Finite Group Databasebeta

← #10698295

Subgroups — #10698295

Order 768.

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

normal subgroups only

This group has 402 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#73310C211yesmaximal, characteristic, complemented
2C3×#1962C211yesmaximal, characteristic
3#71814C212yesmaximal
4#67713C212yesmaximal, complemented
5#67611C211yesmaximal, characteristic, complemented
6#10366—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8C3×#423C2211yescharacteristic
9C2×C3×#104—22no—
10C3×C43C2211yescharacteristic
11#4936C2212yes—
12#4933—24no—
13#4928—22no—
14#4925C2212yes—
15#4537—24no—
16#2864—31no—
17#1962D311yescharacteristic
18#1517—32no—
19#1286—32no—
20#1276—31no—
21C2×C3×#41—21no—
22C2×C3×M16—21no—
23C3×C4⋊Q8Q811yescharacteristic
24C3×#127Q811yescharacteristic
25C3×#124—21no—
26C3×#121—22no—
27C3×C4×D4—22no—
28C2×C3×C42—22no—
29C2×C3×C42D412yes—
30C2×C3×C42C2311yescharacteristic
31C2×C4⋊Dic3—22no—
32C2×Dic3⋊C4—24no—
33C3×C4⋊C8—22no—
34C3×#104—44no—
35C3⋊#95—24no—
36C3⋊#95—22no—
37C3⋊#95D414yes—
38#830—22no—
39C4⋊(C3⋊C8)—22no—
40C4×C3⋊C8—22no—
41#532C2×D311yescharacteristic
42#431—32no—
43#423C2×D311yescharacteristic
44C2×#104—22no—
45#389—62no—
46#387—64no—
47#382—32no—
48C43C2×D311yescharacteristic
49#327—64no—
50C3×#41—42no—
51C3×#41—42no—
52C2×C3×Q8—21no—
53C2×C3×D4—21no—
54C2×C3×D4—22no—
55C2×C3×D4—41no—
56C22×C3×C4—22no—
57C22×C3×C4—41no—
58C22×C3×C4C2×D412yes—
59C22×C3×C4C2×D411yescharacteristic
60C22×Dic3—24no—
61C3×M16—42no—
62C2×C3×C8—41no—
63C3×C4⋊C4—22no—
64C3×C22⋊C4—42no—
65C3×C42—22no—
66C3×C42—22no—
67C3×C42—44no—
68C3×C42C2×Q811yescharacteristic
69C3×C42#4111yescharacteristic
70C3×C42#4112yes—
71C4⋊Dic3—42no—
72C4⋊Dic3—42no—
73Dic3⋊C4—48no—
74C2×C3⋊C8—42no—
75C2×#41—21no—
76C2×M16—21no—
77C4⋊Q8Dic611yescharacteristic
78#127Dic611yescharacteristic
79#124—21no—
80#121—22no—
81C4×D4—22no—
82C2×C4⋊C4—64no—
83C2×C4⋊C4—62no—
84C2×C42—22no—
85C2×C42#6712yes—
86C2×C42C22×D311yescharacteristic
87C4⋊C8—22no—
88C4⋊C8—62no—
89#104—44no—
90C22⋊C8—62no—
91C4×C8—62no—
92#95—34no—
93#95—64no—
94#95—62no—
95C23×C3—41no—
96C23×C3#12011yescharacteristic
97C3×Q8—42no—
98C3×D4—42no—
99C3×D4—82no—
100C3×D4—82no—
101C2×C3×C4—24no—
102C2×C3×C4—42no—
103C2×C3×C4—42no—
104C2×C3×C4—42no—
105C2×C3×C4—41no—
106C2×C3×C4—81no—
107C2×C3×C4—81no—
108C2×C3×C4C22⋊Q812yes—
109C2×C3×C4C22⋊Q811yescharacteristic
110C2×C3×C4#12412yes—
111C2×C3×C4#12411yescharacteristic
112C2×C3×C4#13512yes—
113C2×C3×C4AGL(1,ℤ/8)12yes—
114C2×Dic3—48no—
115C2×Dic3—44no—
116C3×C8—81no—
117C3⋊C8—82no—
118#41—42no—
119#41—42no—
120C2×Q8—21no—
121C2×D4—22no—
122C2×D4—21no—
123C2×D4—41no—
124C22×C4—22no—
125C22×C4—41no—
126C22×C4—64no—
127C22×C4D3×D412yes—
128C22×C4C2×#6711yescharacteristic
129M16—42no—
130C2×C8—41no—
131C2×C8—122no—
132C4⋊C4—22no—
133C4⋊C4—128no—
134C4⋊C4—122no—
135C4⋊C4—122no—
136C22⋊C4—42no—
137C42—22no—
138C42—22no—
139C42—44no—
140C42C2×Dic611yescharacteristic
141C42#23511yescharacteristic
142C42#23712yes—
143C22×C3—22no—
144C22×C3—81no—
145C22×C3—82no—
146C22×C3#39311yescharacteristic
147C22×C3#44912yes—
148C3×C4—22no—
149C3×C4—22no—
150C3×C4—44no—
151C3×C4—81no—
152C3×C4#47812yes—
153C3×C4#48811yescharacteristic
154C3×C4#49011yescharacteristic
155Dic3—84no—
156C23—41no—
157C23#93711yescharacteristic
158Q8—42no—
159D4—42no—
160D4—82no—
161D4—82no—
162C2×C4—24no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—128no—
170C2×C4—124no—
171C2×C4#87812yes—
172C2×C4#88512yes—
173C2×C4#92411yescharacteristic
174C2×C4#93212yes—
175C2×C4#93611yescharacteristic
176C2×C4#95012yes—
177C8—81no—
178C8—242no—
179C2×C3—22no—
180C2×C3—161no—
181C2×C3#202511yescharacteristic
182C2×C3#203112yes—
183C22—22no—
184C22—81no—
185C22—82no—
186C22#494711yescharacteristic
187C22#503712yes—
188C4—22no—
189C4—22no—
190C4—44no—
191C4—81no—
192C4—244no—
193C4#500912yes—
194C4#505611yescharacteristic
195C4#507111yescharacteristic
196C3#1036611yescharacteristic, complemented, Sylow
197C2—22no—
198C2—161no—
199C2#6963911yescharacteristic
200C2#7036812yes—
subgroupG/H
centerC22#4947
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1962C2
socleC22×C3#393
prime radicalC3×#532C22
hypercenter#1962D3
hypocenterC3#10366
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1962

Complements

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

NH ≅ G/Ntype
1. #73310C2semidirect
4. #67713C2semidirect
5. #67611C2semidirect
196. C3#10366semidirect

Maximal subgroups

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

subgroupindex
#733102
C3×#19622
#718142
#677132
#676112
#103663