—64no—
227C8—62no—
228C8—121no—
229C2×C3—41no—
230C2×C3—41no—
231C2×C3—41no—
232C2×C3—41no—
233C2×C3—41no—
234C2×C3—41no—
235C2×C3—41no—
236C22C2×C4×S411yescharacteristic
237C22#591411yescharacteristic
238C22C2×#98311yescharacteristic
239C22Q8×S411yescharacteristic
240C22C2×#98411yescharacteristic
241C22#592511yescharacteristic
242C22#592611yescharacteristic
243C4—31no—
244C4—31no—
245C4—31no—
246C4—31no—
247C4—42no—
248C4—62no—
249C4—124no—
250C4—122no—
251C4—122no—
252C4—241no—
253C4#538814yes—
254C3—41noSylow
255C2#8530211yescharacteristic
256C2C2×#538811yescharacteristic
257C2#8536611yescharacteristic
258C2#8540011yescharacteristic
259C2#8540611yescharacteristic
260C2#8541511yescharacteristic
261C2#8543111yescharacteristic
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. #5391C4semidirect
34. #859Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#53912
C4⋊C4×SL(2,𝔽3)2
#730632
#730592
C4×#8592
#229713
#49724