Finite Group Databasebeta

← #10692572

Subgroups — #10692572

Order 768.

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

normal subgroups only

This group has 730 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, complemented
2#82378C211yesmaximal, characteristic
3#71617C211yesmaximal, characteristic, complemented
4#70774C212yesmaximal, complemented
5#67746C212yesmaximal, complemented
6#9397—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#940—21no—
9C2×#930—22no—
10C2×#874C2211yescharacteristic
11C3×C8×D4—24no—
12C3×#431C2212yes—
13C2×C3×C22⋊C8—22no—
14C2×#833—21no—
15#4988—21no—
16#4960C414yes—
17C4×C4⋊Dic3C2211yescharacteristic
18#4924C2212yes—
19#4530—21no—
20#4485—24no—
21#2972D311yescharacteristic, complemented
22#2541—31no—
23#1800—31no—
24#1545—32no—
25#1303—32no—
26C22×C3×D4D411yescharacteristic
27C23×C3×C4D412yes—
28C22×#67—24no—
29C22×D12—21no—
30C22×D12—21no—
31C22×D12C2×C412yes—
32C22×C3×C8—24no—
33C3×C4×D4—22no—
34C3×C4×D4—22no—
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C22⋊C4C2×C412yes—
37C2×C3×C42C2311yescharacteristic
38#940—42no—
39#940—42no—
40#930—48no—
41C2×(C2×D3)⋊C4—24no—
42C2×C4⋊Dic3—21no—
43C2×C4×Dic3—21no—
44C2×#208—22no—
45#874—22no—
46#874—22no—
47C3×C4⋊C8D414yescomplemented
48C3×C22⋊C8—24no—
49C3×C22⋊C8—22no—
50C3×C4×C8—22no—
51#833—42no—
52C3⋊#95C2×C412yes—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)—22no—
55C2×#127—31no—
56C2×#127—61no—
57C2×#121—62no—
58C2×C4×D4C2×D311yescharacteristic
59C8×D4—24no—
60#431C2×D312yes—
61#423—32no—
62C2×#100—61no—
63C2×C22⋊C8—22no—
64#392—34no—
65#390—61no—
66C4×C4⋊C4—31no—
67#343—61no—
68#331—64no—
69C24×C3C22⋊C412yes—
70C23×D3—22no—
71C2×C3×D4—22no—
72C2×C3×D4—22no—
73C2×C3×D4D812yes—
74C2×C3×D4SD1612yes—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—44no—
78C22×C3×C4C22⋊C412yes—
79C22×C3×C4C22×C411yescharacteristic
80C22×C3×C4C2×D411yescharacteristic
81C22×C3×C4C2×D411yescharacteristic
82C2×#67—48no—
83C2×#67—48no—
84C2×#67—84no—
85C22×Dic3—22no—
86C2×D12—24no—
87C2×D12—44no—
88C2×D12—41no—
89C2×D12—42no—
90C2×D12—41no—
91C2×D12—41no—
92C2×D12—41no—
93C2×D12—42no—
94C2×D12—82no—
95C2×C3×C8—24no—
96C2×C3×C8—84no—
97C3×C4⋊C4—22no—
98C3×C22⋊C4—24no—
99C3×C42C2×D412yes—
100C3×C42#4112yes—
101(C2×D3)⋊C4—48no—
102C4⋊Dic3—24no—
103C4×Dic3—41no—
104C4×Dic3—41no—
105#208—42no—
106#208—42no—
107C2×C3⋊C8—42no—
108C22×D4—32no—
109C22×D4—64no—
110C22×D4—61no—
111C22×D4—61no—
112C22×D4D1211yescharacteristic
113C23×C4#6712yes—
114C2×M16—62no—
115C22×C8—24no—
116#127—62no—
117#127—62no—
118#127—122no—
119#127—122no—
120#121—128no—
121C4×D4—22no—
122C4×D4—22no—
123C2×C4⋊C4—61no—
124C2×C4⋊C4D1211yescharacteristic
125C2×C22⋊C4—64no—
126C2×C22⋊C4C4×D312yes—
127C2×C42—61no—
128C2×C42C22×D311yescharacteristic
129C4⋊C8—62no—
130C4⋊C8—62no—
131C4⋊C8#6714yescomplemented
132#100—122no—
133C22⋊C8—24no—
134C22⋊C8—22no—
135C4×C8—22no—
136#95—32no—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3—42no—
140C23×C3—42no—
141C23×C3#10214yes—
142C23×C3C2×C22⋊C411yescharacteristic
143C22×D3—44no—
144C22×D3—44no—
145C22×D3—42no—
146C22×D3—42no—
147C22×D3—42no—
148C3×D4—24no—
149C3×D4—24no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—41no—
153C2×C3×C4—44no—
154C2×C3×C4—42no—
155C2×C3×C4—82no—
156C2×C3×C4C4×D412yes—
157C2×C3×C4#12011yescharacteristic
