Finite Group Databasebeta

← #11715623

Subgroups — #11715623

Order 768.

339 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 23 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×Q8⋊SL(2,𝔽3)C211yesmaximal, characteristic
2#85552C212yesmaximal, complemented
3#18551—31nomaximal, Sylow
4Q8⋊SL(2,𝔽3)C2211yescharacteristic
5#5420—41nomaximal
6#5419—42nomaximal
7C2×Q82D311yescharacteristic
8#3139—31no—
9#3113—32no—
10#1645—32no—
11#1645—32no—
12#1645—32no—
13#1645—32no—
14#1558—31no—
15#1491—32no—
16C22×SL(2,𝔽3)—41no—
17C22×SL(2,𝔽3)—42no—
18C22×SL(2,𝔽3)—81no—
19C2×#226—42no—
20#859—42no—
21#859—42no—
22Q82—32no—
23Q82—31no—
24Q82—61no—
25Q82C2×D311yescharacteristic
26#533—31no—
27C2×C4⋊Q8—31no—
28C2×C4⋊Q8—61no—
29C2×C4×Q8—32no—
30#491—61no—
31#491—61no—
32#487—32no—
33#487—32no—
34#483—62no—
35#475—32no—
36#475—32no—
37#462—32no—
38#462—32no—
39#451—62no—
40#441—62no—
41#441—62no—
42#432—32no—
43#419—61no—
44#419—61no—
45C4×M16—31no—
46#333—31no—
47#333—31no—
48#333—32no—
49Q8⋊C8—32no—
50Q8⋊C8—32no—
51Q8⋊C8—32no—
52Q8⋊C8—32no—
53C2×SL(2,𝔽3)—42no—
54C2×SL(2,𝔽3)—41no—
55C2×SL(2,𝔽3)—42no—
56C2×SL(2,𝔽3)—81no—
57C2×SL(2,𝔽3)—82no—
58C2×SL(2,𝔽3)—82no—
59#226—82no—
60#217—161no—
61C22×Q8—21no—
62C22×Q8—31no—
63C22×Q8S412yes—
64C22×Q8S411yescharacteristic
65C2×Q16—62no—
66C2×Q16—62no—
67C2×M16—61no—
68C4⋊Q8—31no—
69C4⋊Q8—31no—
70C4⋊Q8—31no—
71C4⋊Q8—31no—
72C4⋊Q8—32no—
73C4⋊Q8—62no—
74C4⋊Q8—61no—
75C4⋊Q8—61no—
76C4⋊Q8—62no—
77C4⋊Q8—62no—
78C4⋊Q8—61no—
79C4⋊Q8—61no—
80#125—61no—
81C22⋊Q8—62no—
82C4×Q8—32no—
83C4×Q8—32no—
84C4×Q8—32no—
85C4×Q8—32no—
86C4×Q8—62no—
87C4×Q8—62no—
88C4×Q8—62no—
89#117—61no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4—62no—
92C2×C42—31no—
93C2×C42—61no—
94#106—62no—
95#106—62no—
96C4⋊C8—32no—
97C4⋊C8—32no—
98#103—62no—
99#103—62no—
100#103—62no—
101#103—62no—
102#103—62no—
103#103—62no—
104#103—62no—
105#103—62no—
106C22⋊C8—62no—
107#97—61no—
108#97—62no—
109C4×C8—32no—
110C23×C3—161no—
111C2×Dic3—162no—
112SL(2,𝔽3)—41no—
113SL(2,𝔽3)—42no—
114SL(2,𝔽3)—81no—
115C2×Q8—22no—
116C2×Q8—21no—
117C2×Q8—32no—
118C2×Q8—31no—
119C2×Q8—31no—
120C2×Q8—31no—
121C2×Q8—42no—
122C2×Q8—61no—
123C2×Q8—62no—
124C2×Q8—62no—
125C2×Q8—61no—
126C2×Q8—82no—
127C2×Q8—122no—
128C2×Q8C2×S412yes—
129C2×Q8C2×S411yescharacteristic
130C22×C4—32no—
131C22×C4—31no—
132C22×C4—61no—
133Q16—122no—
134Q16—124no—
135Q16—122no—
136M16—122no—
137C2×C8—62no—
138C2×C8—62no—
139C2×C8—62no—
140C4⋊C4—32no—
141C4⋊C4—32no—
142C4⋊C4—62no—
143C4⋊C4—62no—
144C4⋊C4—62no—
145C4⋊C4—62no—
146C4⋊C4—62no—
147C4⋊C4—62no—
148C4⋊C4—62no—
149C4⋊C4—122no—
150C22⋊C4—121no—
151C42—31no—
152C42—32no—
153C42—31no—
154C42—61no—
155C42—62no—
156C42—61no—
157C42—121no—
158C22×C3—162no—
159C22×C3—161no—
160C22×C3—322no—
161Dic3—322no—
162C23#102011yescharacteristic
163Q8—32no—
164Q8—34no—
165Q8—41no—
166Q8—62no—
167Q8—81no—
168Q8—122no—
169Q8—122no—
170Q8—121no—
171Q8—121no—
172Q8—121no—
173Q8—241no—
174Q8—241no—
175Q8#98312yes—
176C2×C4—32no—
177C2×C4—32no—
178C2×C4—31no—
179C2×C4—31no—
180C2×C4—62no—
181C2×C4—62no—
182C2×C4—62no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—122no—
186C2×C4—122no—
187C8—122no—
188C8—122no—
189C8—122no—
190C2×C3—162no—
191C2×C3—161no—
192C2×C3—322no—
193C22—22no—
194C22#593312yes—
195C22C2×#102011yescharacteristic
196C4—32no—
197C4—32no—
198C4—62no—
199C4—61no—
200C4—61no—
201C4—121no—
202C4—121no—
203C4—242no—
204C3—161noSylow
205C2—22no—
206C2#8750712yes—
207C2C2×#593311yescharacteristic
subgroupG/H
center, Frattini, socleC22C2×#1020
commutator, cosocle, hypocenterQ8⋊SL(2,𝔽3)C22
FittingC2×Q82D3
prime radicalC2×Q8⋊SL(2,𝔽3)C2
hypercenterC23#1020
LCSG ▷ Q8⋊SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #85552C2semidirect

Maximal subgroups

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

subgroupindex
C2×Q8⋊SL(2,𝔽3)2
#855522
#185513
#54204
#54194