Finite Group Databasebeta

← #11713671

Subgroups — #11713671

Order 768.

451 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 451 conjugacy classes of subgroups — too many to draw. Only its 55 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#3503⋊C3C211yesmaximal, characteristic
2#85337C211yesmaximal, characteristic
3#73076C214yesmaximal
4C4×GU(2,𝔽3)C211yesmaximal, characteristic
5#25830—31nomaximal, Sylow
6Q8×SL(2,𝔽3)—21no—
7#557⋊C3—21no—
8#520⋊C3C2212yes—
9C4×#41⋊C3C2211yescharacteristic
10#5426C2212yes—
11C2×GU(2,𝔽3)C2212yes—
12#5391—21no—
13#5386—21no—
14#5080—41nomaximal
15#3503D311yescharacteristic
16#3165—31no—
17#3079—31no—
18#2868—31no—
19#2060—34no—
20#1997—31no—
21#1994—31no—
22#1859—34no—
23C4×#104—31no—
24C2×#41⋊C3C2311yescharacteristic
25C3×C4⋊Q8—41no—
26#869—41no—
27C4×SL(2,𝔽3)—22no—
28C4×SL(2,𝔽3)D411yescharacteristic
29C4×SL(2,𝔽3)D411yescharacteristic
30GU(2,𝔽3)Q814yescomplemented
31#859—22no—
32#808—44no—
33C4×C3⋊C8—41no—
34Q82—21no—
35#557—21no—
36#554—61no—
37D4×Q8—61no—
38#544—32no—
39#533—31no—
40C2×C4⋊Q8—31no—
41C2×C4⋊Q8—31no—
42C2×C4⋊Q8—62no—
43#520C2×D312yes—
44C4×#41C2×D311yescharacteristic
45#501—62no—
46#500—61no—
47#498—61no—
48#477—61no—
49#476—61no—
50#475—61no—
51#474—61no—
52#428—32no—
53#428—34no—
54#423—31no—
55C2×#104—32no—
56#417—32no—
57C4×M16—31no—
58#384—34no—
59#384—62no—
60C43—31no—
61#343—34no—
62#339—32no—
63#339—34no—
64C2×C3×Q8—81no—
65C2×Dic6—81no—
66#41⋊C3C2×Q812yes—
67C2×SL(2,𝔽3)C2×D411yescharacteristic
68C3×C4⋊C4—42no—
69C3×C4⋊C4—81no—
70C3×C42—41no—
71C4⋊Dic3—42no—
72C4⋊Dic3—81no—
73C2×C3⋊C8—42no—
74C2×#41C22×D311yescharacteristic
75C22×Q8—32no—
76C22×Q8—61no—
77C2×M16—32no—
78C2×M16—32no—
79C4⋊Q8—31no—
80C4⋊Q8—61no—
81C4⋊Q8—61no—
82C4⋊Q8—61no—
83C4⋊Q8—61no—
84C4⋊Q8—61no—
85C4⋊Q8—61no—
86C4⋊Q8—62no—
87C4⋊Q8—61no—
88C4⋊Q8—128no—
89C4⋊Q8S411yescharacteristic
90#125—32no—
91#125—62no—
92C22⋊Q8—61no—
93C22⋊Q8—61no—
94C22⋊Q8—64no—
95C4×Q8—22no—
96C4×Q8—61no—
97C4×Q8—61no—
98C4×Q8#6711yescharacteristic
99C4×Q8#6711yescharacteristic
100C4×D4—31no—
101C4×D4—31no—
102C4×D4—62no—
103#117—31no—
104#117—32no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4—32no—
107C2×C4⋊C4—32no—
108C2×C4⋊C4—61no—
109C2×C4⋊C4—68no—
110C2×C42—31no—
111C2×C42—32no—
112C2×C42—61no—
113C2×C42—61no—
114#107—62no—
115#107—64no—
116#106—62no—
117#106—64no—
118C4⋊C8—61no—
119C4⋊C8—61no—
120#104—34no—
121#104—34no—
122#103—62no—
123#102—62no—
124#97—61no—
125C4×C8—61no—
126C3×Q8—162no—
127C2×C3×C4—41no—
128C2×C3×C4—41no—
129C2×C3×C4—41no—
130C2×C3×C4—82no—
131C2×Dic3—82no—
132Dic6—162no—
133SL(2,𝔽3)C4⋊Q811yescharacteristic, complemented
134C3⋊C8—44no—
135#41—32no—
136#41D3×Q812yes—
137C2×Q8—21no—
138C2×Q8—62no—
139C2×Q8—61no—
140C2×Q8—121no—
141C2×Q8—124no—
142C2×Q8—124no—
143C2×Q8—121no—
144C2×Q8C2×#6711yescharacteristic
145C2×D4—31no—
146C22×C4—31no—
147C22×C4—31no—
148C22×C4—31no—
149C22×C4—34no—
150C22×C4—62no—
151C22×C4—62no—
152M16—34no—
153M16—62no—
154M16—64no—
155C2×C8—62no—
156C2×C8—62no—
157C4⋊C4—21no—
158C4⋊C4—31no—
159C4⋊C4—31no—
160C4⋊C4—32no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—62no—
165C4⋊C4—62no—
166C4⋊C4—61no—
167C4⋊C4—62no—
168C4⋊C4—121no—
169C4⋊C4—1216no—
170C4⋊C4—121no—
171C4⋊C4C2×S412yes—
172C22⋊C4—62no—
173C22⋊C4—61no—
174C42—31no—
175C42—31no—
176C42—31no—
177C42—34no—
178C42—62no—
179C42—62no—
180C42—64no—
181C42—64no—
182C42C2×S411yescharacteristic
183C22×C3—41no—
184C3×C4—42no—
185C3×C4—42no—
186C3×C4—42no—
187C3×C4—162no—
188Dic3—162no—
189C23—31no—
190Q8—31no—
191Q8—42no—
192Q8—122no—
193Q8—242no—
194Q8—242no—
195Q8—242no—
196Q8#94411yescharacteristic
197D4—32no—
198D4—61no—
199C2×C4—22no—
200C2×C4—31no—
201C2×C4—32no—
202C2×C4—32no—
203C2×C4—31no—
204C2×C4—31no—
205C2×C4—64no—
206C2×C4—64no—
207C2×C4—64no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—124no—
212C2×C4—124no—
213C2×C4—122no—
214C2×C4GL(2,ℤ/4)11yescharacteristic
215C2×C4GL(2,ℤ/4)11yescharacteristic
216C2×C4C22×S411yescharacteristic
217C8—64no—
218C8—122no—
219C2×C3—41no—
220C2×C3—41no—
221C2×C3—41no—
222C22—32no—
223C22—61no—
224C22—61no—
225C22C2×GL(2,ℤ/4)11yescharacteristic
226C4—31no—
227C4—31no—
228C4—42no—
229C4—68no—
230C4—61no—
231C4—61no—
232C4—122no—
233C4—242no—
234C4#542912yes—
235C4#543012yes—
236C4Q8×S412yes—
237C3—41noSylow
238C2—62no—
239C2#8550211yescharacteristic
240C2C2×#542911yescharacteristic
241C2C2×#543011yescharacteristic
subgroupG/H
center, socleC22C2×GL(2,ℤ/4)
commutator, prime radical, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#3503D3
hypercenterC4⋊Q8S4
hypocenterSL(2,𝔽3)C4⋊Q8
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
30. GU(2,𝔽3)Q8semidirect
133. 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
#3503⋊C32
#853372
#730762
C4×GU(2,𝔽3)2
#258303
#50804