Finite Group Databasebeta

← #11715635

Subgroups — #11715635

Order 768.

342 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 25 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#41⋊SL(2,𝔽3)C211yesmaximal, characteristic
2#85552C212yesmaximal, complemented
3#18801—31nomaximal, Sylow
4Q8⋊SL(2,𝔽3)C2211yescharacteristic
5#5426—41nomaximal
6#5391—41nomaximal
7C4⋊#226—41nomaximal
8Q8×#41D311yescharacteristic
9#3230—31no—
10#3079—31no—
11C2×#462—31no—
12#1649—32no—
13#1648—32no—
14#1645—32no—
15#1644—32no—
16#1550—31no—
17#1487—31no—
18#1477—31no—
19C2×#41⋊C3—41no—
20C2×#226—42no—
21C4×SL(2,𝔽3)—41no—
22C4×SL(2,𝔽3)—41no—
23C4×SL(2,𝔽3)—81no—
24#859—42no—
25#859—42no—
26C22×Q16—61no—
27Q82C2×D311yescharacteristic
28#557—31no—
29#554—31no—
30#554—61no—
31D4×Q8—31no—
32#533—31no—
33#533—61no—
34C2×C4⋊Q8—31no—
35C4×#41—31no—
36C2×C4×Q8—31no—
37#500—32no—
38#487—32no—
39#484—61no—
40#484—61no—
41#477—61no—
42#477—61no—
43#477—61no—
44#476—61no—
45#475—32no—
46#474—32no—
47#462—32no—
48#462—32no—
49#462—61no—
50#462—61no—
51#431—31no—
52#428—61no—
53#423—31no—
54C2×C4⋊C8—31no—
55#417—61no—
56C2×#103—61no—
57#333—32no—
58#332—32no—
59Q8⋊C8—32no—
60Q8⋊C8—32no—
61Q8⋊C8—32no—
62D4⋊C8—32no—
63#41⋊C3—42no—
64C2×SL(2,𝔽3)—41no—
65C2×SL(2,𝔽3)—41no—
66C2×SL(2,𝔽3)—41no—
67C2×SL(2,𝔽3)—81no—
68#226—82no—
69C4⋊Dic3—161no—
70C2×#41S411yescharacteristic
71C22×Q8—31no—
72C22×Q8—61no—
73C2×Q16—62no—
74C2×Q16—62no—
75C2×Q16—122no—
76C2×Q16—122no—
77C2×M16—61no—
78C22×C8—61no—
79C4⋊Q8—31no—
80C4⋊Q8—31no—
81C4⋊Q8—32no—
82C4⋊Q8—61no—
83C4⋊Q8—61no—
84C4⋊Q8—61no—
85C4⋊Q8—61no—
86C4⋊Q8—61no—
87#125—61no—
88#125—61no—
89#125—61no—
90C22⋊Q8—31no—
91C22⋊Q8—31no—
92C22⋊Q8—61no—
93C22⋊Q8—61no—
94C22⋊Q8—61no—
95C22⋊Q8—61no—
96C22⋊Q8—61no—
97C4×Q8—21no—
98C4×Q8—31no—
99C4×Q8—31no—
100C4×Q8—31no—
101C4×Q8—31no—
102C4×Q8—31no—
103C4×Q8—31no—
104C4×Q8—61no—
105C4×Q8—61no—
106C4×Q8S411yescharacteristic
107C4×Q8S411yescharacteristic
108C4×D4—31no—
109C4×D4—31no—
110C4×D4—61no—
111C4×D4—61no—
112#117—31no—
113#117—61no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4—61no—
117C2×C4⋊C4—61no—
118C2×C42—31no—
119C2×C42—61no—
120#107—62no—
121#107—62no—
122#107—62no—
123#106—62no—
124C4⋊C8—32no—
125C4⋊C8—32no—
126C4⋊C8—61no—
127C4⋊C8—61no—
128C4⋊C8—61no—
129#103—62no—
130#103—62no—
131#103—62no—
132#103—62no—
133#103—62no—
134#102—62no—
135C22⋊C8—61no—
136C4×C8—32no—
137C2×C3×C4—161no—
138C2×Dic3—162no—
139SL(2,𝔽3)—41no—
140SL(2,𝔽3)—41no—
