Finite Group Databasebeta

← #11714095

Subgroups — #11714095

Order 768.

583 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 67 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×#5395C213yesmaximal
2C2×C4⋊#226C213yesmaximal, complemented
3#73167C211yesmaximal, characteristic, complemented
4#33169—31nomaximal, Sylow
5C22×GL(2,𝔽3)C2211yescharacteristic
6C22×#226C2213yes—
7C2×C4×SL(2,𝔽3)C2213yes—
8#5395—26no—
9C4⋊#226—26no—
10#4677—41nomaximal
11C2×#464—33no—
12C2×#462—33no—
13#2658—31no—
14#2276D311yescharacteristic, complemented
15#1983—33no—
16#1981—33no—
17#1843—31no—
18C22×SL(2,𝔽3)C2311yescharacteristic
19C2×GL(2,𝔽3)—23no—
20C2×#226—26no—
21C2×#226—23no—
22C2×(C2×D3)⋊C4—43no—
23C2×C4⋊Dic3—43no—
24C4×SL(2,𝔽3)D416yes—
25C3×#95—41no—
26C22×Q16—33no—
27C22×SD16—31no—
28C2×C4⋊Q8—33no—
29C2×#124—33no—
30C2×C4×Q8C2×D313yes—
31#464—66no—
32#464—66no—
33#462—66no—
34#462—66no—
35C2×C4⋊C8—33no—
36C2×C4⋊C8—33no—
37C2×#103—33no—
38C2×#103—36no—
39C2×#102—33no—
40#399—63no—
41#399—61no—
42C4×C4⋊C4—33no—
43#376—31no—
44C23×D3—41no—
45C22×C3×C4—43no—
46C22×Dic3—43no—
47C2×SL(2,𝔽3)C2×D413yes—
48GL(2,𝔽3)—41no—
49#226—43no—
50(C2×D3)⋊C4—86no—
51C4⋊Dic3—86no—
52C22×Q8—33no—
53C22×Q8C22×D311yescharacteristic
54C22×D4—31no—
55C2×Q16—66no—
56C2×Q16—63no—
57C2×Q16—66no—
58C2×Q16—63no—
59C2×SD16—63no—
60C2×SD16—63no—
61C22×C8—31no—
62C22×C8—33no—
63C4⋊Q8—66no—
64C4⋊Q8—66no—
65#124—66no—
66#124—66no—
67C4×Q8—36no—
68C4×Q8D1216yes—
69C2×C4⋊C4—33no—
70C2×C4⋊C4—63no—
71C2×C4⋊C4—63no—
72C2×C22⋊C4—63no—
73C2×C22⋊C4—63no—
74C2×C42—33no—
75C2×C42—61no—
76C4⋊C8—66no—
77C4⋊C8—66no—
78#103—612no—
79#103—66no—
80#102—66no—
81#95—33no—
82#95S411yescharacteristic
83C23×C3—41no—
84C22×D3—42no—
85C22×D3—83no—
86C22×D3—83no—
87C2×C3×C4—46no—
88C2×C3×C4—86no—
89C2×Dic3—86no—
90C2×Dic3—83no—
91SL(2,𝔽3)#12711yescharacteristic
92C24—61no—
93C2×Q8—33no—
94C2×Q8—63no—
95C2×Q8—66no—
96C2×Q8—63no—
97C2×Q8—66no—
98C2×Q8C2×D1213yes—
99C2×D4—63no—
100C2×D4—63no—
101C22×C4—31no—
102C22×C4—33no—
103C22×C4—63no—
104C22×C4C2×S413yes—
105Q16—123no—
106Q16—126no—
107Q16—123no—
108SD16—123no—
109SD16—121no—
110C2×C8—63no—
111C2×C8—63no—
112C2×C8—66no—
113C4⋊C4—36no—
114C4⋊C4—63no—
115C4⋊C4—126no—
116C4⋊C4—126no—
117C22⋊C4—126no—
118C22⋊C4—126no—
119C42—36no—
120C42—36no—
121C42—241no—
122C22×C3—43no—
123C22×C3—41no—
124C22×C3—43no—
125C2×D3—86no—
126C2×D3—163no—
127C2×D3—161no—
128C3×C4—86no—
129Dic3—163no—
130C23—123no—
131C23—121no—
132C23—123no—
133C23C22×S411yescharacteristic
134Q8—33no—
135Q8—61no—
136Q8—123no—
137Q8—123no—
138Q8—126no—
139Q8#87411yescharacteristic
140D4—121no—
141D4—123no—
142C2×C4—26no—
143C2×C4—33no—
144C2×C4—33no—
145C2×C4—36no—
146C2×C4—63no—
147C2×C4—63no—
148C2×C4—123no—
149C2×C4—126no—
150C2×C4C4⋊S416yes—
151C8—121no—
152C8—123no—
153C2×C3—41no—
154C2×C3—43no—
155C2×C3—43no—
156D3—162no—
157C22—241no—
158C22—243no—
159C22—243no—
160C22C2×C4⋊S413yes—
161C22#592411yescharacteristic
162C22#592713yes—
163C4—26no—
164C4—31no—
165C4—33no—
166C4—66no—
167C4—243no—
168C3—41noSylow
169C2—481no—
170C2#8528911yescharacteristic
171C2#8535913yes—
172C2#8537013yes—
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
Fitting#2276D3
prime radicalC22×GL(2,𝔽3)C22
hypercenter#95S4
hypocenterSL(2,𝔽3)#127
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊#226C2semidirect
3. #73167C2semidirect
14. #2276D3semidirect

Maximal subgroups

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

subgroupindex
C2×#53952
C2×C4⋊#2262
#731672
#331693
#46774