Finite Group Databasebeta

← #10692552

Subgroups — #10692552

Order 768.

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

normal subgroups only

This group has 558 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#82382C211yesmaximal, characteristic
3#71617C211yesmaximal, characteristic
4#70778C212yesmaximal, complemented
5#67746C212yesmaximal, complemented
6#9393—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#936—21no—
9C2×#869C2211yescharacteristic
10C3×C8×D4—24no—
11C3×#431C2212yes—
12C2×C3×C22⋊C8—22no—
13C2×#833—21no—
14#5212—22no—
15#4973—21no—
16#4953C414yes—
17C4×C4⋊Dic3C2211yescharacteristic
18#4924C2212yes—
19#4530—21no—
20#4486—24no—
21#2972D311yescharacteristic
22#2547—31no—
23#1800—31no—
24#1546—32no—
25#1303—32no—
26C22×C3×D4D411yescharacteristic
27C23×C3×C4D412yes—
28C22×Dic6—21no—
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—
36#936—42no—
37#936—42no—
38C2×C4⋊Dic3—21no—
39C2×C4⋊Dic3—21no—
40C2×C4⋊Dic3C2×C412yes—
41C2×C4×Dic3—21no—
42C2×#208—22no—
43#869—22no—
44#869—22no—
45C3×C4⋊C8D414yes—
46C3×C22⋊C8—24no—
47C3×C22⋊C8—22no—
48C3×C4×C8—22no—
49#833—42no—
50C3⋊#95—24no—
51C3⋊#95C2×C412yes—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)—22no—
54C2×C4⋊Q8—31no—
55C2×#124—61no—
56C2×C4×D4C2×D311yescharacteristic
57C8×D4—24no—
58#431C2×D312yes—
59#423—32no—
60C2×#100—61no—
61C2×C22⋊C8—22no—
62#397—34no—
63#391—61no—
64#380—62no—
65C4×C4⋊C4—31no—
66#343—61no—
67#332—64no—
68C24×C3C22⋊C412yes—
69C2×C3×D4—22no—
70C2×C3×D4—22no—
71C2×C3×D4D812yes—
72C2×C3×D4SD1612yes—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4—44no—
76C22×C3×C4C22⋊C412yes—
77C22×C3×C4C22×C411yescharacteristic
78C22×C3×C4C2×D411yescharacteristic
79C22×C3×C4C2×D411yescharacteristic
80C22×Dic3—22no—
81C22×Dic3—22no—
82C2×Dic6—42no—
83C2×Dic6—41no—
84C2×Dic6—41no—
85C2×Dic6—42no—
86C2×C3×C8—24no—
87C2×C3×C8—84no—
88C3×C4⋊C4—22no—
89C3×C22⋊C4—24no—
90C3×C42C2×D412yes—
91C3×C42#4112yes—
92#217—48no—
93C4⋊Dic3—24no—
94C4⋊Dic3—42no—
95C4⋊Dic3—41no—
96C4⋊Dic3—41no—
97C4×Dic3—41no—
98C4×Dic3—41no—
99#208—42no—
100#208—42no—
101C2×C3⋊C8—42no—
102C22×Q8—61no—
103C22×D4D1211yescharacteristic
104C23×C4#6712yes—
105C2×M16—62no—
106C22×C8—24no—
107C4⋊Q8—62no—
108C4⋊Q8—62no—
109#124—122no—
110#124—122no—
111C4×D4—22no—
112C4×D4—22no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—61no—
115C2×C4⋊C4—61no—
116C2×C4⋊C4D1211yescharacteristic
117C2×C22⋊C4—64no—
118C2×C22⋊C4C4×D312yes—
119C2×C42—61no—
120C2×C42C22×D311yescharacteristic
121C4⋊C8—62no—
122C4⋊C8—62no—
123C4⋊C8#6714yes—
124#100—122no—
125C22⋊C8—24no—
126C22⋊C8—22no—
127C4×C8—22no—
128#95—32no—
129#95—64no—
130C23×C3—22no—
131C23×C3—22no—
132C23×C3—42no—
133C23×C3—42no—
134C23×C3#10214yes—
135C23×C3C2×C22⋊C411yescharacteristic
136C3×D4—24no—
137C3×D4—24no—
138C2×C3×C4—22no—
139C2×C3×C4—22no—
140C2×C3×C4—41no—
141C2×C3×C4—44no—
142C2×C3×C4—42no—
