Finite Group Databasebeta

← #10692655

Subgroups — #10692655

Order 768.

552 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 99 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
1#82182C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic, complemented
3C3×#1463C211yesmaximal, characteristic, complemented
4#69977C212yesmaximal
5#67683C212yesmaximal, complemented
6#9414—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C3×#407C2212yes—
10#5213—21no—
11#5213—21no—
12#5212—21no—
13#5211—21no—
14C4×#217C2211yescharacteristic
15C4×#217C2212yes—
16#4991C412yescomplemented
17#4990—22no—
18#4986C412yescomplemented
19#4980—24no—
20C3×#324—22no—
21#4470—24no—
22#2591—31no—
23#2313—31no—
24#1507—32no—
25#1463D311yescharacteristic, complemented
26#1297—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C22×D12C2×C411yescharacteristic
32C2×C3×M16—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4C2×C412yes—
36C2×C3×C4⋊C4D412yes—
37C2×#217—22no—
38C2×#217—21no—
39#930—22no—
40#930—22no—
41C2×(C2×D3)⋊C4—22no—
42C2×(C2×D3)⋊C4—21no—
43C2×(C2×D3)⋊C4—22no—
44C2×C4⋊Dic3—22no—
45C2×C4⋊Dic3C2×C411yescharacteristic
46C2×C4×Dic3—22no—
47C2×C4×Dic3—21no—
48C3×C22⋊C8—22no—
49C3×C22⋊C8—22no—
50C3⋊#95—24no—
51C3⋊#95—22no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95C2×C412yes—
55C3⋊#95D414yes—
56C2×#121—31no—
57C22×C4⋊C4C2×D311yescharacteristic
58#407C2×D312yes—
59#400—32no—
60#394—32no—
61#388—62no—
62#387—64no—
63#386—61no—
64#386—61no—
65#381—61no—
66#380—61no—
67C4×C22⋊C4—32no—
68C4×C22⋊C4—31no—
69#327—64no—
70#324—22no—
71C24×C3C2×D411yescharacteristic
72C23×D3—21no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—42no—
78C22×C3×C4—81no—
79C22×C3×C4—81no—
80C22×C3×C4C22⋊C412yes—
81C22×C3×C4C22⋊C412yes—
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D412yes—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4#4112yes—
86C2×#67—44no—
87C2×#67—41no—
88C2×#67—41no—
89C22×Dic3—21no—
90C22×Dic3—22no—
91C22×Dic3—24no—
92C2×D12—22no—
93C2×D12—81no—
94C3×M16—82no—
95C2×C3×C8—42no—
96C3×C4⋊C4—24no—
97C3×C4⋊C4—24no—
98#217—44no—
99#217—42no—
100(C2×D3)⋊C4—44no—
101(C2×D3)⋊C4—42no—
102(C2×D3)⋊C4—82no—
103C4⋊Dic3—22no—
104C4⋊Dic3—82no—
105C4×Dic3—82no—
106C4×Dic3—81no—
107C22×D4—31no—
108C22×D4—61no—
109C23×C4D1211yescharacteristic
110C23×C4D1211yescharacteristic
111C23×C4C22×D311yescharacteristic
112C2×M16—22no—
113#121—62no—
114#121—62no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4C4×D312yes—
120C2×C4⋊C4#6712yes—
121C2×C22⋊C4—62no—
122C2×C22⋊C4—61no—
123C2×C22⋊C4—62no—
124C2×C22⋊C4—61no—
125C2×C22⋊C4—62no—
126C2×C42—62no—
127C2×C42—61no—
128C22⋊C8—22no—
129C22⋊C8—22no—
130#95—32no—
131#95—34no—
132#95—62no—
133#95—64no—
134#95—62no—
135#95—62no—
136C23×C3—22no—
137C23×C3—22no—
138C23×C3—41no—
139C23×C3C2×C22⋊C411yescharacteristic
140C23×C3#12011yescharacteristic
141C23×C3#12311yescharacteristic
142C22×D3—42no—
143C22×D3—42no—
144C22×D3—41no—
