Finite Group Databasebeta

← #10692567

Subgroups — #10692567

Order 768.

470 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2982C211yesmaximal, characteristic
2#82479C211yesmaximal, characteristic
3#71637C211yesmaximal, characteristic
4#70780C212yesmaximal
5#67746C212yesmaximal
6#9396—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#869C2211yescharacteristic
9C3×C8×Q8—24no—
10C3×#431—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#2982D311yescharacteristic
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×C4×C8—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
54C8×Q8—24no—
55#431—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—22no—
68C2×C3×Q8SD1612yes—
69C2×C3×Q8Q1612yes—
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—42no—
91#208—42no—
92#208—42no—
93C2×C3⋊C8—42no—
94C22×Q8—61no—
95C22×Q8D1211yescharacteristic
96C2×M16—62no—
97C4⋊Q8—62no—
98C4⋊Q8—62no—
99C4×Q8—22no—
100C4×Q8—22no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—61no—
103C2×C4⋊C4—61no—
104C2×C4⋊C4C4×D312yes—
105C2×C4⋊C4D1211yescharacteristic
106C2×C42—61no—
107C2×C42#6712yes—
108C2×C42C22×D311yescharacteristic
109C4⋊C8—24no—
110C4⋊C8—62no—
111C4⋊C8—62no—
112C4⋊C8#6714yes—
113#101—122no—
114C22⋊C8—22no—
115C4×C8—24no—
116C4×C8—22no—
117#95—32no—
118#95—64no—
119#95—64no—
120C23×C3C2×C22⋊C411yescharacteristic
121C3×Q8—24no—
122C3×Q8—24no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—41no—
128C2×C3×C4—42no—
129C2×C3×C4#10314yes—
130C2×C3×C4C4×D412yes—
131C2×C3×C4#12011yescharacteristic
132C2×C3×C4#12112yes—
133C2×C3×C4#12311yescharacteristic
134C2×C3×C4C2×SD1611yescharacteristic
135C2×C3×C4C2×Q1611yescharacteristic
136C2×C3×C4#13512yes—
137C2×Dic3—42no—
138C2×Dic3—44no—
139C2×Dic3—42no—
140C2×Dic3—42no—
141C2×Dic3—42no—
142Dic6—82no—
143Dic6—82no—
144C3×C8—44no—
145C3⋊C8—82no—
146C2×Q8—22no—
147C2×Q8—22no—
148C2×Q8—124no—
149C2×Q8—121no—
150C2×Q8—121no—
151C2×Q8#20412yes—
152C2×Q8Dic1212yes—
153C22×C4—62no—
154C22×C4—62no—
155C22×C4(C2×D3)⋊C412yes—
156C22×C4(C2×D3)⋊C412yes—
157C22×C4C2×C4×D311yescharacteristic
158C22×C4C2×D1211yescharacteristic
159C22×C4C2×#6711yescharacteristic
160M16—122no—
161M16—122no—
162C2×C8—24no—
163C2×C8—122no—
164C4⋊C4—24no—
165C4⋊C4—22no—
166C4⋊C4—64no—
167C4⋊C4—122no—
168C4⋊C4—122no—
169C42—22no—
170C42—22no—
171C42—122no—
172C42#23512yes—
173C42C2×#6712yes—
174C22×C3—21no—
175C22×C3—21no—
176C22×C3#38611yescharacteristic
177C22×C3C2×#10311yescharacteristic
178C22×C3#41611yescharacteristic
179C3×C4—21no—
180C3×C4—24no—
181C3×C4—22no—
182C3×C4—42no—
183C3×C4#45412yes—
184C3×C4#46512yes—
185C3×C4#46712yes—
186Dic3—82no—
187Dic3—82no—
188C23C2×(C2×D3)⋊C411yescharacteristic
189Q8—24no—
190Q8—24no—
191Q8—242no—
192Q8—242no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—122no—
200C2×C4—124no—
201C2×C4—122no—
202C2×C4—122no—
203C2×C4—122no—
204C2×C4#81614yes—
205C2×C4C2×#20411yescharacteristic
206C2×C4#90412yes—
207C2×C4C2×Dic1211yescharacteristic
208C2×C4C4×#6712yes—
209C2×C4#92911yescharacteristic
210C2×C4#93012yes—
211C2×C4#95311yescharacteristic
212C8—44no—
213C8—242no—
214C2×C3—21no—
215C2×C3—21no—
216C2×C3#188811yescharacteristic
217C2×C3#189211yescharacteristic
218C2×C3#191511yescharacteristic
219C22—21no—
220C22—21no—
221C22C2×#81611yescharacteristic
222C22#511311yescharacteristic
223C22#521311yescharacteristic
224C4—24no—
225C4—22no—
226C4—21no—
227C4—42no—
228C4—242no—
229C4—242no—
230C4#511412yes—
231C4#511612yes—
232C4#524512yes—
233C3#939611yescharacteristic, complemented, Sylow
234C2—21no—
235C2—21no—
236C2#7166111yescharacteristic
237C2#7197211yescharacteristic
238C2#7203111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2982C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2982D3
hypocenterC3#9396
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
233. C3#9396semidirect

Maximal subgroups

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

subgroupindex
C3×#29822
#824792
#716372
#707802
#677462
#93963