Finite Group Databasebeta

← #10697434

Subgroups — #10697434

Order 768.

542 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3162C211yesmaximal, characteristic, complemented
2#83355C212yesmaximal
3#71838C212yesmaximal, complemented
4#69823C211yesmaximal, characteristic
5#67661C211yesmaximal, characteristic, complemented
6#10220—31nomaximal, Sylow
7C3×#533C2212yes—
8C2×#874C2211yescharacteristic
9C3×#501—22no—
10C3×#501—22no—
11C3×#427—22no—
12C2×C3×#97C2211yescharacteristic
13C2×#821—22no—
14#5088—22no—
15#5085—22no—
16#5046—24no—
17#5027—24no—
18#4936C2211yescharacteristic
19#4925C2212yes—
20#4538—24no—
21#3208—32no—
22#3162D311yescharacteristic, complemented
23#1961—31no—
24#1518—32no—
25#1281—31no—
26C22×D12—21no—
27C22×D12—22no—
28C22×C3×C8D412yes—
29C3×C4⋊Q8D412yes—
30C3×C4⋊Q8D412yes—
31C3×#125—22no—
32C3×C22⋊Q8—22no—
33C3×C22⋊Q8—22no—
34C3×C4×Q8—22no—
35C3×C4×Q8—22no—
36C3×#117—22no—
37C2×C3×C42C2311yescharacteristic
38C2×#215—28no—
39C2×C4⋊Dic3—21no—
40#874—22no—
41#874—21no—
42#874D412yes—
43C3×#107—24no—
44C3×#106—22no—
45C3×#106—22no—
46C3×#97—22no—
47C3⋊#95—22no—
48#830D414yes—
49#821—44no—
50#809—28no—
51C4⋊(C3⋊C8)—22no—
52C4×C3⋊C8—22no—
53#533C2×D312yes—
54C2×#127—31no—
55#501—22no—
56#501—22no—
57#492—62no—
58#486—62no—
59#460—64no—
60#450—64no—
61#431—32no—
62#427—22no—
63C2×#102—62no—
64C2×#97C2×D311yescharacteristic
65#382—31no—
66#328—64no—
67C23×D3—41no—
68C2×C3×Q8—22no—
69C2×C3×Q8—22no—
70C22×C3×C4C2×D412yes—
71C22×C3×C4C2×D411yescharacteristic
72C22×Dic3—41no—
73C2×D12—24no—
74C2×D12—22no—
75C2×D12—42no—
76C2×D12—42no—
77C2×D12—41no—
78C2×D12—42no—
79C2×D12—84no—
80C2×C3×C8—22no—
81C2×C3×C8—22no—
82C2×C3×C8SD1614yescomplemented
83C3×C4⋊C4—22no—
84C3×C4⋊C4—22no—
85C3×C4⋊C4—22no—
86C3×C4⋊C4—22no—
87C3×C4⋊C4—42no—
88C3×C4⋊C4—42no—
89C3×C22⋊C4—42no—
90C3×C42—42no—
91C3×C42C2×D411yescharacteristic
92C3×C42C2×D412yes—
93C3×C42#4111yescharacteristic
94#215—88no—
95C4⋊Dic3—42no—
96C2×C3⋊C8—24no—
97C22×D4—62no—
98C22×D4—61no—
99C2×SD16—68no—
100C22×C8#6712yes—
101C4⋊Q8#6712yes—
102C4⋊Q8#6712yes—
103#127—32no—
104#127—61no—
105#127—62no—
106#125—22no—
107C22⋊Q8—22no—
108C22⋊Q8—22no—
109C4×Q8—22no—
110C4×Q8—22no—
111#117—22no—
112C2×C4⋊C4—61no—
113C2×C42C22×D311yescharacteristic
114#107—24no—
115#106—22no—
116#106—22no—
117C4⋊C8—62no—
118#102—68no—
119#102—124no—
120C22⋊C8—34no—
121#97—22no—
122C4×C8—62no—
123#95—62no—
124C23×C3#12711yescharacteristic
125C22×D3—42no—
126C22×D3—84no—
127C22×D3—81no—
128C22×D3—81no—
129C3×Q8—44no—
130C3×Q8—82no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4#12012yes—
135C2×C3×C4#12011yescharacteristic
136C2×C3×C4#12111yescharacteristic
137C2×C3×C4#12112yes—
138C2×C3×C4C2×SD1612yes—
139C2×C3×C4#13712yes—
140C2×Dic3—82no—
141C2×Dic3—81no—
142D12—48no—
143D12—44no—
144D12—84no—
145D12—82no—
146C3×C8—24no—
147C3⋊C8—44no—
148C24—121no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×D4—64no—
152C2×D4—62no—
153C2×D4—122no—
154C2×D4—122no—
155C2×D4—122no—
156C2×D4—121no—
157C2×D4—244no—
158C22×C4—121no—
159C22×C4D3×D412yes—
160C22×C4C2×#6711yescharacteristic
161SD16—248no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—64no—
165C2×C8#21514yescomplemented
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4—122no—
173C22⋊C4—42no—
174C42—42no—
175C42D3×D411yescharacteristic
176C42#23711yescharacteristic
177C42C2×#6712yes—
178C22×C3—22no—
179C22×C3#39211yescharacteristic
180C22×C3#49211yescharacteristic
181C22×C3#49711yescharacteristic
182C2×D3—84no—
183C2×D3—82no—
184C2×D3—162no—
185C2×D3—162no—
186C3×C4—22no—
187C3×C4—22no—
188C3×C4—82no—
189C3×C4—82no—
190C3×C4#44812yes—
191C3×C4#46112yes—
192Dic3—161no—
193C23—122no—
194C23—244no—
195C23—241no—
196C23—241no—
197C23#94011yescharacteristic
198Q8—44no—
199Q8—82no—
200D4—128no—
201D4—124no—
202D4—244no—
203D4—242no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—242no—
208C2×C4—241no—
209C2×C4#93712yes—
210C2×C4#93811yescharacteristic
211C2×C4#93912yes—
212C2×C4C2×#21512yes—
213C2×C4#94212yes—
214C2×C4#95311yescharacteristic
215C8—24no—
216C8—124no—
217C2×C3—21no—
218C2×C3—21no—
219C2×C3#200812yes—
220C2×C3#201311yescharacteristic
221D3—162no—
222C22—22no—
223C22—242no—
224C22—244no—
225C22—482no—
226C22—482no—
227C22#508511yescharacteristic
228C22#508611yescharacteristic
229C22#522311yescharacteristic
230C4—22no—
231C4—22no—
232C4—82no—
233C4—82no—
234C4—481no—
235C4#508312yes—
236C4#522712yes—
237C3#1022011yescharacteristic, complemented, Sylow
238C2—21no—
239C2—21no—
240C2—482no—
241C2#7179111yescharacteristic
242C2#7211912yes—
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3162C2
socleC22×C3#392
prime radicalC2×#874C22
hypercenter#3162D3
hypocenterC3#10220
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
1. C3×#3162C2semidirect
3. #71838C2semidirect
5. #67661C2semidirect
22. #3162D3semidirect
82. C2×C3×C8SD16semidirect
165. C2×C8#215semidirect
237. C3#10220semidirect

Maximal subgroups

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

subgroupindex
C3×#31622
#833552
#718382
#698232
#676612
#102203