Finite Group Databasebeta

← #10691215

Subgroups — #10691215

Order 768.

478 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2921C211yesmaximal, characteristic
2#82262C211yesmaximal, characteristic
3#71571C211yesmaximal, characteristic
4#70506C212yesmaximal, complemented
5#67713C212yesmaximal, complemented
6#9196—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#924—21no—
9C2×#869C2211yescharacteristic
10C3×#433—24no—
11C3×#423C2212yes—
12C3×#407—21no—
13C3×C4×M16—21no—
14#4981—21no—
15#4981C412yes—
16#4953—21no—
17#4953C412yes—
18#4936C2211yescharacteristic
19#4929—21no—
20#4925C2212yes—
21#4528—21no—
22#4525—21no—
23#4482—24no—
24#2921D311yescharacteristic
25#2621—31no—
26#1546—32no—
27#1458—31no—
28#1286—32no—
29C23×C3×C4D411yescharacteristic
30C22×Dic6—21no—
31C2×C3×M16—22no—
32C3×#117—22no—
33C3×#117—22no—
34C2×C3×C4⋊C4C2×C411yescharacteristic
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C22⋊C4C2×C411yescharacteristic
37C2×C3×C22⋊C4D411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C3×C42C2311yescharacteristic
40C2×#217—22no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3C2×C412yes—
44#924—44no—
45C2×Dic3⋊C4—22no—
46#869—22no—
47#869—22no—
48C3×C4⋊C8—22no—
49C3×C4⋊C8D414yes—
50C3×C22⋊C8—24no—
51C3×#97—22no—
52C3×C4×C8—22no—
53C3⋊#95—22no—
54C3⋊#95—22no—
55C3⋊#95—21no—
56C3⋊#95C2×C412yes—
57#830—22no—
58C4⋊(C3⋊C8)—22no—
59C4×C3⋊C8—22no—
60C2×C4⋊Q8—31no—
61C2×C22⋊Q8—61no—
62C2×#117C2×D311yescharacteristic
63#433—24no—
64#431—32no—
65#423C2×D312yes—
66#407—21no—
67C4×M16—21no—
68#398—32no—
69#398—61no—
70#397—32no—
71#397—61no—
72#391—61no—
73#382—31no—
74#332—64no—
75#324—61no—
76#323—61no—
77C24×C3C22⋊C411yescharacteristic
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—22no—
81C22×C3×C4—22no—
82C22×C3×C4C22⋊C411yescharacteristic
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C22⋊C411yescharacteristic
85C22×C3×C4C22×C411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C22×Dic3—22no—
89C22×Dic3—22no—
90C2×Dic6—41no—
91C2×Dic6—41no—
92C2×Dic6—42no—
93C2×Dic6—42no—
94C3×M16—42no—
95C3×M16—42no—
96C2×C3×C8—24no—
97C3×C4⋊C4—22no—
98C3×C4⋊C4—22no—
99C3×C22⋊C4—22no—
100C3×C22⋊C4—22no—
101C3×C42—21no—
102C3×C42—21no—
103C3×C42C2×D412yes—
104C3×C42#4112yes—
105#217—44no—
106C4⋊Dic3—24no—
107C4⋊Dic3—42no—
108C4⋊Dic3—42no—
109Dic3⋊C4—44no—
110C2×C3⋊C8—42no—
111C22×Q8—61no—
112C23×C4#6711yescharacteristic
113C2×M16—22no—
114C4⋊Q8—62no—
115C4⋊Q8—62no—
116C22⋊Q8—124no—
117#117—22no—
118#117—22no—
119C2×C4⋊C4—32no—
120C2×C4⋊C4—62no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4—61no—
123C2×C4⋊C4C4×D311yescharacteristic
124C2×C4⋊C4D1211yescharacteristic
125C2×C22⋊C4—62no—
126C2×C22⋊C4C4×D311yescharacteristic
127C2×C22⋊C4D1211yescharacteristic
128C2×C42#6711yescharacteristic
129C2×C42C22×D311yescharacteristic
130C4⋊C8—22no—
131C4⋊C8—62no—
132C4⋊C8#6714yes—
133C22⋊C8—24no—
134C22⋊C8—62no—
135#97—22no—
136C4×C8—22no—
137C4×C8—62no—
138#95—32no—
139#95—62no—
140#95—62no—
141#95—61no—
142C23×C3—21no—
143C23×C3—21no—
144C23×C3—21no—
