Finite Group Databasebeta

← #10708185

Subgroups — #10708185

Order 768.

394 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 63 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×#2582C211yesmaximal, characteristic
2#69713C212yesmaximal
3#69573C211yesmaximal, characteristic
4#67650C212yesmaximal
5#67635C211yesmaximal, characteristic
6#12247—31nomaximal, Sylow
7C2×C3×C4×Q8—21no—
8#5500C2211yescharacteristic
9C3×#398—21no—
10C3×#398C2212yes—
11C3×#391—22no—
12C3×#389—22no—
13C3×#384C2211yescharacteristic
14#4971—22no—
15#4964—21no—
16#4530C2211yescharacteristic
17#4528C2212yes—
18#4523—21no—
19#4452—22no—
20#2582D311yescharacteristic
21#2060—31no—
22#1611—32no—
23#1346—32no—
24#1325—31no—
25C22×C3×Q8—21no—
26C3×C4×Q8—42no—
27C3×C4×Q8—42no—
28C2×C3×C4⋊C4—21no—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4—21no—
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C4⋊C4D412yes—
34C2×C3×C4⋊C4C2311yescharacteristic
35C2×C3×C42—22no—
36C2×C3×C42D411yescharacteristic
37#926—21no—
38#924—22no—
39C2×#208—21no—
40#890—41no—
41#888—41no—
42C3×#95—22no—
43C3×#95—21no—
44C3×#95—22no—
45C3×#95D412yes—
46#830—22no—
47#810—42no—
48#808—41no—
49#808—42no—
50#807—41no—
51#544—31no—
52C2×C4×Q8—21no—
53#483—62no—
54#428—61no—
55#398—21no—
56#398C2×D312yes—
57#391—22no—
58#389—22no—
59#384C2×D311yescharacteristic
60#343—31no—
61#339—61no—
62#328—62no—
63#324—32no—
64C2×C3×Q8—24no—
65C2×C3×Q8—42no—
66C2×C3×Q8—42no—
67C22×C3×C4—21no—
68C22×C3×C4—22no—
69C22×C3×C4—21no—
70C22×C3×C4—22no—
71C22×C3×C4C2×D411yescharacteristic
72C22×C3×C4C2×D412yes—
73C2×Dic6—22no—
74C3×C4⋊C4—41no—
75C3×C4⋊C4—41no—
76C3×C4⋊C4—42no—
77C3×C4⋊C4—44no—
78C3×C4⋊C4—42no—
79C3×C4⋊C4—42no—
80C3×C4⋊C4—42no—
81C3×C4⋊C4—81no—
82C3×C42—21no—
83C3×C42—44no—
84C3×C42SD1612yes—
85#217—41no—
86C4⋊Dic3—21no—
87C4⋊Dic3—22no—
88C4⋊Dic3—21no—
89Dic3⋊C4—42no—
90C4×Dic3—41no—
91#208—22no—
92#208—41no—
93C2×C3⋊C8—41no—
94C2×C3⋊C8—42no—
95C22×Q8—21no—
96C2×M16—61no—
97C4⋊Q8—121no—
98#125—121no—
99C22⋊Q8—62no—
100C4×Q8—42no—
101C4×Q8—42no—
102#117—61no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4#6711yescharacteristic
108C2×C4⋊C4#6712yes—
109C2×C4⋊C4C22×D311yescharacteristic
110C2×C42—22no—
111C2×C42#6711yescharacteristic
112#107—121no—
113#106—121no—
114#106—122no—
115#103—122no—
116C22⋊C8—62no—
117#95—22no—
118#95—21no—
119#95—22no—
120#95#6712yes—
121C23×C3#12011yescharacteristic
122C3×Q8—48no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—24no—
129C2×C3×C4—22no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4—42no—
136C2×C3×C4—42no—
137C2×C3×C4—41no—
138C2×C3×C4C2×SD1611yescharacteristic
139C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
140C2×Dic3—41no—
141C2×Dic3—41no—
142Dic6—82no—
143C3⋊C8—42no—
144C3⋊C8—82no—
145C2×Q8—24no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—62no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4C2×#6711yescharacteristic
154C22×C4C2×#6712yes—
155M16—62no—
156M16—121no—
157C2×C8—121no—
158C2×C8—122no—
159C4⋊C4—41no—
160C4⋊C4—42no—
161C4⋊C4—41no—
162C4⋊C4—42no—
163C4⋊C4—44no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4—61no—
167C4⋊C4—62no—
168C4⋊C4—61no—
169C4⋊C4—81no—
170C4⋊C4—122no—
171C22⋊C4—121no—
172C42—21no—
173C42—44no—
174C42—121no—
175C42#21412yes—
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—44no—
185C3×C4—42no—
186C3×C4—82no—
187C3×C4—81no—
188Dic3—81no—
189Dic3—81no—
190C23#95311yescharacteristic
191Q8—48no—
192Q8—242no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—121no—
209C2×C4—121no—
210C2×C4#93211yescharacteristic
211C2×C4C2×#21411yescharacteristic
212C8—122no—
213C8—242no—
214C2×C3—21no—
215C2×C3—21no—
216C2×C3#160411yescharacteristic
217C2×C3#219811yescharacteristic
218C2×C3#220511yescharacteristic
219C22—21no—
220C22—21no—
221C22#521811yescharacteristic
222C22#524411yescharacteristic
223C22#524611yescharacteristic
224C4—24no—
225C4—21no—
226C4—21no—
227C4—44no—
228C4—42no—
229C4—81no—
230C4—82no—
231C4—241no—
232C4—241no—
233C3#1224711yescharacteristic, complemented, Sylow
234C2—21no—
235C2—21no—
236C2#7188011yescharacteristic
237C2#7194211yescharacteristic
238C2#7194611yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2582C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#2582D3
hypocenterC3#12247
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2582

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
C3×#25822
#697132
#695732
#676502
#676352
#122473