Finite Group Databasebeta

← #11715632

Subgroups — #11715632

Order 768.

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

normal subgroups only

This group has 332 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#85553C212yesmaximal, complemented
3#18789—31nomaximal, Sylow
4Q8⋊SL(2,𝔽3)C2211yescharacteristic
5#5426—41nomaximal
6#5391—41nomaximal
7C4⋊#226—41nomaximal
8Q8×#41D311yescharacteristic
9#3209—31no—
10#3079—31no—
11C2×#477—31no—
12#1637—32no—
13#1636—32no—
14#1633—32no—
15#1632—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—
26Q82C2×D311yescharacteristic
27#557—31no—
28#554—31no—
29#554—61no—
30D4×Q8—31no—
31#533—31no—
32#533—61no—
33C2×C4⋊Q8—31no—
34C4×#41—31no—
35C2×C4×Q8—31no—
36#494—32no—
37#487—32no—
38#477—32no—
39#477—32no—
40#477—32no—
41#477—61no—
42#477—61no—
43#476—32no—
44#475—61no—
45#474—61no—
46#462—61no—
47#462—61no—
48#451—61no—
49#451—61no—
50#431—31no—
51#428—61no—
52C2×#107—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×M16—61no—
76C22×C8—61no—
77C4⋊Q8—31no—
78C4⋊Q8—31no—
79C4⋊Q8—32no—
80C4⋊Q8—61no—
81C4⋊Q8—61no—
82C4⋊Q8—61no—
83C4⋊Q8—61no—
84C4⋊Q8—61no—
85#125—61no—
86#125—61no—
87#125—61no—
88C22⋊Q8—31no—
89C22⋊Q8—31no—
90C22⋊Q8—61no—
91C22⋊Q8—61no—
92C22⋊Q8—61no—
93C22⋊Q8—61no—
94C22⋊Q8—61no—
95C4×Q8—21no—
96C4×Q8—31no—
97C4×Q8—31no—
98C4×Q8—31no—
99C4×Q8—31no—
100C4×Q8—31no—
101C4×Q8—31no—
102C4×Q8—61no—
103C4×Q8—61no—
104C4×Q8S411yescharacteristic
105C4×Q8S411yescharacteristic
106C4×D4—31no—
107C4×D4—31no—
108C4×D4—61no—
109C4×D4—61no—
110#117—31no—
111#117—61no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—61no—
114C2×C4⋊C4—61no—
115C2×C4⋊C4—61no—
116C2×C42—31no—
117C2×C42—61no—
118#107—62no—
119#107—62no—
120#107—62no—
121#106—62no—
122C4⋊C8—32no—
123C4⋊C8—32no—
124C4⋊C8—61no—
125C4⋊C8—61no—
126C4⋊C8—61no—
127#103—62no—
128#103—62no—
129#103—62no—
130#103—62no—
131#103—62no—
132#102—62no—
133C22⋊C8—61no—
134C4×C8—32no—
135C2×C3×C4—161no—
136C2×Dic3—162no—
137SL(2,𝔽3)—41no—
138SL(2,𝔽3)—41no—
139SL(2,𝔽3)—41no—
140SL(2,𝔽3)—81no—
141#41—32no—
142#41#22612yescomplemented
143C2×Q8—21no—
144C2×Q8—31no—
145C2×Q8—31no—
146C2×Q8—61no—
147C2×Q8—62no—
148C2×Q8—61no—
149C2×Q8—61no—
150C2×Q8—121no—
151C2×Q8—121no—
152C2×Q8—242no—
153C2×Q8C2×S411yescharacteristic
154C2×Q8C2×S411yescharacteristic
155C2×Q8C2×S411yescharacteristic
156C2×D4—31no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—61no—
161C22×C4—61no—
162C22×C4—121no—
163Q16—242no—
164Q16—242no—
165M16—122no—
166C2×C8—62no—
167C2×C8—62no—
168C2×C8—62no—
169C2×C8—122no—
170C4⋊C4—31no—
171C4⋊C4—31no—
172C4⋊C4—31no—
173C4⋊C4—31no—
174C4⋊C4—31no—
175C4⋊C4—31no—
176C4⋊C4—61no—
177C4⋊C4—61no—
178C4⋊C4—61no—
179C4⋊C4—61no—
180C4⋊C4—61no—
181C4⋊C4—61no—
182C4⋊C4—61no—
183C4⋊C4—62no—
184C4⋊C4—62no—
185C4⋊C4—61no—
186C4⋊C4—121no—
187C4⋊C4—121no—
188C22⋊C4—61no—
189C22⋊C4—61no—
190C22⋊C4—61no—
191C42—31no—
192C42—31no—
193C42—31no—
194C42—31no—
195C42—61no—
196C42—61no—
197C42—61no—
198C22×C3—161no—
199C3×C4—162no—
200Dic3—322no—
201C23—31no—
202Q8—31no—
203Q8—31no—
204Q8—41no—
205Q8—61no—
206Q8—61no—
207Q8—81no—
208Q8—121no—
209Q8—121no—
210Q8—121no—
211Q8—122no—
212Q8—122no—
213Q8—242no—
214Q8C2×#22611yescharacteristic
215Q8#98411yescharacteristic
216D4—32no—
217D4—61no—
218C2×C4—31no—
219C2×C4—31no—
220C2×C4—32no—
221C2×C4—32no—
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—31no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4—121no—
230C2×C4—121no—
231C2×C4—121no—
232C2×C4—122no—
233C2×C4—241no—
234C2×C4—242no—
235C2×C4#102011yescharacteristic
236C8—122no—
237C8—122no—
238C8—122no—
239C2×C3—161no—
240C2×C3—161no—
241C2×C3—161no—
242C22—32no—
243C22—61no—
244C22—61no—
245C22C2×#102011yescharacteristic
246C4—31no—
247C4—31no—
248C4—31no—
249C4—31no—
250C4—61no—
251C4—61no—
252C4—61no—
253C4—121no—
254C4—121no—
255C4—242no—
256C4#593012yes—
257C3—161noSylow
258C2—62no—
259C2C2×#593011yescharacteristic
260C2#8750811yescharacteristic
261C2#8752211yescharacteristic
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. #85553C2semidirect
142. #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
#855532
#187893
#54264
#53914
C4⋊#2264