Finite Group Databasebeta

← #11713696

Subgroups — #11713696

Order 768.

533 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 63 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic
2C2×#5391C211yesmaximal, characteristic, complemented
3#73151C214yesmaximal, complemented
4C4×#859C211yesmaximal, characteristic, complemented
5#25883—31nomaximal, Sylow
6C22×#41⋊C3C2212yes—
7C4⋊(#41⋊C3)—22no—
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×#859C2212yes—
10C2×#859C2212yes—
11#5391—22no—
12#4977—41nomaximal
13C2×#550D311yescharacteristic
14C2×#486—31no—
15C2×#477—31no—
16#2563—31no—
17#2033—34no—
18#1982—31no—
19#1954—31no—
20#1788—34no—
21C4×#103—31no—
22C2×#41⋊C3—22no—
23C2×#41⋊C3—24no—
24C22×SL(2,𝔽3)C2311yescharacteristic
25C2×C3×C4⋊C4—41no—
26C2×C4⋊Dic3—41no—
27C2×C4×Dic3—41no—
28C4×SL(2,𝔽3)D412yes—
29#859—22no—
30#859—22no—
31C3⋊#95—44no—
32C22×#41C2×D312yes—
33#550—22no—
34#550—62no—
35C2×C4⋊Q8—31no—
36C2×#127—31no—
37C2×#121—32no—
38C2×C4×Q8C2×D311yescharacteristic
39C2×C4×D4—31no—
40#486—64no—
41#477—62no—
42#477—62no—
43C2×#107—32no—
44C2×C4⋊C8—31no—
45C2×#103—32no—
46C2×#103—32no—
47C2×#102—34no—
48C2×#102—34no—
49C2×C22⋊C8—34no—
50C2×C4×C8—31no—
51#393—62no—
52#390—61no—
53#382—61no—
54#380—34no—
55C4×C4⋊C4—31no—
56#340—32no—
57C22×C3×C4—41no—
58C22×C3×C4—42no—
59C22×Dic3—42no—
60C22×Dic3—42no—
61#41⋊C3—42no—
62C2×SL(2,𝔽3)C2×D411yescharacteristic
63C2×SL(2,𝔽3)#4112yes—
64C3×C4⋊C4—82no—
65C4⋊Dic3—82no—
66C4×Dic3—44no—
67C2×#41—24no—
68C2×#41—22no—
69C2×#41—62no—
70C2×#41—64no—
71C22×Q8—61no—
72C22×Q8C22×D311yescharacteristic
73C22×D4—32no—
74C22×D4—31no—
75C22×D4—61no—
76C23×C4—32no—
77C22×C8—32no—
78C22×C8—32no—
79C4⋊Q8—62no—
80C4⋊Q8—62no—
81#127—34no—
82#127—62no—
83#121—64no—
84#121—64no—
85C4×Q8—32no—
86C4×Q8#6712yes—
87C4×D4—62no—
88C4×D4—62no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—61no—
93C2×C4⋊C4S411yescharacteristic
94C2×C22⋊C4—32no—
95C2×C22⋊C4—64no—
96C2×C42—31no—
97C2×C42—61no—
98#107—64no—
99C4⋊C8—62no—
100#103—62no—
101#103—62no—
102#103—62no—
103#103—62no—
104#102—68no—
105#102—124no—
106C22⋊C8—124no—
107C4×C8—62no—
108#95—61no—
109#95—64no—
110C23×C3—41no—
111C2×C3×C4—42no—
112C2×C3×C4—42no—
113C2×C3×C4—42no—
114C2×C3×C4—82no—
115C2×C3×C4—84no—
116C2×Dic3—44no—
117C2×Dic3—82no—
118C2×Dic3—82no—
119C2×Dic3—82no—
120C2×Dic3—84no—
121SL(2,𝔽3)#12411yescharacteristic
122C24—32no—
123#41—42no—
124#41—124no—
125#41—124no—
126#41—122no—
127C2×Q8—31no—
128C2×Q8—32no—
129C2×Q8—122no—
130C2×Q8—242no—
131C2×Q8#23712yes—
132C2×Q8C2×#6711yescharacteristic
133C2×D4—34no—
134C2×D4—64no—
135C2×D4—64no—
136C2×D4—62no—
137C2×D4—64no—
138C2×D4—62no—
139C2×D4—62no—
140C2×D4—62no—
141C2×D4—242no—
142C22×C4—31no—
143C22×C4—31no—
144C22×C4—64no—
145C22×C4—62no—
146C22×C4—62no—
147C22×C4—64no—
148C22×C4—62no—
149C22×C4—62no—
150C22×C4C2×S412yes—
151C22×C4C2×S411yescharacteristic
152C2×C8—62no—
153C2×C8—62no—
154C2×C8—64no—
155C2×C8—62no—
156C2×C8—62no—
157C4⋊C4—22no—
158C4⋊C4—32no—
159C4⋊C4—61no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C4⋊C4—122no—
165C22⋊C4—64no—
166C22⋊C4—244no—
167C42—32no—
168C42—32no—
169C42—122no—
170C22×C3—42no—
171C22×C3—41no—
172C22×C3—41no—
173C22×C3—42no—
174C22×C3—41no—
175C3×C4—44no—
176C3×C4—162no—
177Dic3—84no—
178Dic3—162no—
179C23—64no—
180C23—62no—
181C23—62no—
182C23—64no—
183C23—62no—
184C23C22×S411yescharacteristic
185Q8—32no—
186Q8—31no—
187Q8—61no—
188Q8—244no—
189Q8#93611yescharacteristic
190D4—68no—
191D4—122no—
192D4—122no—
193D4—124no—
194D4—244no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—32no—
198C2×C4—32no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—32no—
202C2×C4—62no—
203C2×C4—122no—
204C2×C4—122no—
205C2×C4—124no—
206C2×C4—122no—
207C2×C4—122no—
208C2×C4—122no—
209C2×C4—124no—
210C2×C4—122no—
211C2×C4#98412yes—
212C2×C4#98512yes—
213C2×C4GL(2,ℤ/4)12yes—
214C8—122no—
215C8—122no—
216C2×C3—41no—
217C2×C3—41no—
218C2×C3—42no—
219C2×C3—42no—
220C2×C3—41no—
221C22—122no—
222C22—122no—
223C22—124no—
224C22—124no—
225C22—122no—
226C22#591412yes—
227C22C2×#98411yescharacteristic
228C22C2×#98511yescharacteristic
229C22#592512yes—
230C22C2×GL(2,ℤ/4)11yescharacteristic
231C4—31no—
232C4—32no—
233C4—31no—
234C4—42no—
235C4—62no—
236C4—242no—
237C4—242no—
238C4#542514yes—
239C3—41noSylow
240C2—242no—
241C2#8540612yes—
242C2#8540812yes—
243C2#8549211yescharacteristic
244C2C2×#542511yescharacteristic
245C2#8553011yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#550D3
prime radicalC2×C4⋊(#41⋊C3)C2
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C22×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
2. C2×#5391C2semidirect
3. #73151C2semidirect
4. C4×#859C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(#41⋊C3)2
C2×#53912
#731512
C4×#8592
#258833
#49774