143C2×C3×C4—82no—
144C2×C3×C4C4×D412yes—
145C2×C3×C4#12011yescharacteristic
146C2×C3×C4#12112yes—
147C2×C3×C4#12311yescharacteristic
148C2×C3×C4C2×D811yescharacteristic
149C2×C3×C4C2×SD1611yescharacteristic
150C2×C3×C4#13512yes—
151C2×Dic3—42no—
152C2×Dic3—42no—
153C2×Dic3—42no—
154C2×Dic3—42no—
155C2×Dic3—42no—
156C2×Dic3—42no—
157Dic6—82no—
158Dic6—82no—
159C3×C8—44no—
160C3⋊C8—82no—
161C24(C2×D3)⋊C412yes—
162C2×Q8—121no—
163C2×Q8—121no—
164C2×Q8—122no—
165C2×Q8—122no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4#20412yes—
169C2×D4D2412yes—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—44no—
173C22×C4—62no—
174C22×C4—62no—
175C22×C4(C2×D3)⋊C412yes—
176C22×C4C2×C4×D311yescharacteristic
177C22×C4C2×D1211yescharacteristic
178C22×C4C2×#6711yescharacteristic
179M16—122no—
180M16—122no—
181C2×C8—24no—
182C2×C8—84no—
183C2×C8—122no—
184C4⋊C4—22no—
185C4⋊C4—64no—
186C4⋊C4—122no—
187C4⋊C4—121no—
188C4⋊C4—121no—
189C22⋊C4—24no—
190C22⋊C4—128no—
191C42—121no—
192C42—121no—
193C42#23512yes—
194C42C2×#6712yes—
195C22×C3—21no—
196C22×C3—21no—
197C22×C3—24no—
198C22×C3—44no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—82no—
202C22×C3#38611yescharacteristic
203C22×C3C2×#10211yescharacteristic
204C22×C3#41611yescharacteristic
205C3×C4—21no—
206C3×C4—22no—
207C3×C4—42no—
208C3×C4#45512yes—
209C3×C4#46512yes—
210C3×C4#46612yes—
211Dic3—82no—
212Dic3—82no—
213C23—22no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23#82114yes—
218C23C2×(C2×D3)⋊C411yescharacteristic
219Q8—242no—
220Q8—242no—
221D4—24no—
222D4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—82no—
229C2×C4—122no—
230C2×C4—122no—
231C2×C4—122no—
232C2×C4—122no—
233C2×C4—122no—
234C2×C4—122no—
235C2×C4C2×#20411yescharacteristic
236C2×C4C2×D2411yescharacteristic
237C2×C4#90412yes—
238C2×C4C4×#6712yes—
239C2×C4#92911yescharacteristic
240C2×C4#93012yes—
241C2×C4#95311yescharacteristic
242C8—44no—
243C8—242no—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3—44no—
247C2×C3#188511yescharacteristic
248C2×C3#189311yescharacteristic
249C2×C3#191511yescharacteristic
250C22—21no—
251C22—21no—
252C22—24no—
253C22—44no—
254C22—42no—
255C22—42no—
256C22—82no—
257C22C2×#82111yescharacteristic
258C22#511311yescharacteristic
259C22#521311yescharacteristic
260C4—22no—
261C4—21no—
262C4—42no—
263C4—242no—
264C4—242no—
265C4#511412yes—
266C4#511512yes—
267C4#524212yes—
268C3#939311yescharacteristic, complemented, Sylow
269C2—21no—
270C2—21no—
271C2—44no—
272C2#7164911yescharacteristic
273C2#7197311yescharacteristic
274C2#7203111yescharacteristic
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#9393
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. #70778C2semidirect
5. #67746C2semidirect
268. C3#9393semidirect

Maximal subgroups

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

subgroupindex
C3×#29722
#823822
#716172
#707782
#677462
#93933