Finite Group Databasebeta

← #10708187

Subgroups — #10708187

Order 768.

394 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 59 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#74343C211yesmaximal, characteristic
2#69573C211yesmaximal, characteristic
3#68140C212yesmaximal
4C3×#1325C211yesmaximal, characteristic
5#67458C212yesmaximal
6#12247—31nomaximal, Sylow
7#5500C2211yescharacteristic
8C2×C4×Dic6—21no—
9C3×#384C2211yescharacteristic
10#4984—22no—
11#4981—21no—
12#4981C2212yes—
13#4973—22no—
14#4964—21no—
15#4721—22no—
16C3×#324C2212yes—
17#4530C2211yescharacteristic
18#4523—21no—
19#4467—22no—
20#2582—31no—
21#2060—31no—
22#1611—32no—
23#1346—32no—
24#1325D311yescharacteristic
25C22×Dic6—21no—
26C2×C3×C4⋊C4—22no—
27C2×C3×C4⋊C4D411yescharacteristic
28C2×C3×C4⋊C4C2311yescharacteristic
29C2×C3×C42D411yescharacteristic
30#926—21no—
31C2×C4⋊Dic3—21no—
32C2×C4⋊Dic3—21no—
33C2×C4⋊Dic3D412yes—
34#924—22no—
35C2×Dic3⋊C4—22no—
36C2×C4×Dic3—22no—
37C2×#208—21no—
38#890—41no—
39#888—41no—
40C4×Dic6—42no—
41C4×Dic6—42no—
42C3×C22⋊C8—22no—
43C3⋊#95—21no—
44C3⋊#95—22no—
45C3⋊#95—22no—
46C3⋊#95D412yes—
47#817—42no—
48#816—42no—
49#808—41no—
50#807—41no—
51#544—31no—
52C2×C4×Q8—61no—
53#483—62no—
54#428—61no—
55#398—32no—
56#398—61no—
57#391—62no—
58#389—62no—
59#384C2×D311yescharacteristic
60#343—31no—
61#339—61no—
62#328—62no—
63#324C2×D312yes—
64C22×C3×C4—21no—
65C22×C3×C4—21no—
66C22×C3×C4C2×D411yescharacteristic
67C22×C3×C4C2×D412yes—
68C22×Dic3—22no—
69C22×Dic3—22no—
70C2×Dic6—24no—
71C2×Dic6—22no—
72C2×Dic6—42no—
73C2×Dic6—42no—
74C2×C3×C8—42no—
75C3×C4⋊C4—41no—
76C3×C4⋊C4—41no—
77C3×C4⋊C4—44no—
78C3×C42—21no—
79C3×C42SD1612yes—
80#217—41no—
81C4⋊Dic3—21no—
82C4⋊Dic3—21no—
83C4⋊Dic3—22no—
84C4⋊Dic3—42no—
85C4⋊Dic3—42no—
86C4⋊Dic3—81no—
87Dic3⋊C4—42no—
88Dic3⋊C4—42no—
89Dic3⋊C4—42no—
90C4×Dic3—44no—
91C4×Dic3—41no—
92#208—22no—
93#208—41no—
94C2×C3⋊C8—41no—
95C22×Q8—61no—
96C2×M16—61no—
97C4⋊Q8—121no—
98#125—121no—
99C22⋊Q8—62no—
100C4×Q8—122no—
101C4×Q8—122no—
102#117—61no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4—61no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4—61no—
108C2×C4⋊C4D1211yescharacteristic
109C2×C4⋊C4C22×D311yescharacteristic
110C2×C42—62no—
111C2×C42D1211yescharacteristic
112#107—121no—
113#106—121no—
114#106—122no—
115#103—122no—
116C22⋊C8—22no—
117#95—32no—
118#95—62no—
119#95—61no—
120#95—62no—
121C23×C3#12011yescharacteristic
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—22no—
128C2×C3×C4—42no—
129C2×C3×C4—42no—
130C2×C3×C4—41no—
131C2×C3×C4C2×SD1611yescharacteristic
132C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
133C2×Dic3—24no—
134C2×Dic3—42no—
135C2×Dic3—42no—
136C2×Dic3—42no—
137C2×Dic3—42no—
138C2×Dic3—41no—
139C2×Dic3—41no—
140C2×Dic3—42no—
141Dic6—48no—
142Dic6—82no—
143C3×C8—82no—
144C3⋊C8—42no—
145C2×Q8—64no—
146C2×Q8—62no—
147C2×Q8—122no—
148C2×Q8—122no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—62no—
152C22×C4—62no—
153C22×C4C2×D1211yescharacteristic
154C22×C4D3×D412yes—
155M16—62no—
156M16—121no—
157C2×C8—42no—
158C2×C8—121no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—44no—
162C4⋊C4—61no—
163C4⋊C4—62no—
164C4⋊C4—61no—
165C4⋊C4—122no—
166C4⋊C4—122no—
167C4⋊C4—122no—
168C4⋊C4—122no—
169C4⋊C4—122no—
170C4⋊C4—241no—
171C22⋊C4—121no—
172C42—21no—
173C42—124no—
174C42—121no—
175C42#20412yes—
176C22×C3—21no—
177C22×C3—21no—
178C22×C3#45011yescharacteristic
179C22×C3#45611yescharacteristic
180C22×C3B(4,𝔽2)11yescharacteristic
181C3×C4—24no—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4—42no—
185C3×C4—81no—
186Dic3—44no—
187Dic3—82no—
188Dic3—81no—
189Dic3—81no—
190C23#88211yescharacteristic
191Q8—128no—
192Q8—242no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—22no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—64no—
203C2×C4—122no—
204C2×C4—122no—
205C2×C4—122no—
206C2×C4—121no—
207C2×C4—121no—
208C2×C4—122no—
209C2×C4—122no—
210C2×C4C2×#20411yescharacteristic
211C2×C4#90811yescharacteristic
212C8—82no—
213C8—122no—
214C2×C3—21no—
215C2×C3—21no—
216C2×C3#160411yescharacteristic
217C2×C3#219811yescharacteristic
218C2×C3#220511yescharacteristic
219C22—21no—
220C22—21no—
221C22#473111yescharacteristic
222C22#474111yescharacteristic
223C22#475211yescharacteristic
224C4—24no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—81no—
229C4—124no—
230C4—241no—
231C4—242no—
232C4—241no—
233C3#1224711yescharacteristic, complemented, Sylow
234C2—21no—
235C2—21no—
236C2#6813311yescharacteristic
237C2#6914811yescharacteristic
238C2#6916911yescharacteristic
subgroupG/H
centerC22#4741
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1325C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#1325D3
hypocenterC3#12247
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
233. C3#12247semidirect

Maximal subgroups

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

subgroupindex
#743432
#695732
#681402
C3×#13252
#674582
#122473