Finite Group Databasebeta

← #10692577

Subgroups — #10692577

Order 768.

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

normal subgroups only

This group has 574 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, complemented
2#82476C211yesmaximal, characteristic
3#71637C211yesmaximal, characteristic, complemented
4#70774C212yesmaximal
5#67746C212yesmaximal, complemented
6#9398—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#947—21no—
9C2×#874C2211yescharacteristic
10C3×C8×Q8—24no—
11C3×#431—22no—
12C3×#431C2212yes—
13C2×#836—21no—
14#4991C414yes—
15#4988—21no—
16#4939—22no—
17C4×C4⋊Dic3C2211yescharacteristic
18#4924C2212yes—
19#4530—21no—
20#4485—24no—
21#2982D311yescharacteristic, complemented
22#2548—31no—
23#1801—31no—
24#1545—32no—
25#1303—32no—
26C22×C3×Q8D411yescharacteristic
27C22×D12—21no—
28C22×D12—21no—
29C22×D12C2×C412yes—
30C3×C4×Q8—22no—
31C3×C4×Q8—22no—
32C2×C3×C4⋊C4C2×C412yes—
33C2×C3×C4⋊C4D411yescharacteristic
34C2×C3×C42D412yes—
35C2×C3×C42C2311yescharacteristic
36#947—42no—
37#947—42no—
38C2×(C2×D3)⋊C4—24no—
39C2×(C2×D3)⋊C4—24no—
40C2×C4⋊Dic3—21no—
41C2×C4×Dic3—21no—
42C2×#208—22no—
43#874—22no—
44#874—22no—
45C3×C4⋊C8—24no—
46C3×C4⋊C8D414yes—
47C3×C22⋊C8—22no—
48C3×C4×C8—24no—
49C3×C4×C8—22no—
50#836—42no—
51C3⋊#95C2×C412yes—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)—22no—
54C2×#127—31no—
55C2×#124—61no—
56C2×C4×Q8C2×D311yescharacteristic
57C8×Q8—24no—
58#431—22no—
59#431C2×D312yes—
60#423—32no—
61C2×#101—61no—
62#394—34no—
63#390—61no—
64#390—62no—
65C4×C4⋊C4—31no—
66#343—61no—
67#331—64no—
68C23×D3—22no—
69C2×C3×Q8—22no—
70C2×C3×Q8—22no—
71C2×C3×Q8D812yes—
72C2×C3×Q8SD1612yes—
73C22×C3×C4C22⋊C412yes—
74C22×C3×C4C22⋊C412yes—
75C22×C3×C4C22×C411yescharacteristic
76C22×C3×C4C2×D411yescharacteristic
77C22×C3×C4C2×D411yescharacteristic
78C22×Dic3—22no—
79C2×D12—24no—
80C2×D12—44no—
81C2×D12—41no—
82C2×D12—42no—
83C2×D12—41no—
84C2×D12—41no—
85C2×D12—41no—
86C2×D12—42no—
87C2×D12—82no—
88C2×C3×C8—24no—
89C3×C4⋊C4—24no—
90C3×C4⋊C4—22no—
91C3×C42—22no—
92C3×C42—22no—
93C3×C42C2×D412yes—
94C3×C42#4112yes—
95(C2×D3)⋊C4—48no—
96(C2×D3)⋊C4—84no—
97C4⋊Dic3—42no—
98C4×Dic3—41no—
99C4×Dic3—41no—
100#208—42no—
101#208—42no—
102C2×C3⋊C8—42no—
103C22×Q8D1211yescharacteristic
104C22×D4—32no—
105C22×D4—61no—
106C22×D4—61no—
107C2×M16—62no—
108#127—62no—
109#127—62no—
110#124—122no—
111#124—122no—
112C4×Q8—22no—
113C4×Q8—22no—
114C2×C4⋊C4—61no—
115C2×C4⋊C4C4×D312yes—
116C2×C4⋊C4D1211yescharacteristic
117C2×C22⋊C4—64no—
118C2×C22⋊C4—64no—
119C2×C42—61no—
120C2×C42#6712yes—
121C2×C42C22×D311yescharacteristic
122C4⋊C8—24no—
123C4⋊C8—62no—
124C4⋊C8—62no—
125C4⋊C8#6714yes—
126#101—122no—
127C22⋊C8—22no—
128C4×C8—24no—
129C4×C8—22no—
130#95—32no—
131C23×C3C2×C22⋊C411yescharacteristic
132C22×D3—44no—
133C22×D3—44no—
134C22×D3—42no—
135C22×D3—42no—
136C22×D3—42no—
137C3×Q8—24no—
138C3×Q8—24no—
139C2×C3×C4—22no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—41no—
144C2×C3×C4—42no—
