Finite Group Databasebeta

← #11711250

Subgroups — #11711250

Order 768.

331 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 22 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×#5930C212yesmaximal
2#1446⋊C3C211yesmaximal, characteristic
3#9967—31nomaximal, Sylow
4C2×C22⋊SL(2,𝔽3)C2211yescharacteristic
5#5930—22no—
6C4⋊SL(2,ℤ/4)—41nomaximal
7C2×#451—32no—
8#2659—31no—
9C2×#328—32no—
10#1446D311yescharacteristic
11#1280—31no—
12C22⋊SL(2,𝔽3)D411yescharacteristic
13C2×C4×A4—41no—
14C2×SL(2,ℤ/4)—42no—
15C2×#226—82no—
16C23×Q8C2×D311yescharacteristic
17C22×Q16—62no—
18C2×C22⋊Q8—32no—
19#451—64no—
20#451—62no—
21#451—62no—
22C2×#103—62no—
23C2×C22⋊C8—32no—
24#398—62no—
25#395—62no—
26#380—61no—
27#380—61no—
28C2×#95—31no—
29C2×#95—31no—
30#328—64no—
31C22×A4—41no—
32C2×SL(2,𝔽3)—82no—
33C4×A4—42no—
34SL(2,ℤ/4)—82no—
35#226—84no—
36C4⋊Dic3—161no—
37C22×Q8—31no—
38C22×Q8—32no—
39C22×Q8—62no—
40C22×Q8—62no—
41C22×Q8D1211yescharacteristic
42C23×C4—31no—
43C23×C4—31no—
44C23×C4S411yescharacteristic
45C2×Q16—64no—
46C2×Q16—64no—
47C2×Q16—122no—
48C2×Q16—122no—
49C2×Q16—122no—
50C2×Q16—122no—
51C22×C8—62no—
52C22⋊Q8—62no—
53C22⋊Q8—62no—
54C22⋊Q8—62no—
55C22⋊Q8—62no—
56C2×C4⋊C4—32no—
57C2×C4⋊C4—62no—
58C2×C4⋊C4—61no—
59C2×C4⋊C4—61no—
60C2×C4⋊C4—61no—
61C2×C4⋊C4—61no—
62C2×C22⋊C4—62no—
63#103—64no—
64#103—64no—
65C22⋊C8—34no—
66C22⋊C8—62no—
67#95—62no—
68#95—62no—
69#95—62no—
70#95—62no—
71C2×A4—41no—
72C2×A4—41no—
73C2×A4—41no—
74C2×C3×C4—161no—
75C2×Dic3—162no—
76SL(2,𝔽3)—82no—
77C24C2×S411yescharacteristic
78C2×Q8—22no—
79C2×Q8—62no—
80C2×Q8—62no—
81C2×Q8—62no—
82C2×Q8—62no—
83C2×Q8—62no—
84C2×Q8—62no—
85C2×Q8—62no—
86C2×Q8—122no—
87C2×Q8—122no—
88C2×Q8—122no—
89C2×Q8—122no—
90C22×C4—32no—
91C22×C4—32no—
92C22×C4—31no—
93C22×C4—31no—
94C22×C4—32no—
95C22×C4—61no—
96C22×C4—61no—
97C22×C4—61no—
98C22×C4—61no—
99C22×C4—62no—
100C22×C4—62no—
101C22×C4—121no—
102C22×C4—121no—
103C22×C4—121no—
104C22×C4—121no—
105C22×C4#22612yescomplemented
106Q16—124no—
107Q16—124no—
108Q16—124no—
109Q16—124no—
110C2×C8—64no—
111C2×C8—122no—
112C2×C8—122no—
113C4⋊C4—62no—
114C4⋊C4—62no—
115C4⋊C4—122no—
116C4⋊C4—122no—
117C4⋊C4—241no—
118C4⋊C4—241no—
119C4⋊C4—241no—
120C4⋊C4—241no—
121C22⋊C4—122no—
122C22⋊C4—122no—
123C22×C3—161no—
124A4—41no—
125C3×C4—162no—
126Dic3—322no—
127C23—31no—
128C23—31no—
129C23—31no—
130C23—31no—
131C23C4⋊S411yescharacteristic
132C23C2×#22611yescharacteristic
133C23#98411yescharacteristic
134Q8—22no—
135Q8—62no—
136Q8—62no—
137Q8—62no—
138Q8—122no—
139Q8—124no—
140Q8—242no—
141C2×C4—32no—
142C2×C4—32no—
143C2×C4—41no—
144C2×C4—62no—
145C2×C4—62no—
146C2×C4—62no—
147C2×C4—62no—
148C2×C4—121no—
149C2×C4—121no—
150C2×C4—122no—
151C2×C4—121no—
152C2×C4—121no—
153C2×C4—121no—
154C2×C4—122no—
155C2×C4—122no—
156C2×C4—122no—
157C2×C4—122no—
158C2×C4—122no—
159C2×C4—122no—
160C2×C4—122no—
161C2×C4—241no—
162C8—124no—
163C2×C3—161no—
164C2×C3—161no—
165C2×C3—161no—
166C22—31no—
167C22—31no—
168C22—31no—
169C22—31no—
170C22—31no—
171C22—31no—
172C22—31no—
173C22—31no—
174C22—31no—
175C22—61no—
176C22C4⋊#22611yescharacteristic
177C22#1020⋊C211yescharacteristic
178C4—42no—
179C4—62no—
180C4—62no—
181C4—122no—
182C4—242no—
183C4—242no—
184C3—161noSylow
185C2—31no—
186C2—31no—
187C2—31no—
188C2—31no—
189C2#7299111yescharacteristic
190C2#7299811yescharacteristic
191C2#7300111yescharacteristic
subgroupG/H
center, hypercenterC22#1020⋊C2
commutator, prime radical, cosocleC2×C22⋊SL(2,𝔽3)C22
Frattini, socleC24C2×S4
Fitting#1446D3
hypocenterC22⋊SL(2,𝔽3)D4
LCSG ▷ C2×C22⋊SL(2,𝔽3) ▷ C22⋊SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
105. C22×C4#226semidirect

Maximal subgroups

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

subgroupindex
C2×#59302
#1446⋊C32
#99673
C4⋊SL(2,ℤ/4)4