Finite Group Databasebeta

← #10698425

Subgroups — #10698425

Order 768.

414 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 73 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#74088C211yesmaximal, characteristic
2#69975C212yesmaximal
3#69534C211yesmaximal, characteristic
4C3×#1306C211yesmaximal, characteristic
5#67682C212yesmaximal
6#10387—31nomaximal, Sylow
7C2×#926C2211yescharacteristic
8C3×#407C2212yes—
9C3×#381C2211yescharacteristic
10#5211C2211yescharacteristic
11C4×#217C2212yes—
12#4949—21no—
13#4947—21no—
14#4655—22no—
15#4646—22no—
16#4555—22no—
17#4555—22no—
18#4527—21no—
19#4527—21no—
20#2607—31no—
21#1826—32no—
22#1820—31no—
23#1306D311yescharacteristic
24#1284—32no—
25C23×C3×C4C2311yescharacteristic
26C22×Dic6—21no—
27C2×C3×M16—22no—
28C2×C3×C22⋊C4D411yescharacteristic
29C2×C3×C22⋊C4D411yescharacteristic
30C2×#217—22no—
31C2×#217—21no—
32C2×#217—21no—
33#926—22no—
34#926—22no—
35C2×C4⋊Dic3—21no—
36C2×C4⋊Dic3Q812yescomplemented
37C2×Dic3⋊C4—22no—
38C2×C4×Dic3—21no—
39C2×C4×Dic3D412yes—
40C2×C4×Dic3D412yes—
41C3×C22⋊C8—22no—
42C3×C22⋊C8—22no—
43C3×#95—22no—
44C3⋊#95—21no—
45C3⋊#95—22no—
46C3⋊#95—21no—
47C3⋊#95—22no—
48C3⋊#95—22no—
49C2×#117—31no—
50#407C2×D312yes—
51#399—61no—
52#393—61no—
53#391—62no—
54#387—62no—
55#381—31no—
56#381C2×D311yescharacteristic
57C4×C22⋊C4—32no—
58#342—61no—
59#342—61no—
60#339—62no—
61#339—62no—
62C24×C3C2×D411yescharacteristic
63C22×C3×C4—21no—
64C22×C3×C4—21no—
65C22×C3×C4—41no—
66C22×C3×C4—41no—
67C22×C3×C4C2×D412yes—
68C22×C3×C4C2×Q811yescharacteristic
69C22×C3×C4#4112yes—
70C22×C3×C4#4111yescharacteristic
71C22×Dic3—22no—
72C22×Dic3—22no—
73C22×Dic3—22no—
74C22×Dic3—41no—
75C2×Dic6—44no—
76C2×Dic6—81no—
77C3×M16—42no—
78C3×M16—42no—
79C2×C3×C8—42no—
80C3×C22⋊C4—41no—
81C3×C22⋊C4—41no—
82#217—44no—
83#217—41no—
84#217—41no—
85#217—41no—
86#217—41no—
87C4⋊Dic3—24no—
88C4⋊Dic3—81no—
89Dic3⋊C4—44no—
90C4×Dic3—42no—
91C4×Dic3—42no—
92C4×Dic3—42no—
93C22×Q8—61no—
94C23×C4C22×D311yescharacteristic
95C2×M16—22no—
96#117—62no—
97#117—62no—
98C2×C4⋊C4—32no—
99C2×C4⋊C4—62no—
100C2×C4⋊C4—61no—
101C2×C22⋊C4—61no—
102C2×C22⋊C4—62no—
103C2×C22⋊C4—61no—
104C2×C22⋊C4D1211yescharacteristic
105C2×C22⋊C4D1211yescharacteristic
106C2×C42—32no—
107C2×C42—32no—
108C2×C42—61no—
109C22⋊C8—22no—
110C22⋊C8—22no—
111#95—22no—
112#95—61no—
113#95—62no—
114#95—61no—
115#95—62no—
116#95—62no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—42no—
120C23×C3#12011yescharacteristic
121C23×C3C22⋊Q811yescharacteristic
122C23×C3#12411yescharacteristic
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—41no—
126C2×C3×C4—42no—
127C2×C3×C4—41no—
128C2×C3×C4—42no—
129C2×C3×C4—42no—
130C2×C3×C4—41no—
131C2×C3×C4—41no—
132C2×C3×C4C22⋊Q812yes—
133C2×C3×C4#12412yes—
134C2×Dic3—24no—
135C2×Dic3—42no—
136C2×Dic3—42no—
137C2×Dic3—42no—
138C2×Dic3—42no—
139C2×Dic3—42no—
140C2×Dic3—42no—
141C2×Dic3—82no—
142C2×Dic3—81no—
143C2×Dic3—82no—
144Dic6—84no—
145C3×C8—82no—
146C24C2×D1211yescharacteristic
147C2×Q8—124no—
148C2×Q8—241no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—41no—
152C22×C4—41no—
153C22×C4—62no—
154C22×C4—62no—
155C22×C4—62no—
156C22×C4—121no—
157C22×C4#23512yes—
158C22×C4D3×D412yes—
159C22×C4#23711yescharacteristic
160C22×C4D3×Q811yescharacteristic
161M16—42no—
162M16—42no—
163C2×C8—42no—
164C4⋊C4—64no—
165C4⋊C4—124no—
166C4⋊C4—241no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C22⋊C4—124no—
170C22⋊C4—121no—
171C22⋊C4—121no—
172C22⋊C4—121no—
173C22⋊C4—121no—
174C42—122no—
175C42—122no—
176C42—122no—
177C22×C3—22no—
178C22×C3—21no—
179C22×C3—21no—
180C22×C3—22no—
181C22×C3—22no—
182C22×C3—81no—
183C22×C3—81no—
184C22×C3#39311yescharacteristic
185C22×C3#45811yescharacteristic
186C22×C3#45811yescharacteristic
187C3×C4—22no—
188C3×C4—22no—
189C3×C4—81no—
190C3×C4—81no—
191Dic3—44no—
192Dic3—82no—
193Dic3—82no—
194Dic3—161no—
195C23—21no—
196C23—21no—
197C23—42no—
198C23#87511yescharacteristic
199C23#88211yescharacteristic
200C23#89711yescharacteristic
201Q8—244no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—64no—
212C2×C4—122no—
213C2×C4—122no—
214C2×C4—122no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4—242no—
219C2×C4—241no—
220C2×C4—242no—
221C2×C4#88512yes—
222C2×C4#89612yes—
223C8—82no—
224C2×C3—21no—
225C2×C3—21no—
226C2×C3—41no—
227C2×C3—41no—
228C2×C3#203011yescharacteristic
229C2×C3#203112yes—
230C22—22no—
231C22—21no—
232C22—21no—
233C22—22no—
234C22—22no—
235C22—81no—
236C22—81no—
237C22#467311yescharacteristic
238C22#474211yescharacteristic
239C22#474211yescharacteristic
240C4—22no—
241C4—22no—
242C4—81no—
243C4—81no—
244C4—124no—
245C4—242no—
246C4—242no—
247C4—481no—
248C3#1038711yescharacteristic, complemented, Sylow
249C2—21no—
250C2—21no—
251C2—41no—
252C2—41no—
253C2#6895611yescharacteristic
254C2#6904212yes—
subgroupG/H
centerC22#4673
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1306C2
socleC22×C3#393
prime radicalC24×C3C2×D4
hypercenter#1306D3
hypocenterC3#10387
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
36. C2×C4⋊Dic3Q8semidirect
248. C3#10387semidirect

Maximal subgroups

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

subgroupindex
#740882
#699752
#695342
C3×#13062
#676822
#103873