145C2×C3×C4#10214yes—
146C2×C3×C4C4×D412yes—
147C2×C3×C4#12011yescharacteristic
148C2×C3×C4#12112yes—
149C2×C3×C4#12311yescharacteristic
150C2×C3×C4C2×D811yescharacteristic
151C2×C3×C4C2×SD1611yescharacteristic
152C2×C3×C4#13512yes—
153C2×Dic3—42no—
154C2×Dic3—42no—
155C2×Dic3—42no—
156D12—84no—
157D12—82no—
158D12—82no—
159D12—82no—
160D12—82no—
161C3×C8—44no—
162C3⋊C8—82no—
163C24—62no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8#20412yes—
167C2×Q8D2412yes—
168C2×D4—64no—
169C2×D4—124no—
170C2×D4—121no—
171C2×D4—121no—
172C2×D4—122no—
173C2×D4—121no—
174C2×D4—122no—
175C2×D4—121no—
176C2×D4—242no—
177C22×C4—62no—
178C22×C4(C2×D3)⋊C412yes—
179C22×C4(C2×D3)⋊C412yes—
180C22×C4C2×C4×D311yescharacteristic
181C22×C4C2×D1211yescharacteristic
182C22×C4C2×#6711yescharacteristic
183M16—122no—
184M16—122no—
185C2×C8—24no—
186C2×C8—122no—
187C4⋊C4—24no—
188C4⋊C4—22no—
189C4⋊C4—122no—
190C22⋊C4—128no—
191C22⋊C4—244no—
192C42—22no—
193C42—22no—
194C42—121no—
195C42—121no—
196C42#23512yes—
197C42C2×#6712yes—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3#38611yescharacteristic
201C22×C3C2×#10211yescharacteristic
202C22×C3#41611yescharacteristic
203C2×D3—84no—
204C2×D3—84no—
205C2×D3—82no—
206C2×D3—82no—
207C2×D3—82no—
208C3×C4—21no—
209C3×C4—24no—
210C3×C4—22no—
211C3×C4—42no—
212C3×C4#45412yes—
213C3×C4#46512yes—
214C3×C4#46612yes—
215Dic3—82no—
216C23—124no—
217C23—124no—
218C23—122no—
219C23—122no—
220C23—122no—
221C23C2×(C2×D3)⋊C411yescharacteristic
222Q8—24no—
223Q8—24no—
224D4—244no—
225D4—242no—
226D4—242no—
227D4—242no—
228D4—242no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—122no—
236C2×C4—122no—
237C2×C4—122no—
238C2×C4#82114yes—
239C2×C4C2×#20411yescharacteristic
240C2×C4C2×D2411yescharacteristic
241C2×C4#90412yes—
242C2×C4C4×#6712yes—
243C2×C4#92911yescharacteristic
244C2×C4#93012yes—
245C2×C4#95311yescharacteristic
246C8—44no—
247C8—242no—
248C2×C3—21no—
249C2×C3—21no—
250C2×C3#188811yescharacteristic
251C2×C3#189311yescharacteristic
252C2×C3#191411yescharacteristic
253D3—162no—
254C22—21no—
255C22—21no—
256C22—244no—
257C22—244no—
258C22—242no—
259C22—242no—
260C22—242no—
261C22C2×#82111yescharacteristic
262C22#511311yescharacteristic
263C22#521311yescharacteristic
264C4—24no—
265C4—22no—
266C4—21no—
267C4—42no—
268C4—242no—
269C4#511412yes—
270C4#511512yes—
271C4#524512yes—
272C3#939811yescharacteristic, complemented, Sylow
273C2—21no—
274C2—21no—
275C2—482no—
276C2#7166111yescharacteristic
277C2#7197311yescharacteristic
278C2#7203011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2982C2
socleC22×C3#386
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#2982D3
hypocenterC3#9398
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
1. C3×#2982C2semidirect
3. #71637C2semidirect
5. #67746C2semidirect
21. #2982D3semidirect
272. C3#9398semidirect

Maximal subgroups

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

subgroupindex
C3×#29822
#824762
#716372
#707742
#677462
#93983