Finite Group Databasebeta

← #10698327

Subgroups — #10698327

Order 768.

364 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C3×#2402C211yesmaximal, characteristic
2#69471C212yesmaximal
3#69441C211yesmaximal, characteristic
4#67620C212yesmaximal
5#67614C211yesmaximal, characteristic
6#10373—31nomaximal, Sylow
7#5470C2211yescharacteristic
8C3×#400—22no—
9C3×#398—22no—
10C3×C4×C4⋊C4—22no—
11C3×C4×C4⋊C4C2212yes—
12C3×#376C2211yescharacteristic
13#4925C2212yes—
14#4924C2211yescharacteristic
15#4523—22no—
16#4452—22no—
17#4452—22no—
18#2402D311yescharacteristic
19#1995—31no—
20#1503—32no—
21#1303—32no—
22#1274—31no—
23C2×C3×C4⋊C4—22no—
24C2×C3×C4⋊C4—22no—
25C2×C3×C4⋊C4—22no—
26C2×C3×C42—22no—
27C2×C3×C42D412yes—
28C2×C3×C42C2311yescharacteristic
29#926—21no—
30#924—22no—
31C2×#208—21no—
32#870—21no—
33#869Q812yes—
34C4×Dic6—22no—
35C3×#95—22no—
36C3×#95—22no—
37C3×#95—22no—
38C3×#95D412yes—
39C3×#95D412yes—
40#830—22no—
41C4⋊(C3⋊C8)—22no—
42C4⋊(C3⋊C8)—22no—
43C4×C3⋊C8—22no—
44#533—31no—
45#431—32no—
46#423—31no—
47#400—22no—
48#398—22no—
49C4×C4⋊C4—22no—
50C4×C4⋊C4C2×D312yes—
51#376C2×D311yescharacteristic
52#339—62no—
53#328—62no—
54#328—62no—
55C22×C3×C4—22no—
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4C2×D412yes—
59C22×C3×C4C2×D411yescharacteristic
60C2×Dic6—22no—
61C3×C4⋊C4—44no—
62C3×C4⋊C4—44no—
63C3×C4⋊C4—44no—
64C3×C42—22no—
65C3×C42—22no—
66C3×C42—42no—
67C3×C42—42no—
68C3×C42C2×Q811yescharacteristic
69C3×C42#4112yes—
70C3×C42#4111yescharacteristic
71#217—41no—
72C4⋊Dic3—22no—
73C4⋊Dic3—22no—
74Dic3⋊C4—42no—
75C4×Dic3—41no—
76#208—42no—
77C2×C3⋊C8—42no—
78C2×C3⋊C8—41no—
79C2×M16—61no—
80C4⋊Q8—32no—
81#125—61no—
82C22⋊Q8—62no—
83C4×Q8—62no—
84#117—61no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C42—22no—
89C2×C42#6712yes—
90C2×C42C22×D311yescharacteristic
91C4⋊C8—62no—
92C4⋊C8—62no—
93C22⋊C8—62no—
94C4×C8—62no—
95#95—22no—
96#95—22no—
97#95—22no—
98#95#6712yes—
99#95#6712yes—
100C23×C3#12011yescharacteristic
101C2×C3×C4—24no—
102C2×C3×C4—42no—
103C2×C3×C4—42no—
104C2×C3×C4—42no—
105C2×C3×C4—42no—
106C2×C3×C4—42no—
107C2×C3×C4—42no—
108C2×C3×C4—42no—
109C2×C3×C4—42no—
110C2×C3×C4—42no—
111C2×C3×C4C22⋊Q812yes—
112C2×C3×C4C22⋊Q811yescharacteristic
113C2×C3×C4#12412yes—
114C2×C3×C4#12411yescharacteristic
115C2×C3×C4#13512yes—
116C2×C3×C4#13712yes—
117C2×Dic3—42no—
118Dic6—82no—
119C3⋊C8—82no—
120C3⋊C8—81no—
121C2×Q8—62no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4C2×#6712yes—
126C22×C4C2×#6711yescharacteristic
127M16—122no—
128C2×C8—122no—
129C2×C8—121no—
130C4⋊C4—44no—
131C4⋊C4—44no—
132C4⋊C4—44no—
133C4⋊C4—62no—
134C4⋊C4—62no—
135C4⋊C4—122no—
136C22⋊C4—121no—
137C42—22no—
138C42—22no—
139C42—42no—
140C42—42no—
141C42—121no—
142C42D3×Q811yescharacteristic
143C42#23911yescharacteristic
144C42#23912yes—
145C22×C3—22no—
146C22×C3#39311yescharacteristic
147C22×C3#45212yes—
148C3×C4—22no—
149C3×C4—22no—
150C3×C4—44no—
151C3×C4—82no—
152C3×C4—82no—
153C3×C4#47912yes—
154C3×C4#48811yescharacteristic
155C3×C4#49111yescharacteristic
156Dic3—82no—
157C23#95311yescharacteristic
158Q8—242no—
159C2×C4—24no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—122no—
170C2×C4#94612yes—
171C2×C4#94611yescharacteristic
172C2×C4#94712yes—
173C2×C4#94711yescharacteristic
174C2×C4#95012yes—
175C2×C4#95112yes—
176C8—242no—
177C8—241no—
178C2×C3—22no—
179C2×C3#202611yescharacteristic
180C2×C3#203112yes—
181C22—22no—
182C22#523111yescharacteristic
183C22#523912yes—
184C4—22no—
185C4—22no—
186C4—44no—
187C4—82no—
188C4—82no—
189C4—242no—
190C4#506312yes—
191C4#506811yescharacteristic
192C4#506911yescharacteristic
193C3#1037311yescharacteristic, complemented, Sylow
194C2—22no—
195C2#7174411yescharacteristic
196C2#7184212yes—
subgroupG/H
centerC2×C4#947
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2402C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#2402D3
hypocenterC3#10373
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2402

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
C3×#24022
#694712
#694412
#676202
#676142
#103733