Finite Group Databasebeta

← #10698311

Subgroups — #10698311

Order 768.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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#73310C211yesmaximal, characteristic
2#70793C212yesmaximal
3#69396C211yesmaximal, characteristic
4C3×#1276C211yesmaximal, characteristic
5#67738C212yesmaximal
6#10369—31nomaximal, Sylow
7#5470C2211yescharacteristic
8C3×#431C2212yes—
9C3×C43C2211yescharacteristic
10#4936C2212yes—
11#4933—24no—
12#4928—22no—
13#4924C2211yescharacteristic
14#4523—22no—
15#4467—24no—
16#2864—31no—
17#1994—31no—
18#1518—32no—
19#1286—32no—
20#1276D311yescharacteristic
21C2×C3×C42—22no—
22C2×C3×C42D412yes—
23C2×C3×C42C2311yescharacteristic
24#926—21no—
25C2×C4⋊Dic3—22no—
26#924—22no—
27C2×Dic3⋊C4—24no—
28C2×#208—21no—
29#870—21no—
30#869Q812yes—
31C4×Dic6—22no—
32C3×C4⋊C8—22no—
33C3×C22⋊C8—22no—
34C3×C4×C8—22no—
35C3⋊#95—24no—
36C3⋊#95—22no—
37C3⋊#95D414yes—
38C4⋊(C3⋊C8)—22no—
39#533—31no—
40#431C2×D312yes—
41#423—31no—
42#389—62no—
43#387—64no—
44#382—32no—
45C43C2×D311yescharacteristic
46#339—62no—
47#328—64no—
48C22×C3×C4—22no—
49C22×C3×C4C2×D411yescharacteristic
50C22×C3×C4C2×D412yes—
51C22×Dic3—24no—
52C2×Dic6—22no—
53C2×C3×C8—42no—
54C3×C42—22no—
55C3×C42—22no—
56C3×C42—44no—
57C3×C42C2×Q811yescharacteristic
58C3×C42#4112yes—
59C3×C42#4111yescharacteristic
60#217—41no—
61C4⋊Dic3—22no—
62C4⋊Dic3—22no—
63C4⋊Dic3—42no—
64C4⋊Dic3—42no—
65Dic3⋊C4—48no—
66Dic3⋊C4—42no—
67C4×Dic3—41no—
68#208—42no—
69C2×C3⋊C8—41no—
70C2×M16—61no—
71C4⋊Q8—32no—
72#125—61no—
73C22⋊Q8—62no—
74C4×Q8—62no—
75#117—61no—
76C2×C4⋊C4—64no—
77C2×C4⋊C4—62no—
78C2×C42—22no—
79C2×C42D1212yes—
80C2×C42C22×D311yescharacteristic
81C4⋊C8—22no—
82C4⋊C8—62no—
83C22⋊C8—22no—
84C4×C8—22no—
85#95—34no—
86#95—64no—
87#95—62no—
88C23×C3#12011yescharacteristic
89C2×C3×C4—24no—
90C2×C3×C4—42no—
91C2×C3×C4—42no—
92C2×C3×C4—42no—
93C2×C3×C4C22⋊Q812yes—
94C2×C3×C4C22⋊Q811yescharacteristic
95C2×C3×C4#12411yescharacteristic
96C2×C3×C4#12412yes—
97C2×C3×C4#13512yes—
98C2×C3×C4#13712yes—
99C2×Dic3—48no—
100C2×Dic3—42no—
101C2×Dic3—44no—
102Dic6—82no—
103C3×C8—82no—
104C3⋊C8—81no—
105C2×Q8—62no—
106C22×C4—22no—
107C22×C4—64no—
108C22×C4C2×D1211yescharacteristic
109C22×C4D3×D412yes—
110M16—122no—
111C2×C8—42no—
112C2×C8—121no—
113C4⋊C4—62no—
114C4⋊C4—62no—
115C4⋊C4—128no—
116C4⋊C4—122no—
117C4⋊C4—122no—
118C4⋊C4—122no—
119C22⋊C4—121no—
120C42—22no—
121C42—22no—
122C42—44no—
123C42—121no—
124C42#23512yes—
125C42#23711yescharacteristic
126C42D3×Q811yescharacteristic
127C22×C3—22no—
128C22×C3#39311yescharacteristic
129C22×C3#45212yes—
130C3×C4—22no—
131C3×C4—22no—
132C3×C4—44no—
133C3×C4#47912yes—
134C3×C4#48811yescharacteristic
135C3×C4#49111yescharacteristic
136Dic3—84no—
137Dic3—82no—
138C23#88211yescharacteristic
139Q8—242no—
140C2×C4—24no—
141C2×C4—42no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—128no—
145C2×C4—122no—
146C2×C4—124no—
147C2×C4#87511yescharacteristic
148C2×C4#88512yes—
149C2×C4#89612yes—
150C2×C4#89711yescharacteristic
151C2×C4#90412yes—
152C2×C4#90912yes—
153C8—82no—
154C8—241no—
155C2×C3—22no—
156C2×C3#202611yescharacteristic
157C2×C3#203112yes—
158C22—22no—
159C22#467311yescharacteristic
160C22#473312yes—
161C4—22no—
162C4—22no—
163C4—44no—
164C4—244no—
165C4—242no—
166C4#469711yescharacteristic
167C4#470311yescharacteristic
168C4#482912yes—
169C3#1036911yescharacteristic, complemented, Sylow
170C2—22no—
171C2#6869011yescharacteristic
172C2#6877912yes—
subgroupG/H
centerC22#4673
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1276C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#1276D3
hypocenterC3#10369
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
169. C3#10369semidirect

Maximal subgroups

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

subgroupindex
#733102
#707932
#693962
C3×#12762
#677382
#103693