Finite Group Databasebeta

← #10692542

Subgroups — #10692542

Order 768.

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

normal subgroups only

This group has 538 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×#2972C211yesmaximal, characteristic
2#82384C211yesmaximal, characteristic
3#71617C211yesmaximal, characteristic
4#70779C212yesmaximal, complemented
5#67746C212yesmaximal, complemented
6#9391—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#869C2211yescharacteristic
9C3×C8×D4—24no—
10C3×#431C2212yes—
11C2×C3×C22⋊C8—22no—
12#5221—21no—
13C2×#833—21no—
14#5212—22no—
15#4977—21no—
16#4947C414yes—
17C4×C4⋊Dic3C2211yescharacteristic
18#4924C2212yes—
19#4530—21no—
20#4481—24no—
21#2972D311yescharacteristic
22#2563—31no—
23#1800—31no—
24#1548—32no—
25#1303—32no—
26C22×C3×D4D411yescharacteristic
27C23×C3×C4D412yes—
28C22×Dic6C2×C412yes—
29C22×C3×C8—24no—
30C3×C4×D4—22no—
31C3×C4×D4—22no—
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C22⋊C4C2×C412yes—
34C2×C3×C42C2311yescharacteristic
35C2×#217—24no—
36C2×C4⋊Dic3—21no—
37C2×C4⋊Dic3—21no—
38C2×C4⋊Dic3—21no—
39C2×C4×Dic3—21no—
40C2×#208—22no—
41#869—22no—
42#869—22no—
43C3×C4⋊C8D414yes—
44C3×C22⋊C8—24no—
45C3×C22⋊C8—22no—
46C3×C4×C8—22no—
47#833—42no—
48C3⋊#95—24no—
49C3⋊#95C2×C412yes—
50C4⋊(C3⋊C8)—22no—
51C4⋊(C3⋊C8)—22no—
52C2×C4⋊Q8—31no—
53C2×C4×D4C2×D311yescharacteristic
54C8×D4—24no—
55#431C2×D312yes—
56#423—32no—
57C2×#100—61no—
58C2×C22⋊C8—22no—
59#393—34no—
60#390—61no—
61#382—61no—
62#380—62no—
63C4×C4⋊C4—31no—
64#343—61no—
65#333—64no—
66C24×C3C22⋊C412yes—
67C2×C3×D4—22no—
68C2×C3×D4—22no—
69C2×C3×D4SD1612yes—
70C2×C3×D4Q1612yes—
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4—44no—
74C22×C3×C4C22⋊C412yes—
75C22×C3×C4C22×C411yescharacteristic
76C22×C3×C4C2×D411yescharacteristic
77C22×C3×C4C2×D411yescharacteristic
78C22×Dic3—22no—
79C22×Dic3—22no—
80C2×Dic6—24no—
81C2×Dic6—82no—
82C2×C3×C8—24no—
83C2×C3×C8—84no—
84C3×C4⋊C4—22no—
85C3×C22⋊C4—24no—
86C3×C42C2×D412yes—
87C3×C42#4112yes—
88#217—84no—
89C4⋊Dic3—42no—
90C4⋊Dic3—41no—
91C4⋊Dic3—42no—
92C4⋊Dic3—41no—
93C4×Dic3—42no—
94#208—42no—
95#208—42no—
96C2×C3⋊C8—42no—
97C22×Q8—32no—
98C22×D4D1211yescharacteristic
99C23×C4#6712yes—
100C2×M16—62no—
101C22×C8—24no—
102C4⋊Q8—62no—
103C4⋊Q8—62no—
104C4×D4—22no—
105C4×D4—22no—
106C2×C4⋊C4—61no—
107C2×C4⋊C4—61no—
108C2×C4⋊C4—61no—
109C2×C4⋊C4D1211yescharacteristic
110C2×C22⋊C4—64no—
111C2×C22⋊C4C4×D312yes—
112C2×C42—61no—
113C2×C42C22×D311yescharacteristic
114C4⋊C8—62no—
115C4⋊C8—62no—
116C4⋊C8#6714yes—
117#100—122no—
118C22⋊C8—24no—
119C22⋊C8—22no—
120C4×C8—22no—
121#95—32no—
122#95—64no—
123C23×C3—22no—
124C23×C3—22no—
125C23×C3—42no—
126C23×C3—42no—
127C23×C3#10314yes—
128C23×C3C2×C22⋊C411yescharacteristic
129C3×D4—24no—
130C3×D4—24no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—41no—
134C2×C3×C4—44no—
135C2×C3×C4—42no—