145C23×C3—41no—
146C23×C3—41no—
147C23×C3C2×C22⋊C411yescharacteristic
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—21no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4C4×D412yes—
160C2×C3×C4#12011yescharacteristic
161C2×C3×C4#12112yes—
162C2×C3×C4#12311yescharacteristic
163C2×C3×C4#13512yes—
164C2×C3×C4AGL(1,ℤ/8)12yes—
165C2×Dic3—42no—
166C2×Dic3—42no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169C2×Dic3—42no—
170C2×Dic3—42no—
171Dic6—82no—
172Dic6—82no—
173C3×C8—44no—
174C3⋊C8—82no—
175C24(C2×D3)⋊C411yescharacteristic
176C2×Q8—121no—
177C2×Q8—122no—
178C2×Q8—122no—
179C2×Q8—121no—
180C22×C4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—62no—
185C22×C4—62no—
186C22×C4(C2×D3)⋊C411yescharacteristic
187C22×C4(C2×D3)⋊C411yescharacteristic
188C22×C4(C2×D3)⋊C411yescharacteristic
189C22×C4C2×C4×D311yescharacteristic
190C22×C4C2×D1211yescharacteristic
191C22×C4C2×#6711yescharacteristic
192M16—42no—
193M16—42no—
194C2×C8—24no—
195C2×C8—122no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—64no—
199C4⋊C4—124no—
200C4⋊C4—122no—
201C4⋊C4—122no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—124no—
205C42—21no—
206C42—21no—
207C42#23512yes—
208C42C2×#6712yes—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—42no—
212C22×C3—41no—
213C22×C3—41no—
214C22×C3—41no—
215C22×C3—81no—
216C22×C3#38611yescharacteristic
217C22×C3#41611yescharacteristic
218C22×C3#41811yescharacteristic
219C3×C4—22no—
220C3×C4—22no—
221C3×C4—41no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4#44912yes—
225C3×C4#48212yes—
226Dic3—82no—
227Dic3—82no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—41no—
232C23—41no—
233C23C2×(C2×D3)⋊C411yescharacteristic
234Q8—242no—
235Q8—242no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—122no—
248C2×C4—122no—
249C2×C4—122no—
250C2×C4—122no—
251C2×C4—122no—
252C2×C4—122no—
253C2×C4C4×#6712yes—
254C2×C4#92911yescharacteristic
255C2×C4#93012yes—
256C2×C4#94912yes—
257C2×C4#95012yes—
258C2×C4#95311yescharacteristic
259C8—44no—
260C8—242no—
261C2×C3—21no—
262C2×C3—21no—
263C2×C3—81no—
264C2×C3#187511yescharacteristic
265C2×C3#190911yescharacteristic
266C2×C3#191511yescharacteristic
267C22—21no—
268C22—21no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22#500111yescharacteristic
275C22#500211yescharacteristic
276C22#521311yescharacteristic
277C4—22no—
278C4—22no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—242no—
283C4—242no—
284C4#500312yes—
285C4#523812yes—
286C3#919611yescharacteristic, complemented, Sylow
287C2—21no—
288C2—21no—
289C2—81no—
290C2#7158611yescharacteristic
291C2#7202411yescharacteristic
292C2#7203111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2921C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2921D3
hypocenterC3#9196
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
4. #70506C2semidirect
5. #67713C2semidirect
286. C3#9196semidirect

Maximal subgroups

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

subgroupindex
C3×#29212
#822622
#715712
#705062
#677132
#91963