Finite Group Databasebeta

← #10691205

Subgroups — #10691205

Order 768.

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

normal subgroups only

This group has 458 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#82264C211yesmaximal, characteristic
3#71571C211yesmaximal, characteristic
4#70508C212yesmaximal, complemented
5#67713C212yesmaximal, complemented
6#9194—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#869C2211yescharacteristic
9C3×#433—24no—
10C3×#423C2212yes—
11C3×#407—21no—
12C3×C4×M16—21no—
13#5212—21no—
14#4986C412yes—
15#4979—21no—
16#4949C412yes—
17#4947—21no—
18#4936C2211yescharacteristic
19#4928—21no—
20#4925C2212yes—
21#4528—21no—
22#4525—21no—
23#4486—24no—
24#2921D311yescharacteristic
25#2628—31no—
26#1549—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—
44C2×Dic3⋊C4—22no—
45#869—22no—
46#869—22no—
47C3×C4⋊C8—22no—
48C3×C4⋊C8D414yes—
49C3×C22⋊C8—24no—
50C3×#97—22no—
51C3×C4×C8—22no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95—21no—
55C3⋊#95C2×C412yes—
56#830—22no—
57C4⋊(C3⋊C8)—22no—
58C4×C3⋊C8—22no—
59C2×C4⋊Q8—31no—
60C2×#117C2×D311yescharacteristic
61#433—24no—
62#431—32no—
63#423C2×D312yes—
64#407—21no—
65C4×M16—21no—
66#400—32no—
67#399—32no—
68#395—61no—
69#393—61no—
70#389—61no—
71#382—31no—
72#380—61no—
73#332—64no—
74#324—61no—
75#323—61no—
76C24×C3C22⋊C411yescharacteristic
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4C22⋊C411yescharacteristic
82C22×C3×C4C22⋊C411yescharacteristic
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C22×C411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C22×Dic3—22no—
88C22×Dic3—22no—
89C2×Dic6—42no—
90C2×Dic6—82no—
91C3×M16—42no—
92C3×M16—42no—
93C2×C3×C8—24no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—22no—
96C3×C22⋊C4—22no—
97C3×C22⋊C4—22no—
98C3×C42—21no—
99C3×C42—21no—
100C3×C42C2×D412yes—
101C3×C42#4112yes—
102#217—82no—
103C4⋊Dic3—24no—
104C4⋊Dic3—41no—
105C4⋊Dic3—41no—
106C4⋊Dic3—42no—
107Dic3⋊C4—82no—
108C2×C3⋊C8—42no—
109C22×Q8—61no—
110C23×C4#6711yescharacteristic
111C2×M16—22no—
112C4⋊Q8—62no—
113C4⋊Q8—62no—
114#117—22no—
115#117—22no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—61no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4C4×D311yescharacteristic
121C2×C4⋊C4D1211yescharacteristic
122C2×C22⋊C4—62no—
123C2×C22⋊C4C4×D311yescharacteristic
124C2×C22⋊C4D1211yescharacteristic
125C2×C42#6711yescharacteristic
126C2×C42C22×D311yescharacteristic
127C4⋊C8—22no—
128C4⋊C8—62no—
129C4⋊C8#6714yes—
130C22⋊C8—24no—
131C22⋊C8—62no—
132#97—22no—
133C4×C8—22no—
134C4×C8—62no—
135#95—32no—
136#95—62no—
137#95—62no—
138#95—61no—
139C23×C3—21no—
140C23×C3—21no—
141C23×C3—21no—
142C23×C3—41no—
143C23×C3—41no—
144C23×C3C2×C22⋊C411yescharacteristic
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—41no—
151C2×C3×C4—41no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—41no—
155C2×C3×C4—41no—
156C2×C3×C4C4×D412yes—
157C2×C3×C4#12011yescharacteristic
158C2×C3×C4#12112yes—
159C2×C3×C4#12311yescharacteristic
160C2×C3×C4#13512yes—
161C2×C3×C4#13712yes—
162C2×Dic3—42no—
163C2×Dic3—42no—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—42no—
167C2×Dic3—42no—
168Dic6—84no—
169C3×C8—44no—
170C3⋊C8—82no—
171C24(C2×D3)⋊C411yescharacteristic
172C2×Q8—122no—
173C2×Q8—242no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—21no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4(C2×D3)⋊C411yescharacteristic
181C22×C4(C2×D3)⋊C411yescharacteristic
182C22×C4(C2×D3)⋊C411yescharacteristic
183C22×C4C2×C4×D311yescharacteristic
184C22×C4C2×D1211yescharacteristic
185C22×C4C2×#6711yescharacteristic
186M16—42no—
187M16—42no—
188C2×C8—24no—
189C2×C8—122no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—64no—
193C4⋊C4—121no—
194C4⋊C4—122no—
195C4⋊C4—121no—
196C4⋊C4—242no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—242no—
200C42—21no—
201C42—21no—
202C42#23512yes—
203C42C2×#6712yes—
204C22×C3—21no—
205C22×C3—21no—
206C22×C3—42no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—81no—
211C22×C3#38611yescharacteristic
212C22×C3#41611yescharacteristic
213C22×C3#41911yescharacteristic
214C3×C4—22no—
215C3×C4—22no—
216C3×C4—41no—
217C3×C4—41no—
218C3×C4—41no—
219C3×C4#45212yes—
220C3×C4#48512yes—
221Dic3—82no—
222Dic3—82no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—41no—
227C23—41no—
228C23C2×(C2×D3)⋊C411yescharacteristic
229Q8—244no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—122no—
242C2×C4—122no—
243C2×C4—122no—
244C2×C4—122no—
245C2×C4—122no—
246C2×C4—122no—
247C2×C4C4×#6712yes—
248C2×C4#92911yescharacteristic
249C2×C4#93012yes—
250C2×C4#95012yes—
251C2×C4#95112yes—
252C2×C4#95311yescharacteristic
253C8—44no—
254C8—242no—
255C2×C3—21no—
256C2×C3—21no—
257C2×C3—81no—
258C2×C3#187311yescharacteristic
259C2×C3#191011yescharacteristic
260C2×C3#191511yescharacteristic
261C22—21no—
262C22—21no—
263C22—42no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—81no—
268C22#500211yescharacteristic
269C22#500411yescharacteristic
270C22#521311yescharacteristic
271C4—22no—
272C4—22no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4—242no—
277C4—242no—
278C4#499612yes—
279C4#523912yes—
280C3#919411yescharacteristic, complemented, Sylow
281C2—21no—
282C2—21no—
283C2—81no—
284C2#7158511yescharacteristic
285C2#7202511yescharacteristic
286C2#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#9194
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. #70508C2semidirect
5. #67713C2semidirect
280. C3#9194semidirect

Maximal subgroups

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

subgroupindex
C3×#29212
#822642
#715712
#705082
#677132
#91943