141SL(2,𝔽3)—41no—
142SL(2,𝔽3)—81no—
143#41—32no—
144#41#22612yescomplemented
145C2×Q8—21no—
146C2×Q8—31no—
147C2×Q8—31no—
148C2×Q8—61no—
149C2×Q8—62no—
150C2×Q8—61no—
151C2×Q8—61no—
152C2×Q8—121no—
153C2×Q8—121no—
154C2×Q8—122no—
155C2×Q8—122no—
156C2×Q8C2×S411yescharacteristic
157C2×Q8C2×S411yescharacteristic
158C2×Q8C2×S411yescharacteristic
159C2×D4—31no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—31no—
163C22×C4—61no—
164C22×C4—61no—
165C22×C4—121no—
166Q16—122no—
167Q16—124no—
168Q16—122no—
169M16—122no—
170C2×C8—62no—
171C2×C8—62no—
172C2×C8—62no—
173C2×C8—122no—
174C4⋊C4—31no—
175C4⋊C4—31no—
176C4⋊C4—31no—
177C4⋊C4—31no—
178C4⋊C4—31no—
179C4⋊C4—31no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4—61no—
183C4⋊C4—61no—
184C4⋊C4—61no—
185C4⋊C4—61no—
186C4⋊C4—61no—
187C4⋊C4—62no—
188C4⋊C4—62no—
189C4⋊C4—61no—
190C4⋊C4—121no—
191C4⋊C4—121no—
192C22⋊C4—61no—
193C22⋊C4—61no—
194C22⋊C4—61no—
195C42—31no—
196C42—31no—
197C42—31no—
198C42—31no—
199C42—61no—
200C42—61no—
201C42—61no—
202C22×C3—161no—
203C3×C4—162no—
204Dic3—322no—
205C23—31no—
206Q8—31no—
207Q8—31no—
208Q8—41no—
209Q8—61no—
210Q8—61no—
211Q8—81no—
212Q8—121no—
213Q8—121no—
214Q8—122no—
215Q8—122no—
216Q8—121no—
217Q8—242no—
218Q8C2×#22611yescharacteristic
219Q8#98611yescharacteristic
220D4—32no—
221D4—61no—
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—32no—
225C2×C4—32no—
226C2×C4—31no—
227C2×C4—31no—
228C2×C4—31no—
229C2×C4—61no—
230C2×C4—61no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—121no—
234C2×C4—121no—
235C2×C4—121no—
236C2×C4—122no—
237C2×C4—241no—
238C2×C4—242no—
239C2×C4#102011yescharacteristic
240C8—122no—
241C8—122no—
242C8—122no—
243C2×C3—161no—
244C2×C3—161no—
245C2×C3—161no—
246C22—32no—
247C22—61no—
248C22—61no—
249C22C2×#102011yescharacteristic
250C4—31no—
251C4—31no—
252C4—31no—
253C4—31no—
254C4—61no—
255C4—61no—
256C4—61no—
257C4—121no—
258C4—121no—
259C4—242no—
260C4#593012yes—
261C3—161noSylow
262C2—62no—
263C2C2×#593011yescharacteristic
264C2#8751011yescharacteristic
265C2#8751811yescharacteristic
subgroupG/H
center, Frattini, socleC22C2×#1020
commutator, cosocle, hypocenterQ8⋊SL(2,𝔽3)C22
FittingQ8×#41D3
prime radical#41⋊SL(2,𝔽3)C2
hypercenterC2×C4#1020
LCSG ▷ Q8⋊SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #85552C2semidirect
144. #41#226semidirect

Maximal subgroups

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

subgroupindex
#41⋊SL(2,𝔽3)2
#855522
#188013
#54264
#53914
C4⋊#2264