Finite Group Databasebeta

← #11715731

Subgroups — #11715731

Order 768.

544 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 27 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#87502C212yesmaximal
2C8×C22⋊A4C211yesmaximal, characteristic
3#32931—31nomaximal, Sylow
4C4×C22⋊A4C2211yescharacteristic
5#5936—22no—
6#5398—43nomaximal
7C24×C8D311yescharacteristic
8C2×#467—33no—
9#2848—32no—
10#1892—36no—
11#1808—33no—
12C2×C22⋊A4D411yescharacteristic
13A4⋊Q8—46no—
14C8×A4—43no—
15C8×A4—81no—
16C24×C4C2×D311yescharacteristic
17C22×Q16—31no—
18C23×C8—33no—
19C23×C8—33no—
20C23×C8—61no—
21C23×C8—61no—
22C2×C22⋊Q8—36no—
23#467—63no—
24#467—63no—
25#467—63no—
26#467—63no—
27C2×#107—33no—
28C2×#103—36no—
29#385—66no—
30#380—36no—
31#379—62no—
32#340—36no—
33C22⋊A4Q1611yescharacteristic, complemented
34C4×A4—43no—
35C4×A4—81no—
36SL(2,ℤ/4)—86no—
37Dic12—161no—
38C25D1211yescharacteristic
39C22×Q8—32no—
40C23×C4—33no—
41C23×C4—33no—
42C23×C4—61no—
43C23×C4—61no—
44C2×Q16—63no—
45C2×Q16—63no—
46C22×C8—21no—
47C22×C8—31no—
48C22×C8—33no—
49C22×C8—33no—
50C22×C8—33no—
51C22×C8—33no—
52C22×C8—66no—
53C22×C8—63no—
54C22×C8—63no—
55C22×C8—61no—
56C22×C8—63no—
57C22×C8S413yes—
58C22⋊Q8—66no—
59C22⋊Q8—66no—
60C22⋊Q8—66no—
61C22⋊Q8—66no—
62C2×C4⋊C4—36no—
63C2×C4⋊C4—66no—
64C2×C22⋊C4—66no—
65C2×C22⋊C4—66no—
66#107—63no—
67#107—63no—
68#103—66no—
69#103—66no—
70#95—66no—
71C2×A4—43no—
72C2×A4—81no—
73Dic6—162no—
74C3×C8—161no—
75C24—33no—
76C24—33no—
77C24—61no—
78C24—61no—
79C24Dic1211yescharacteristic, complemented
80C2×Q8—66no—
81C2×Q8—66no—
82C22×C4—21no—
83C22×C4—31no—
84C22×C4—33no—
85C22×C4—33no—
86C22×C4—33no—
87C22×C4—33no—
88C22×C4—63no—
89C22×C4—66no—
90C22×C4—63no—
91C22×C4—61no—
92C22×C4—63no—
93C22×C4—66no—
94C22×C4—62no—
95C22×C4C2×S413yes—
96Q16—123no—
97Q16—121no—
98C2×C8—33no—
99C2×C8—33no—
100C2×C8—33no—
101C2×C8—33no—
102C2×C8—63no—
103C2×C8—66no—
104C2×C8—63no—
105C2×C8—61no—
106C2×C8—61no—
107C4⋊C4—66no—
108C4⋊C4—66no—
109C4⋊C4—126no—
110C4⋊C4—126no—
111C22⋊C4—126no—
112C22⋊C4—126no—
113C22⋊C4—126no—
114C22⋊C4—126no—
115A4—43no—
116A4—81no—
117C3×C4—161no—
118Dic3—322no—
119C23—21no—
120C23—31no—
121C23—33no—
122C23—33no—
123C23—33no—
124C23—33no—
125C23—33no—
126C23—61no—
127C23—61no—
128C23—63no—
129C23—66no—
130C23—63no—
131C23—63no—
132C23C4⋊S413yes—
133Q8—126no—
134Q8—122no—
135C2×C4—33no—
136C2×C4—33no—
137C2×C4—33no—
138C2×C4—33no—
139C2×C4—63no—
140C2×C4—66no—
141C2×C4—63no—
142C2×C4—61no—
143C2×C4—61no—
144C2×C4—126no—
145C2×C4—126no—
146C2×C4—1212no—
147C8—33no—
148C8—61no—
149C8#102011yescharacteristic
150C2×C3—161no—
151C22—21no—
152C22—33no—
153C22—33no—
154C22—31no—
155C22—33no—
156C22—33no—
157C22—33no—
158C22—63no—
159C22—66no—
160C22—63no—
161C22—63no—
162C22—61no—
163C22—61no—
164C22#539813yescomplemented
165C4—33no—
166C4—61no—
167C4—246no—
168C4—242no—
169C4C2×#102011yescharacteristic
170C3—161noSylow
171C2—33no—
172C2—33no—
173C2—61no—
174C2—61no—
175C2C4⋊#102011yescharacteristic
subgroupG/H
centerC2C4⋊#1020
commutator, cosocleC4×C22⋊A4C22
FrattiniC4C2×#1020
FittingC24×C8D3
socleC25D12
prime radicalC2×C22⋊A4D4
hypercenterC8#1020
hypocenterC22⋊A4Q16
LCSG ▷ C4×C22⋊A4 ▷ C2×C22⋊A4 ▷ C22⋊A4
UCSC1 ◁ C2 ◁ C4 ◁ C8

Complements

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

NH ≅ G/Ntype
33. C22⋊A4Q16semidirect
79. C24Dic12semidirect
164. C22#5398semidirect

Maximal subgroups

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

subgroupindex
#875022
C8×C22⋊A42
#329313
#53984