145C22×D3—42no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—42no—
151C2×C3×C4—41no—
152C2×C3×C4—42no—
153C2×C3×C4—41no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—81no—
159C2×C3×C4C4×D412yes—
160C2×C3×C4#12112yes—
161#67—82no—
162#67—82no—
163C2×Dic3—44no—
164C2×Dic3—42no—
165C2×Dic3—42no—
166C2×Dic3—44no—
167C2×Dic3—42no—
168C2×Dic3—41no—
169C2×Dic3—42no—
170C2×Dic3—44no—
171D12—82no—
172C3×C8—82no—
173C24—61no—
174C24C2×D1211yescharacteristic
175C2×D4—62no—
176C2×D4—124no—
177C2×D4—121no—
178C2×D4—121no—
179C2×D4—241no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—42no—
185C22×C4—64no—
186C22×C4—62no—
187C22×C4—61no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4(C2×D3)⋊C412yes—
191C22×C4(C2×D3)⋊C412yes—
192C22×C4C2×C4×D311yescharacteristic
193C22×C4D3×D412yes—
194C22×C4#23712yes—
195C22×C4C2×#6711yescharacteristic
196M16—82no—
197C2×C8—42no—
198C4⋊C4—24no—
199C4⋊C4—24no—
200C4⋊C4—62no—
201C4⋊C4—242no—
202C22⋊C4—124no—
203C22⋊C4—124no—
204C22⋊C4—122no—
205C22⋊C4—122no—
206C22⋊C4—242no—
207C42—242no—
208C42—241no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—44no—
217C22×C3#38611yescharacteristic
218C22×C3#42112yes—
219C22×C3#45412yes—
220C2×D3—84no—
221C2×D3—81no—
222C2×D3—81no—
223C2×D3—81no—
224C3×C4—22no—
225C3×C4—22no—
226C3×C4—42no—
227C3×C4—42no—
228Dic3—84no—
229Dic3—82no—
230Dic3—81no—
231C23—22no—
232C23—22no—
233C23—41no—
234C23—122no—
235C23—122no—
236C23—122no—
237C23—121no—
238C23#88211yescharacteristic
239C23#88611yescharacteristic
240C23C2×(C2×D3)⋊C411yescharacteristic
241D4—242no—
242D4—242no—
243D4—242no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—81no—
257C2×C4—122no—
258C2×C4—124no—
259C2×C4—124no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—121no—
264C2×C4—124no—
265C2×C4#67⋊C412yes—
266C2×C4#93912yes—
267C8—82no—
268C2×C3—22no—
269C2×C3—22no—
270C2×C3—41no—
271C2×C3#188811yescharacteristic
272C2×C3#189711yescharacteristic
273C2×C3#191311yescharacteristic
274D3—161no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—44no—
283C22—244no—
284C22—241no—
285C22—241no—
286C22—241no—
287C22#475312yes—
288C22#495811yescharacteristic
289C22#513912yes—
290C4—22no—
291C4—22no—
292C4—42no—
293C4—42no—
294C4—244no—
295C4—242no—
296C4—241no—
297C3#941411yescharacteristic, complemented, Sylow
298C2—22no—
299C2—22no—
300C2—41no—
301C2—481no—
302C2#6996611yescharacteristic
303C2#7002411yescharacteristic
304C2#7010411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1463C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenter#1463D3
hypocenterC3#9414
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
2. #82155C2semidirect
3. C3×#1463C2semidirect
5. #67683C2semidirect
16. #4991C4semidirect
18. #4986C4semidirect
25. #1463D3semidirect
297. C3#9414semidirect

Maximal subgroups

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

subgroupindex
#821822
#821552
C3×#14632
#699772
#676832
#94143