Finite Group Databasebeta

← #11713698

Subgroups — #11713698

Order 768.

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

normal subgroups only

This group has 559 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×#5386C211yesmaximal, characteristic, complemented
3#73151C214yesmaximal, complemented
4C4×#859C211yesmaximal, characteristic, complemented
5#25887—31nomaximal, Sylow
6C22×#41⋊C3C2212yes—
7C4⋊(#41⋊C3)—22no—
8C2×C4×SL(2,𝔽3)C2211yescharacteristic
9C2×#859C2212yes—
10C2×#859C2212yes—
11#5386—22no—
12#4973—41nomaximal
13C2×#550D311yescharacteristic
14C2×#486—31no—
15C2×#475—31no—
16#2547—31no—
17#2034—34no—
18#1982—31no—
19#1954—31no—
20#1788—34no—
21C4×#103—31no—
22C22×Dic6—41no—
23C2×#41⋊C3—22no—
24C2×#41⋊C3—24no—
25C22×SL(2,𝔽3)C2311yescharacteristic
26C2×C3×C4⋊C4—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×#124—61no—
38C2×#121—32no—
39C2×C4×Q8C2×D311yescharacteristic
40C2×C4×D4—31no—
41#486—64no—
42#475—62no—
43#475—62no—
44C2×#106—32no—
45C2×C4⋊C8—31no—
46C2×#103—32no—
47C2×#103—32no—
48C2×#102—34no—
49C2×#102—34no—
50C2×C22⋊C8—34no—
51C2×C4×C8—31no—
52#397—62no—
53#391—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—
61C2×Dic6—44no—
62C2×Dic6—82no—
63C2×Dic6—82no—
64#41⋊C3—42no—
65C2×SL(2,𝔽3)C2×D411yescharacteristic
66C2×SL(2,𝔽3)#4112yes—
67C3×C4⋊C4—82no—
68C4×Dic3—82no—
69C2×#41—24no—
70C2×#41—22no—
71C2×#41—62no—
72C2×#41—64no—
73C22×Q8—61no—
74C22×Q8C22×D311yescharacteristic
75C22×D4—32no—
76C22×D4—31no—
77C22×D4—61no—
78C23×C4—32no—
79C22×C8—32no—
80C22×C8—32no—
81C4⋊Q8—62no—
82C4⋊Q8—62no—
83#127—34no—
84#127—62no—
85#124—124no—
86#121—64no—
87#121—64no—
88C4×Q8—32no—
89C4×Q8#6712yes—
90C4×D4—62no—
91C4×D4—62no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4—61no—
96C2×C4⋊C4S411yescharacteristic
97C2×C22⋊C4—32no—
98C2×C22⋊C4—64no—
99C2×C42—31no—
100C2×C42—61no—
101#106—64no—
102C4⋊C8—62no—
103#103—62no—
104#103—62no—
105#103—62no—
106#103—62no—
107#102—68no—
108#102—124no—
109C22⋊C8—124no—
110C4×C8—62no—
111#95—61no—
112#95—64no—
113C23×C3—41no—
114C2×C3×C4—42no—
115C2×C3×C4—42no—
116C2×C3×C4—42no—
117C2×C3×C4—82no—
118C2×C3×C4—84no—
119C2×Dic3—44no—
120C2×Dic3—44no—
121C2×Dic3—82no—
122C2×Dic3—82no—
123C2×Dic3—82no—
124C2×Dic3—82no—
125Dic6—88no—
126SL(2,𝔽3)#12411yescharacteristic, complemented
127C24—32no—
128#41—42no—
129#41—124no—
130#41—124no—
131#41—122no—
132C2×Q8—31no—
133C2×Q8—32no—
134C2×Q8—122no—
135C2×Q8—122no—
136C2×Q8—122no—
137C2×Q8#23712yes—
138C2×Q8C2×#6711yescharacteristic
139C2×D4—34no—
140C2×D4—64no—
141C2×D4—64no—
142C2×D4—62no—
143C2×D4—64no—
144C2×D4—62no—
145C2×D4—62no—
146C2×D4—62no—
147C2×D4—242no—
148C22×C4—31no—
149C22×C4—31no—
150C22×C4—64no—
151C22×C4—62no—
152C22×C4—62no—
153C22×C4—64no—
154C22×C4—62no—
155C22×C4—62no—
156C22×C4C2×S412yes—
157C22×C4C2×S411yescharacteristic
158C2×C8—62no—
159C2×C8—62no—
160C2×C8—64no—
161C2×C8—62no—
162C2×C8—62no—
163C4⋊C4—22no—
164C4⋊C4—32no—
165C4⋊C4—61no—
166C4⋊C4—62no—
167C4⋊C4—62no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—122no—
171C22⋊C4—64no—
172C22⋊C4—128no—
173C42—32no—
174C42—32no—
175C42—122no—
176C22×C3—42no—
177C22×C3—41no—
178C22×C3—41no—
179C22×C3—42no—
180C22×C3—41no—
181C3×C4—44no—
182C3×C4—162no—
183Dic3—84no—
184Dic3—84no—
185C23—64no—
186C23—62no—
187C23—62no—
188C23—64no—
189C23—62no—
190C23C22×S411yescharacteristic
191Q8—32no—
192Q8—31no—
193Q8—61no—
194Q8—244no—
195Q8#93611yescharacteristic
196D4—68no—
197D4—122no—
198D4—122no—
199D4—124no—
200D4—244no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—32no—
204C2×C4—32no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—32no—
208C2×C4—62no—
209C2×C4—122no—
210C2×C4—122no—
211C2×C4—124no—
212C2×C4—122no—
213C2×C4—122no—
214C2×C4—122no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4#98512yes—
219C2×C4#98612yes—
220C2×C4GL(2,ℤ/4)12yes—
221C8—122no—
222C8—122no—
223C2×C3—41no—
224C2×C3—41no—
225C2×C3—42no—
226C2×C3—42no—
227C2×C3—41no—
228C22—122no—
229C22—122no—
230C22—124no—
231C22—124no—
232C22—122no—
233C22#591412yes—
234C22C2×#98511yescharacteristic
235C22C2×#98611yescharacteristic
236C22#592412yes—
237C22C2×GL(2,ℤ/4)11yescharacteristic
238C4—31no—
239C4—32no—
240C4—31no—
241C4—42no—
242C4—62no—
243C4—242no—
244C4—242no—
245C4#542714yes—
246C3—41noSylow
247C2—242no—
248C2#8540312yes—
249C2#8541212yes—
250C2#8549211yescharacteristic
251C2C2×#542711yescharacteristic
252C2#8552711yescharacteristic
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×#5386C2semidirect
3. #73151C2semidirect
4. C4×#859C2semidirect
126. SL(2,𝔽3)#124semidirect

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×#53862
#731512
C4×#8592
#258873
#49734