158C2×C3×C4#12112yes—
159C2×C3×C4#12311yescharacteristic
160C2×C3×C4C2×D811yescharacteristic
161C2×C3×C4C2×SD1611yescharacteristic
162C2×C3×C4#13512yes—
163#67—816no—
164C2×Dic3—24no—
165C2×Dic3—42no—
166C2×Dic3—42no—
167D12—84no—
168D12—82no—
169D12—82no—
170D12—82no—
171D12—82no—
172C3×C8—44no—
173C3⋊C8—82no—
174C24—62no—
175C24(C2×D3)⋊C412yes—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—64no—
179C2×D4—124no—
180C2×D4—128no—
181C2×D4—128no—
182C2×D4—121no—
183C2×D4—121no—
184C2×D4—122no—
185C2×D4—121no—
186C2×D4—122no—
187C2×D4—121no—
188C2×D4—242no—
189C2×D4—244no—
190C2×D4#20412yes—
191C2×D4D2412yes—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—44no—
195C22×C4—62no—
196C22×C4(C2×D3)⋊C412yes—
197C22×C4C2×C4×D311yescharacteristic
198C22×C4C2×D1211yescharacteristic
199C22×C4C2×#6711yescharacteristic
200M16—122no—
201M16—122no—
202C2×C8—24no—
203C2×C8—84no—
204C2×C8—122no—
205C4⋊C4—22no—
206C4⋊C4—64no—
207C22⋊C4—24no—
208C22⋊C4—128no—
209C42—121no—
210C42—121no—
211C42#23512yes—
212C42C2×#6712yes—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—24no—
216C22×C3—44no—
217C22×C3—42no—
218C22×C3—42no—
219C22×C3—82no—
220C22×C3#38611yescharacteristic
221C22×C3C2×#10211yescharacteristic
222C22×C3#41611yescharacteristic
223C2×D3—84no—
224C2×D3—84no—
225C2×D3—82no—
226C2×D3—82no—
227C2×D3—82no—
228C3×C4—21no—
229C3×C4—22no—
230C3×C4—42no—
231C3×C4#45312yes—
232C3×C4#46512yes—
233C3×C4#46612yes—
234Dic3—44no—
235C23—22no—
236C23—22no—
237C23—42no—
238C23—42no—
239C23—124no—
240C23—124no—
241C23—122no—
242C23—122no—
243C23—122no—
244C23#82114yes—
245C23C2×(C2×D3)⋊C411yescharacteristic
246D4—24no—
247D4—24no—
248D4—244no—
249D4—242no—
250D4—2416no—
251D4—242no—
252D4—242no—
253D4—242no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—64no—
260C2×C4—82no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4C2×#20411yescharacteristic
264C2×C4C2×D2411yescharacteristic
265C2×C4#90412yes—
266C2×C4C4×#6712yes—
267C2×C4#92911yescharacteristic
268C2×C4#93012yes—
269C2×C4#95311yescharacteristic
270C8—44no—
271C8—242no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—44no—
275C2×C3#188711yescharacteristic
276C2×C3#189311yescharacteristic
277C2×C3#191411yescharacteristic
278D3—162no—
279C22—21no—
280C22—21no—
281C22—24no—
282C22—44no—
283C22—42no—
284C22—42no—
285C22—82no—
286C22—244no—
287C22—244no—
288C22—242no—
289C22—242no—
290C22—242no—
291C22C2×#82111yescharacteristic
292C22#511311yescharacteristic
293C22#521311yescharacteristic
294C4—22no—
295C4—21no—
296C4—42no—
297C4—124no—
298C4#511412yes—
299C4#511512yes—
300C4#524112yes—
301C3#939711yescharacteristic, complemented, Sylow
302C2—21no—
303C2—21no—
304C2—44no—
305C2—482no—
306C2#7165011yescharacteristic
307C2#7197311yescharacteristic
308C2#7203011yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2972C2
socleC22×C3#386
prime radical#82378C2
cosocleC2×C3×C42C23
hypercenter#2972D3
hypocenterC3#9397
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
1. C3×#2972C2semidirect
3. #71617C2semidirect
4. #70774C2semidirect
5. #67746C2semidirect
21. #2972D3semidirect
47. C3×C4⋊C8D4semidirect
131. C4⋊C8#67semidirect
301. C3#9397semidirect

Maximal subgroups

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

subgroupindex
C3×#29722
#823782
#716172
#707742
#677462
#93973