Finite Group Databasebeta

← #10696478

Subgroups — #10696478

Order 768.

552 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 97 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×#2822C211yesmaximal, characteristic, complemented
2C2×#5067C212yesmaximal
3C2×#4543C212yesmaximal, complemented
4#69553C211yesmaximal, characteristic
5#67630C211yesmaximal, characteristic, complemented
6#10080—31nomaximal, Sylow
7C2×C3×#125C2212yes—
8C2×#874C2211yescharacteristic
9C3×#400—22no—
10C3×#398—22no—
11C3×#398—22no—
12C3×#382C2211yescharacteristic
13#5067—24no—
14#5067—24no—
15C2×#809—22no—
16C2×#809—22no—
17C2×#809—22no—
18C2×#803C2211yescharacteristic
19C2×#803C2212yes—
20#4543—24no—
21C2×#490—32no—
22#2822D311yescharacteristic, complemented
23#1961—31no—
24C2×#330—32no—
25#1295—31no—
26C22×D12—22no—
27C22×D12—21no—
28C3×#125—22no—
29C3×#125—22no—
30C3×#125—22no—
31C3×#125—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4D412yes—
36C2×C3×C4⋊C4D412yes—
37C2×C3×C4⋊C4D412yes—
38C2×C3×C42C2311yescharacteristic
39C22×C3⋊C8—21no—
40C22×C3⋊C8—22no—
41#874—22no—
42#874—21no—
43#874D412yes—
44C3×#95—24no—
45C3×#95—22no—
46C3×#95—22no—
47#809—28no—
48#809—28no—
49#809—44no—
50#803—22no—
51#803—22no—
52#803D414yes—
53C2×#127—31no—
54C2×#125C2×D312yes—
55#490—64no—
56#490—64no—
57C2×#102—62no—
58C2×#102—62no—
59C2×#102—62no—
60C2×#97—32no—
61C2×#97—31no—
62#400—22no—
63#398—22no—
64#398—22no—
65#382C2×D311yescharacteristic
66#330—64no—
67C23×D3—41no—
68C22×C3×C4—22no—
69C22×C3×C4—22no—
70C22×C3×C4—22no—
71C22×C3×C4C2×D412yes—
72C22×C3×C4C2×D411yescharacteristic
73C2×D12—24no—
74C2×D12—22no—
75C2×D12—42no—
76C2×D12—48no—
77C2×D12—42no—
78C2×D12—42no—
79C2×D12—41no—
80C3×C4⋊C4—22no—
81C3×C4⋊C4—22no—
82C3×C4⋊C4—22no—
83C3×C4⋊C4—22no—
84C3×C4⋊C4—42no—
85C3×C4⋊C4—42no—
86C3×C4⋊C4—42no—
87C3×C4⋊C4—42no—
88C3×C4⋊C4—42no—
89C3×C4⋊C4—82no—
90C3×C42C2×D411yescharacteristic
91C3×C42C2×D412yes—
92C3×C42#4111yescharacteristic
93C2×C3⋊C8—22no—
94C2×C3⋊C8—24no—
95C2×C3⋊C8—44no—
96C2×C3⋊C8—41no—
97C2×C3⋊C8—41no—
98C22×D4—62no—
99C22×D4—61no—
100C22×C8—61no—
101C22×C8—62no—
102#127—32no—
103#127—62no—
104#127—61no—
105#125—22no—
106#125—22no—
107#125—22no—
108#125—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4#6712yes—
113C2×C4⋊C4#6712yes—
114C2×C4⋊C4#6712yes—
115C2×C42C22×D311yescharacteristic
116#102—68no—
117#102—68no—
118#102—124no—
119#97—34no—
120#97—62no—
121#97—62no—
122#95—24no—
123#95—22no—
124#95—22no—
125C23×C3#12011yescharacteristic
126C22×D3—42no—
127C22×D3—81no—
128C22×D3—84no—
129C22×D3—81no—
130C2×C3×C4—21no—
131C2×C3×C4—24no—
132C2×C3×C4—21no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4—42no—
136C2×C3×C4—42no—
137C2×C3×C4—44no—
138C2×C3×C4—44no—
139C2×C3×C4—42no—
140C2×C3×C4#12012yes—
141C2×C3×C4#12111yescharacteristic
142C2×C3×C4#12112yes—
143C2×C3×C4#12711yescharacteristic
144D12—48no—
145D12—44no—
146D12—84no—
147D12—82no—
148C3⋊C8—44no—
149C3⋊C8—42no—
150C24—121no—
151C2×D4—64no—
152C2×D4—62no—
153C2×D4—122no—
154C2×D4—128no—
155C2×D4—122no—
156C2×D4—122no—
157C2×D4—121no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4C2×#6712yes—
162C22×C4C2×#6711yescharacteristic
163C2×C8—62no—
164C2×C8—64no—
165C2×C8—124no—
166C2×C8—121no—
167C2×C8—121no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—42no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—82no—
178C42D3×D411yescharacteristic
179C42D3×D412yes—
180C42#23711yescharacteristic
181C22×C3#39211yescharacteristic
182C22×C3#45311yescharacteristic
183C22×C3#45414yes—
184C22×C3#45411yescharacteristic
185C2×D3—84no—
186C2×D3—82no—
187C2×D3—162no—
188C2×D3—162no—
189C3×C4—22no—
190C3×C4—22no—
191C3×C4—21no—
192C3×C4—21no—
193C3×C4—82no—
194C3×C4—82no—
195C3×C4—82no—
196C23—122no—
197C23—241no—
198C23—244no—
199C23—241no—
200C23#95311yescharacteristic
201D4—128no—
202D4—124no—
203D4—244no—
204D4—242no—
205C2×C4—21no—
206C2×C4—24no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—44no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4#93712yes—
216C2×C4#93811yescharacteristic
217C2×C4#93912yes—
218C2×C4#94011yescharacteristic
219C8—124no—
220C8—122no—
221C2×C3#165712yes—
222C2×C3#165812yes—
223C2×C3#200312yes—
224C2×C3#200611yescharacteristic
225D3—162no—
226C22—244no—
227C22—242no—
228C22—482no—
229C22—482no—
230C22#522311yescharacteristic
231C22#524111yescharacteristic
232C22#524514yes—
233C22#524511yescharacteristic
234C4—22no—
235C4—22no—
236C4—21no—
237C4—21no—
238C4—82no—
239C4—82no—
240C4—82no—
241C3#1008011yescharacteristic, complemented, Sylow
242C2—482no—
243C2#7186712yes—
244C2#7187911yescharacteristic
245C2#7195512yes—
246C2#7196012yes—
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2822C2
socleC23×C3#120
prime radicalC2×#874C22
hypercenter#2822D3
hypocenterC3#10080
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2822

Complements

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

NH ≅ G/Ntype
1. C3×#2822C2semidirect
3. C2×#4543C2semidirect
5. #67630C2semidirect
22. #2822D3semidirect
241. C3#10080semidirect

Maximal subgroups

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

subgroupindex
C3×#28222
C2×#50672
C2×#45432
#695532
#676302
#100803