Finite Group Databasebeta

← #10692607

Subgroups — #10692607

Order 768.

450 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 119 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×#2983C211yesmaximal, characteristic
2#82479C211yesmaximal, characteristic
3#71637C211yesmaximal, characteristic
4#70780C212yesmaximal
5#67746C212yesmaximal
6#9404—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#869C2211yescharacteristic
9C3×#446—24no—
10C3×#432—22no—
11C3×#431C2212yes—
12#5229—21no—
13C2×#836—21no—
14#4986C414yes—
15#4977—21no—
16C4×C4⋊Dic3C2211yescharacteristic
17#4929—22no—
18#4924C2212yes—
19#4530—21no—
20#4482—24no—
21#2983D311yescharacteristic
22#2568—31no—
23#1801—31no—
24#1549—32no—
25#1303—32no—
26C22×C3×Q8D411yescharacteristic
27C22×Dic6—21no—
28C3×C4×Q8—22no—
29C3×C4×Q8—22no—
30C2×C3×C4⋊C4C2×C412yes—
31C2×C3×C4⋊C4D411yescharacteristic
32C2×C3×C42D412yes—
33C2×C3×C42C2311yescharacteristic
34C2×C4⋊Dic3—21no—
35C2×C4⋊Dic3—21no—
36C2×C4⋊Dic3C2×C412yes—
37C2×C4×Dic3—21no—
38C2×#208—22no—
39#869—22no—
40#869—22no—
41C3×C4⋊C8—24no—
42C3×C4⋊C8D414yes—
43C3×C22⋊C8—22no—
44C3×#97—24no—
45C3×C4×C8—22no—
46#836—42no—
47C3⋊#95—24no—
48C3⋊#95—24no—
49C3⋊#95C2×C412yes—
50C4⋊(C3⋊C8)—22no—
51C4⋊(C3⋊C8)—22no—
52C2×C4⋊Q8—31no—
53C2×C4×Q8C2×D311yescharacteristic
54#446—24no—
55#432—22no—
56#431C2×D312yes—
57#423—32no—
58C2×#101—61no—
59#400—34no—
60#391—61no—
61#391—62no—
62#382—61no—
63C4×C4⋊C4—31no—
64#343—61no—
65#332—64no—
66C2×C3×Q8—22no—
67C2×C3×Q8—21no—
68C2×C3×Q8—21no—
69C2×C3×Q8—22no—
70C22×C3×C4C22⋊C412yes—
71C22×C3×C4C22⋊C412yes—
72C22×C3×C4C22×C411yescharacteristic
73C22×C3×C4C2×D411yescharacteristic
74C22×C3×C4C2×D411yescharacteristic
75C22×Dic3—22no—
76C22×Dic3—22no—
77C2×Dic6—44no—
78C2×Dic6—41no—
79C2×Dic6—41no—
80C2×C3×C8—24no—
81C3×C4⋊C4—24no—
82C3×C4⋊C4—22no—
83C3×C42—22no—
84C3×C42—22no—
85C3×C42C2×D412yes—
86C3×C42#4112yes—
87C4⋊Dic3—24no—
88C4⋊Dic3—42no—
89C4⋊Dic3—42no—
90C4×Dic3—41no—
91C4×Dic3—41no—
92#208—42no—
93#208—42no—
94C2×C3⋊C8—42no—
95C22×Q8—61no—
96C22×Q8D1211yescharacteristic
97C2×M16—62no—
98C4⋊Q8—62no—
99C4⋊Q8—62no—
100C4×Q8—22no—
101C4×Q8—22no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—61no—
104C2×C4⋊C4—61no—
105C2×C4⋊C4C4×D312yes—
106C2×C4⋊C4D1211yescharacteristic
107C2×C42—61no—
108C2×C42#6712yes—
109C2×C42C22×D311yescharacteristic
110C4⋊C8—24no—
111C4⋊C8—62no—
112C4⋊C8—62no—
113C4⋊C8#6714yes—
114#101—122no—
115C22⋊C8—22no—
116#97—24no—
117C4×C8—22no—
118#95—32no—
119#95—64no—
120#95—64no—
121C23×C3C2×C22⋊C411yescharacteristic
122C3×Q8—42no—
123C3×Q8—42no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—22no—
129C2×C3×C4—41no—
130C2×C3×C4—42no—
131C2×C3×C4C4×D412yes—
132C2×C3×C4#12011yescharacteristic
133C2×C3×C4#12112yes—
134C2×C3×C4#12311yescharacteristic
135C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
136C2×C3×C4#13712yes—
137C2×C3×C4#13711yescharacteristic
138C2×Dic3—42no—
139C2×Dic3—44no—
140C2×Dic3—42no—
141C2×Dic3—42no—
142C2×Dic3—42no—
143Dic6—82no—
144Dic6—82no—
145C3×C8—44no—
146C3⋊C8—82no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—21no—
150C2×Q8—22no—
151C2×Q8—124no—
152C2×Q8—121no—
153C2×Q8—121no—
154C22×C4—62no—
155C22×C4—62no—
156C22×C4(C2×D3)⋊C412yes—
157C22×C4(C2×D3)⋊C412yes—
158C22×C4C2×C4×D311yescharacteristic
159C22×C4C2×D1211yescharacteristic
160C22×C4C2×#6711yescharacteristic
161M16—122no—
162M16—122no—
163C2×C8—24no—
164C2×C8—122no—
165C4⋊C4—24no—
166C4⋊C4—22no—
167C4⋊C4—64no—
168C4⋊C4—122no—
169C4⋊C4—122no—
170C42—22no—
171C42—22no—
172C42—121no—
173C42—121no—
174C42#23512yes—
175C42C2×#6712yes—
176C22×C3—21no—
177C22×C3—21no—
178C22×C3#38611yescharacteristic
179C22×C3#41711yescharacteristic
180C22×C3#41911yescharacteristic
181C3×C4—21no—
182C3×C4—22no—
183C3×C4—42no—
184C3×C4—42no—
185C3×C4#45412yes—
186C3×C4#46812yes—
187C3×C4#47012yes—
188Dic3—82no—
189Dic3—82no—
190C23C2×(C2×D3)⋊C411yescharacteristic
191Q8—42no—
192Q8—42no—
193Q8—242no—
194Q8—242no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—122no—
203C2×C4—124no—
204C2×C4—122no—
205C2×C4—122no—
206C2×C4—122no—
207C2×C4#90811yescharacteristic
208C2×C4#90912yes—
209C2×C4#90911yescharacteristic
210C2×C4C4×#6712yes—
211C2×C4#92911yescharacteristic
212C2×C4#93012yes—
213C2×C4#95311yescharacteristic
214C8—44no—
215C8—242no—
216C2×C3—21no—
217C2×C3—21no—
218C2×C3#188811yescharacteristic
219C2×C3#189411yescharacteristic
220C2×C3#191511yescharacteristic
221C22—21no—
222C22—21no—
223C22#512011yescharacteristic
224C22#513311yescharacteristic
225C22#521311yescharacteristic
226C4—22no—
227C4—21no—
228C4—42no—
229C4—42no—
230C4—242no—
231C4—242no—
232C4#513412yes—
233C4#513712yes—
234C4#524512yes—
235C3#940411yescharacteristic, complemented, Sylow
236C2—21no—
237C2—21no—
238C2#7166111yescharacteristic
239C2#7198911yescharacteristic
240C2#7203111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2983C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2983D3
hypocenterC3#9404
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
235. C3#9404semidirect

Maximal subgroups

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

subgroupindex
C3×#29832
#824792
#716372
#707802
#677462
#94043