Finite Group Databasebeta

← #10698330

Subgroups — #10698330

Order 768.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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#73398C211yesmaximal, characteristic
2C3×#1995C211yesmaximal, characteristic
3#71835C212yesmaximal
4#67715C212yesmaximal
5#67614C211yesmaximal, characteristic
6#10373—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C3×#423C2211yescharacteristic
9C3×#376C2211yescharacteristic
10C4×C4⋊Dic3C2212yes—
11C4×Dic3⋊C4—22no—
12#4925C2212yes—
13#4660—22no—
14#4657—22no—
15C3×#339—22no—
16#4538—22no—
17#4538—22no—
18#2402—31no—
19#1995D311yescharacteristic
20#1503—32no—
21#1303—32no—
22#1274—31no—
23C2×C3×M16—21no—
24C3×C4⋊Q8Q812yes—
25C3×#125—21no—
26C3×C22⋊Q8—22no—
27C3×C4×Q8—22no—
28C3×#117—21no—
29C2×C3×C42D412yes—
30C2×C3×C42C2311yescharacteristic
31C2×C4⋊Dic3—22no—
32C2×Dic3⋊C4—22no—
33C2×Dic3⋊C4—22no—
34C2×C4×Dic3—22no—
35C3×C4⋊C8—22no—
36C3×#95—22no—
37C3⋊#95—22no—
38C3⋊#95—22no—
39C3⋊#95D412yes—
40C3⋊#95D412yes—
41#830—22no—
42C4⋊(C3⋊C8)—22no—
43C4×C3⋊C8—22no—
44#533C2×D311yescharacteristic
45#431—32no—
46#423C2×D311yescharacteristic
47#400—62no—
48#398—62no—
49C4×C4⋊C4—32no—
50C4×C4⋊C4—62no—
51#376C2×D311yescharacteristic
52#339—22no—
53#328—62no—
54#328—62no—
55C2×C3×Q8—22no—
56C22×C3×C4—22no—
57C22×C3×C4C2×D412yes—
58C22×C3×C4C2×D411yescharacteristic
59C22×Dic3—22no—
60C22×Dic3—22no—
61C3×M16—42no—
62C2×C3×C8—41no—
63C3×C4⋊C4—22no—
64C3×C4⋊C4—22no—
65C3×C4⋊C4—42no—
66C3×C22⋊C4—41no—
67C3×C42—22no—
68C3×C42—22no—
69C3×C42—41no—
70C3×C42C2×Q811yescharacteristic
71C3×C42#4111yescharacteristic
72C3×C42#4112yes—
73C4⋊Dic3—44no—
74Dic3⋊C4—44no—
75Dic3⋊C4—44no—
76C4×Dic3—42no—
77C4×Dic3—42no—
78C2×C3⋊C8—42no—
79C2×M16—21no—
80C4⋊Q8Dic612yes—
81#125—21no—
82C22⋊Q8—22no—
83C4×Q8—22no—
84#117—21no—
85C2×C4⋊C4—62no—
86C2×C4⋊C4—62no—
87C2×C4⋊C4—62no—
88C2×C42—62no—
89C2×C42#6712yes—
90C2×C42C22×D311yescharacteristic
91C4⋊C8—22no—
92C4⋊C8—62no—
93C22⋊C8—62no—
94C4×C8—62no—
95#95—22no—
96#95—32no—
97#95—32no—
98#95—62no—
99#95—62no—
100C23×C3#12011yescharacteristic
101C3×Q8—82no—
102C2×C3×C4—42no—
103C2×C3×C4—42no—
104C2×C3×C4—42no—
105C2×C3×C4—42no—
106C2×C3×C4—42no—
107C2×C3×C4C22⋊Q812yes—
108C2×C3×C4C22⋊Q811yescharacteristic
109C2×C3×C4#12412yes—
110C2×C3×C4#12411yescharacteristic
111C2×C3×C4#13512yes—
112C2×C3×C4#13712yes—
113C2×Dic3—24no—
114C2×Dic3—42no—
115C2×Dic3—42no—
116C2×Dic3—42no—
117C2×Dic3—42no—
118C2×Dic3—42no—
119C3×C8—81no—
120C3⋊C8—82no—
121C2×Q8—22no—
122C22×C4—22no—
123C22×C4—62no—
124C22×C4—62no—
125C22×C4D3×D412yes—
126C22×C4C2×#6711yescharacteristic
127M16—42no—
128C2×C8—41no—
129C2×C8—122no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—42no—
133C4⋊C4—124no—
134C4⋊C4—124no—
135C4⋊C4—124no—
136C22⋊C4—41no—
137C42—22no—
138C42—22no—
139C42—41no—
140C42—122no—
141C42—122no—
142C42C2×Dic611yescharacteristic
143C42#23511yescharacteristic
144C42#23712yes—
145C22×C3—22no—
146C22×C3#39311yescharacteristic
147C22×C3#45212yes—
148C3×C4—22no—
149C3×C4—22no—
150C3×C4—82no—
151C3×C4—82no—
152C3×C4#47912yes—
153C3×C4#48811yescharacteristic
154C3×C4#49111yescharacteristic
155Dic3—44no—
156Dic3—82no—
157C23#93711yescharacteristic
158Q8—82no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—64no—
165C2×C4—122no—
166C2×C4—122no—
167C2×C4—122no—
168C2×C4—122no—
169C2×C4—122no—
170C2×C4#87812yes—
171C2×C4#88512yes—
172C2×C4#92411yescharacteristic
173C2×C4#93611yescharacteristic
174C2×C4#94212yes—
175C2×C4#95012yes—
176C8—81no—
177C8—242no—
178C2×C3—22no—
179C2×C3#202611yescharacteristic
180C2×C3#203112yes—
181C22—22no—
182C22#494711yescharacteristic
183C22#504712yes—
184C4—22no—
185C4—22no—
186C4—82no—
187C4—82no—
188C4—124no—
189C4—242no—
190C4#502212yes—
191C4#505611yescharacteristic
192C4#508211yescharacteristic
193C3#1037311yescharacteristic, complemented, Sylow
194C2—22no—
195C2#6966911yescharacteristic
196C2#7039412yes—
subgroupG/H
centerC2×C4#936
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1995C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#1995D3
hypocenterC3#10373
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1995

Complements

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

NH ≅ G/Ntype
193. C3#10373semidirect

Maximal subgroups

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

subgroupindex
#733982
C3×#19952
#718352
#677152
#676142
#103733