Finite Group Databasebeta

← #11715634

Subgroups — #11715634

Order 768.

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

normal subgroups only

This group has 354 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#18800—31nomaximal, Sylow
4C2×#984—41nomaximal
5Q8⋊SL(2,𝔽3)C2211yescharacteristic
6#5386—41nomaximal
7#5386—41nomaximal
8Q8×#41D311yescharacteristic
9#3229—31no—
10C2×#475—31no—
11#3039—31no—
12#1653—32no—
13#1651—32no—
14#1645—32no—
15#1643—32no—
16#1550—31no—
17#1487—31no—
18#1477—31no—
19C2×#41⋊C3—41no—
20#984—44no—
21C2×#226—42no—
22C4×SL(2,𝔽3)—41no—
23C4×SL(2,𝔽3)—41no—
24C4×SL(2,𝔽3)—81no—
25#859—42no—
26#859—42no—
27C2×#137—61no—
28Q82C2×D311yescharacteristic
29#557—31no—
30#554—31no—
31#554—61no—
32D4×Q8—31no—
33#533—31no—
34#533—61no—
35C2×C4⋊Q8—31no—
36C4×#41—31no—
37C2×C4×Q8—31no—
38#498—32no—
39#487—32no—
40#483—61no—
41#483—61no—
42#475—32no—
43#475—32no—
44#475—61no—
45#475—61no—
46#475—61no—
47#475—61no—
48#462—32no—
49#462—61no—
50#461—32no—
51#461—61no—
52#431—31no—
53C2×#106—61no—
54#423—31no—
55C2×C4⋊C8—31no—
56#419—61no—
57C2×#103—61no—
58#333—32no—
59#332—32no—
60Q8⋊C8—32no—
61Q8⋊C8—32no—
62Q8⋊C8—32no—
63D4⋊C8—32no—
64C2×Dic6—161no—
65#41⋊C3—42no—
66C2×SL(2,𝔽3)—41no—
67C2×SL(2,𝔽3)—41no—
68C2×SL(2,𝔽3)—41no—
69C2×SL(2,𝔽3)—81no—
70#226—44no—
71C2×#41S411yescharacteristic
72C22×Q8—31no—
73C22×Q8—61no—
74#137—122no—
75#137—122no—
76C2×Q16—62no—
77C2×SD16—62no—
78C2×M16—61no—
79C22×C8—61no—
80C4⋊Q8—31no—
81C4⋊Q8—31no—
82C4⋊Q8—32no—
83C4⋊Q8—61no—
84C4⋊Q8—61no—
85C4⋊Q8—61no—
86C4⋊Q8—61no—
87C4⋊Q8—61no—
88#125—61no—
89#125—61no—
90#125—61no—
91C22⋊Q8—31no—
92C22⋊Q8—31no—
93C22⋊Q8—61no—
94C22⋊Q8—61no—
95C22⋊Q8—61no—
96C22⋊Q8—61no—
97C22⋊Q8—61no—
98C4×Q8—21no—
99C4×Q8—31no—
100C4×Q8—31no—
101C4×Q8—31no—
102C4×Q8—31no—
103C4×Q8—31no—
104C4×Q8—31no—
105C4×Q8—61no—
106C4×Q8—61no—
107C4×Q8S411yescharacteristic
108C4×Q8S411yescharacteristic
109C4×D4—31no—
110C4×D4—31no—
111C4×D4—61no—
112C4×D4—61no—
113#117—31no—
114#117—61no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—61no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—61no—
119C2×C42—31no—
120C2×C42—61no—
121#106—62no—
122#106—62no—
123#106—62no—
124#106—62no—
125C4⋊C8—32no—
126C4⋊C8—32no—
127C4⋊C8—61no—
128C4⋊C8—61no—
129C4⋊C8—61no—
130#103—62no—
131#103—62no—
132#103—62no—
133#103—62no—
134#103—62no—
135#103—62no—
136C22⋊C8—61no—
137C4×C8—32no—
138C2×C3×C4—161no—
139C2×Dic3—162no—
140Dic6—164no—
141SL(2,𝔽3)—41no—
142SL(2,𝔽3)—41no—
143SL(2,𝔽3)—41no—
144SL(2,𝔽3)—81no—
145#41—32no—
146#41GL(2,𝔽3)12yes—
147C2×Q8—21no—
148C2×Q8—31no—
149C2×Q8—31no—
150C2×Q8—61no—
151C2×Q8—62no—
152C2×Q8—61no—
153C2×Q8—61no—
154C2×Q8—122no—
155C2×Q8—122no—
156C2×Q8—121no—
157C2×Q8—121no—
158C2×Q8C2×S411yescharacteristic
159C2×Q8C2×S411yescharacteristic
160C2×Q8C2×S411yescharacteristic
161C2×D4—31no—
162C22×C4—31no—
163C22×C4—31no—
164C22×C4—31no—
165C22×C4—61no—
166C22×C4—61no—
167C22×C4—121no—
168Q16—122no—
169Q16—122no—
170SD16—124no—
171M16—122no—
172C2×C8—62no—
173C2×C8—62no—
174C2×C8—62no—
175C2×C8—122no—
176C4⋊C4—31no—
177C4⋊C4—31no—
178C4⋊C4—31no—
179C4⋊C4—31no—
180C4⋊C4—31no—
181C4⋊C4—31no—
182C4⋊C4—61no—
183C4⋊C4—61no—
184C4⋊C4—61no—
185C4⋊C4—61no—
186C4⋊C4—61no—
187C4⋊C4—61no—
188C4⋊C4—61no—
189C4⋊C4—62no—
190C4⋊C4—62no—
191C4⋊C4—61no—
192C4⋊C4—121no—
193C4⋊C4—121no—
194C22⋊C4—61no—
195C22⋊C4—61no—
196C22⋊C4—61no—
197C42—31no—
198C42—31no—
199C42—31no—
200C42—31no—
201C42—61no—
202C42—61no—
203C42—61no—
204C22×C3—161no—
205C3×C4—162no—
206Dic3—164no—
207C23—31no—
208Q8—31no—
209Q8—31no—
210Q8—41no—
211Q8—61no—
212Q8—61no—
213Q8—81no—
214Q8—121no—
215Q8—121no—
216Q8—122no—
217Q8—122no—
218Q8—121no—
219Q8—242no—
220Q8C2×GL(2,𝔽3)11yescharacteristic
221Q8#98411yescharacteristic, complemented
222D4—32no—
223D4—61no—
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—32no—
227C2×C4—32no—
228C2×C4—31no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—61no—
234C2×C4—61no—
235C2×C4—121no—
236C2×C4—121no—
237C2×C4—121no—
238C2×C4—122no—
239C2×C4—241no—
240C2×C4—242no—
241C2×C4#102011yescharacteristic
242C8—122no—
243C8—122no—
244C8—122no—
245C2×C3—161no—
246C2×C3—161no—
247C2×C3—161no—
248C22—32no—
249C22—61no—
250C22—61no—
251C22C2×#102011yescharacteristic
252C4—31no—
253C4—31no—
254C4—31no—
255C4—31no—
256C4—61no—
257C4—61no—
258C4—61no—
259C4—121no—
260C4—121no—
261C4—242no—
262C4#593112yes—
263C3—161noSylow
264C2—62no—
265C2C2×#593111yescharacteristic
266C2#8750811yescharacteristic
267C2#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
221. Q8#984semidirect

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
#188003
C2×#9844
#53864