136C2×C3×C4—82no—
137C2×C3×C4C4×D412yes—
138C2×C3×C4#12011yescharacteristic
139C2×C3×C4#12112yes—
140C2×C3×C4#12311yescharacteristic
141C2×C3×C4C2×SD1611yescharacteristic
142C2×C3×C4C2×Q1611yescharacteristic
143C2×C3×C4#13512yes—
144C2×Dic3—42no—
145C2×Dic3—42no—
146C2×Dic3—44no—
147C2×Dic3—42no—
148C2×Dic3—42no—
149Dic6—84no—
150C3×C8—44no—
151C3⋊C8—82no—
152C24(C2×D3)⋊C412yes—
153C2×Q8—64no—
154C2×Q8—242no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4#20412yes—
158C2×D4Dic1212yes—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—44no—
162C22×C4—62no—
163C22×C4—62no—
164C22×C4(C2×D3)⋊C412yes—
165C22×C4C2×C4×D311yescharacteristic
166C22×C4C2×D1211yescharacteristic
167C22×C4C2×#6711yescharacteristic
168M16—122no—
169M16—122no—
170C2×C8—24no—
171C2×C8—84no—
172C2×C8—122no—
173C4⋊C4—22no—
174C4⋊C4—122no—
175C4⋊C4—121no—
176C4⋊C4—122no—
177C4⋊C4—121no—
178C22⋊C4—24no—
179C22⋊C4—244no—
180C42—122no—
181C42#23512yes—
182C42C2×#6712yes—
183C22×C3—21no—
184C22×C3—21no—
185C22×C3—24no—
186C22×C3—44no—
187C22×C3—42no—
188C22×C3—42no—
189C22×C3—82no—
190C22×C3#38611yescharacteristic
191C22×C3C2×#10311yescharacteristic
192C22×C3#41611yescharacteristic
193C3×C4—21no—
194C3×C4—22no—
195C3×C4—42no—
196C3×C4#45512yes—
197C3×C4#46512yes—
198C3×C4#46712yes—
199Dic3—82no—
200Dic3—82no—
201C23—22no—
202C23—22no—
203C23—42no—
204C23—42no—
205C23#81614yes—
206C23C2×(C2×D3)⋊C411yescharacteristic
207Q8—244no—
208D4—24no—
209D4—24no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—44no—
214C2×C4—42no—
215C2×C4—82no—
216C2×C4—122no—
217C2×C4—124no—
218C2×C4—122no—
219C2×C4—122no—
220C2×C4—122no—
221C2×C4C2×#20411yescharacteristic
222C2×C4#90412yes—
223C2×C4C2×Dic1211yescharacteristic
224C2×C4C4×#6712yes—
225C2×C4#92911yescharacteristic
226C2×C4#93012yes—
227C2×C4#95311yescharacteristic
228C8—44no—
229C8—242no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—44no—
233C2×C3#188511yescharacteristic
234C2×C3#189211yescharacteristic
235C2×C3#191411yescharacteristic
236C22—21no—
237C22—21no—
238C22—24no—
239C22—44no—
240C22—42no—
241C22—42no—
242C22—82no—
243C22C2×#81611yescharacteristic
244C22#511311yescharacteristic
245C22#521311yescharacteristic
246C4—22no—
247C4—21no—
248C4—42no—
249C4—242no—
250C4—242no—
251C4#511412yes—
252C4#511612yes—
253C4#524212yes—
254C3#939111yescharacteristic, complemented, Sylow
255C2—21no—
256C2—21no—
257C2—44no—
258C2#7164911yescharacteristic
259C2#7197211yescharacteristic
260C2#7203011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2972C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#2972D3
hypocenterC3#9391
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
4. #70779C2semidirect
5. #67746C2semidirect
254. C3#9391semidirect

Maximal subgroups

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

subgroupindex
C3×#29722
#823842
#716172
#707792
#677462
#93913