Finite Group Databasebeta

← #10698280

Subgroups — #10698280

Order 768.

416 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 99 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#73308C211yesmaximal, characteristic
2C3×#1994C211yesmaximal, characteristic
3#71838C212yesmaximal
4#67713C212yesmaximal
5#67611C211yesmaximal, characteristic
6#10363—31nomaximal, Sylow
7C3×#533C2211yescharacteristic
8C2×#869—21no—
9C3×#423C2211yescharacteristic
10C3×C43C2211yescharacteristic
11#4936—21no—
12#4936C2212yes—
13#4929—24no—
14#4925C2212yes—
15C3×#339—21no—
16C3×#339—21no—
17#4538—24no—
18#2862—31no—
19#1994D311yescharacteristic
20#1518—32no—
21#1286—32no—
22#1276—31no—
23C22×Dic6—22no—
24C2×C3×M16—21no—
25C3×C4⋊Q8Q812yescomplemented
26C3×#125—21no—
27C3×C22⋊Q8—22no—
28C3×C4×Q8—22no—
29C3×#117—21no—
30C2×C3×C42—22no—
31C2×C3×C42D411yescharacteristic
32C2×C3×C42D411yescharacteristic
33C2×C3×C42C2311yescharacteristic
34C2×C4⋊Dic3—22no—
35C2×C4⋊Dic3—22no—
36#869—42no—
37#869—42no—
38C3×C4⋊C8—22no—
39C3⋊#95—24no—
40C3⋊#95—22no—
41C3⋊#95D414yes—
42#830—22no—
43C4⋊(C3⋊C8)—22no—
44C4×C3⋊C8—22no—
45#533C2×D311yescharacteristic
46C2×C4⋊Q8—61no—
47#431—32no—
48#423C2×D311yescharacteristic
49#391—64no—
50#382—32no—
51#382—61no—
52C43C2×D311yescharacteristic
53#339—21no—
54#339—21no—
55#328—64no—
56C2×C3×Q8—22no—
57C22×C3×C4—21no—
58C22×C3×C4—21no—
59C22×C3×C4C2×D412yes—
60C22×C3×C4C2×D411yescharacteristic
61C22×Dic3—24no—
62C2×Dic6—44no—
63C2×Dic6—48no—
64C3×M16—41no—
65C3×M16—41no—
66C2×C3×C8—41no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—22no—
69C3×C4⋊C4—42no—
70C3×C22⋊C4—41no—
71C3×C42—21no—
72C3×C42—21no—
73C3×C42—44no—
74C3×C42—41no—
75C3×C42SD1612yes—
76C3×C42Q1612yes—
77C3×C42C2×Q811yescharacteristic
78C3×C42#4111yescharacteristic
79C3×C42#4112yes—
80C4⋊Dic3—44no—
81C4⋊Dic3—42no—
82C4⋊Dic3—42no—
83C2×C3⋊C8—42no—
84C22×Q8—62no—
85C2×M16—21no—
86C4⋊Q8—122no—
87C4⋊Q8—122no—
88C4⋊Q8Dic612yescomplemented
89#125—21no—
90C22⋊Q8—22no—
91C4×Q8—22no—
92#117—21no—
93C2×C4⋊C4—62no—
94C2×C4⋊C4—62no—
95C2×C42—22no—
96C2×C42#6711yescharacteristic
97C2×C42#6711yescharacteristic
98C2×C42C22×D311yescharacteristic
99C4⋊C8—22no—
100C4⋊C8—62no—
101C22⋊C8—62no—
102C4×C8—62no—
103#95—34no—
104#95—64no—
105#95—62no—
106C23×C3#12011yescharacteristic
107C3×Q8—82no—
108C2×C3×C4—24no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—42no—
112C2×C3×C4—42no—
113C2×C3×C4—42no—
114C2×C3×C4C22⋊Q812yes—
115C2×C3×C4C22⋊Q811yescharacteristic
116C2×C3×C4#12412yes—
117C2×C3×C4#12411yescharacteristic
118C2×C3×C4C2×SD1611yescharacteristic
119C2×C3×C4C2×Q1611yescharacteristic
120C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
121C2×C3×C4#13711yescharacteristic
122C2×Dic3—48no—
123C2×Dic3—44no—
124Dic6—88no—
125C3×C8—81no—
126C3⋊C8—82no—
127C2×Q8—22no—
128C2×Q8—124no—
129C2×Q8—128no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—64no—
133C22×C4D3×D412yes—
134C22×C4C2×#6711yescharacteristic
135M16—41no—
136M16—41no—
137C2×C8—41no—
138C2×C8—122no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—42no—
142C4⋊C4—124no—
143C4⋊C4—122no—
144C4⋊C4—122no—
145C22⋊C4—41no—
146C42—21no—
147C42—21no—
148C42—44no—
149C42—41no—
150C42#21512yes—
151C42#21612yes—
152C42C2×Dic611yescharacteristic
153C42#23511yescharacteristic
154C42#23712yes—
155C22×C3—22no—
156C22×C3#39311yescharacteristic
157C22×C3#45011yescharacteristic
158C22×C3#45111yescharacteristic
159C3×C4—24no—
160C3×C4—22no—
161C3×C4—22no—
162C3×C4—42no—
163C3×C4—82no—
164C3×C4#47511yescharacteristic
165C3×C4#47711yescharacteristic
166C3×C4#48711yescharacteristic
167C3×C4#48911yescharacteristic
168Dic3—84no—
169C23#93711yescharacteristic
170Q8—82no—
171Q8—248no—
172C2×C4—24no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—128no—
179C2×C4—124no—
180C2×C4#87812yes—
181C2×C4#88512yes—
182C2×C4#92411yescharacteristic
183C2×C4#93611yescharacteristic
184C2×C4C2×#21511yescharacteristic
185C2×C4C2×#21611yescharacteristic
186C2×C4#94911yescharacteristic
187C2×C4#95111yescharacteristic
188C8—81no—
189C8—242no—
190C2×C3—21no—
191C2×C3—21no—
192C2×C3#202411yescharacteristic
193C2×C3#203112yes—
194C22—22no—
195C22#494711yescharacteristic
196C22#504611yescharacteristic
197C22#505011yescharacteristic
198C4—24no—
199C4—22no—
200C4—22no—
201C4—42no—
202C4—82no—
203C4—244no—
204C4#502011yescharacteristic
205C4#502111yescharacteristic
206C4#505511yescharacteristic
207C4#507911yescharacteristic
208C3#1036311yescharacteristic, complemented, Sylow
209C2—21no—
210C2—21no—
211C2#6966811yescharacteristic
212C2#7036812yes—
subgroupG/H
centerC22#4947
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1994C2
socleC22×C3#393
prime radicalC23×C3#120
hypercenter#1994D3
hypocenterC3#10363
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1994

Complements

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

NH ≅ G/Ntype
25. C3×C4⋊Q8Q8semidirect
88. C4⋊Q8Dic6semidirect
208. C3#10363semidirect

Maximal subgroups

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

subgroupindex
#733082
C3×#19942
#718382
#677132
#676112
#103633