Finite Group Databasebeta

← #10704592

Subgroups — #10704592

Order 768.

434 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 77 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#74225C211yesmaximal, characteristic
2#68144C212yesmaximal
3C3×#1324C211yesmaximal, characteristic, complemented
4#67632C211yesmaximal, characteristic, complemented
5#67453C212yesmaximal, complemented
6#11514—31nomaximal, Sylow
7C2×#892C411yescharacteristic
8C2×D3×C4⋊C4C2211yescharacteristic
9C2×#872C411yescharacteristic
10C3×#382C2211yescharacteristic
11#4977C411yescharacteristic
12#4975C2211yescharacteristic
13#4935C411yescharacteristic
14#4726—22no—
15#4668—22no—
16#4647—22no—
17C3×#324C2212yes—
18#4528C2212yes—
19#4469—22no—
20#2317—31no—
21#1463—32no—
22#1325—32no—
23#1324—31no—
24#1324D311yescharacteristic, complemented
25C22×C4×D3C2×C411yescharacteristic
26C22×C4×D3D412yes—
27C2×C3×C4⋊C4C2×C411yescharacteristic
28C2×C3×C4⋊C4C2311yescharacteristic
29C2×C3×C42C2×C411yescharacteristic
30C2×(C2×D3)⋊C4—21no—
31C2×(C2×D3)⋊C4—21no—
32C2×C4⋊Dic3—22no—
33C2×C4⋊Dic3C2×C411yescharacteristic
34C2×C4⋊Dic3C2×C411yescharacteristic
35C2×Dic3⋊C4—21no—
36C2×Dic3⋊C4—22no—
37C2×Dic3⋊C4D412yes—
38C2×C4×Dic3—21no—
39C2×C4×Dic3C2×C411yescharacteristic
40C2×ΓL(1,𝔽25)—42no—
41#892—42no—
42D3×C4⋊C4—42no—
43#872—22no—
44#872—22no—
45C3×C22⋊C8—22no—
46C3×#95—22no—
47C3⋊#95—22no—
48#830—22no—
49(C2×D3)⋊C8—42no—
50C2×#117—31no—
51C2×#117—31no—
52C22×C4⋊C4—31no—
53#407—62no—
54#389—62no—
55#385—62no—
56#384—31no—
57#384—31no—
58#382—31no—
59#382C2×D311yescharacteristic
60#324—32no—
61#324—62no—
62#324C2×D312yes—
63C23×D3C22⋊C411yescharacteristic
64C22×C3×C4—21no—
65C22×C3×C4—21no—
66C22×C3×C4C22×C411yescharacteristic
67C22×C3×C4C2×D412yes—
68C22×Dic3—21no—
69C22×Dic3—21no—
70C22×Dic3C22⋊C411yescharacteristic
71C22×Dic3C22⋊C412yes—
72C2×C4×D3—22no—
73C2×C4×D3—21no—
74C2×C4×D3—21no—
75C2×C4×D3—22no—
76C2×C4×D3—22no—
77C2×C4×D3—22no—
78C2×C4×D3—82no—
79C2×C3×C8—42no—
80C3×C4⋊C4—41no—
81C3×C4⋊C4—41no—
82C3×C42—21no—
83C3×C42M1612yescomplemented
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—42no—
86C4⋊Dic3—41no—
87C4⋊Dic3—41no—
88C4⋊Dic3—82no—
89Dic3⋊C4—24no—
90Dic3⋊C4—42no—
91Dic3⋊C4—82no—
92C4×Dic3—21no—
93C4×Dic3—42no—
94C4×Dic3M1612yescomplemented
95C2×C3⋊C8—42no—
96ΓL(1,𝔽25)—84no—
97C23×C4—31no—
98C23×C4—32no—
99C2×M16—122no—
100#117—62no—
101#117—62no—
102#117—122no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4—31no—
106C2×C4⋊C4—61no—
107C2×C4⋊C4—62no—
108C2×C4⋊C4—62no—
109C2×C4⋊C4—122no—
110C2×C4⋊C4C4×D311yescharacteristic
111C2×C4⋊C4C22×D311yescharacteristic
112C2×C22⋊C4—61no—
113C2×C22⋊C4—61no—
114C2×C42—31no—
115C2×C42—61no—
116C2×C42C4×D311yescharacteristic
117C22⋊C8—22no—
118C22⋊C8—62no—
119C22⋊C8—122no—
120#95—22no—
121#95—62no—
122C23×C3C2×C22⋊C411yescharacteristic
123C22×D3—21no—
124C22×D3—21no—
125C22×D3—41no—
126C22×D3—41no—
127C22×D3#9912yes—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—41no—
133C2×C3×C4—41no—
134C2×C3×C4—42no—
135C2×C3×C4—42no—
136C2×C3×C4—41no—
137C2×C3×C4C2×M1611yescharacteristic
138C2×C3×C4C2×M1611yescharacteristic
139C2×Dic3—21no—
140C2×Dic3—21no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—41no—
144C2×Dic3—41no—
145C2×Dic3—42no—
146C2×Dic3—42no—
147C2×Dic3—41no—
148C2×Dic3#10112yes—
149C4×D3—24no—
150C4×D3—24no—
151C4×D3—44no—
152C4×D3—82no—
153C3×C8—82no—
154C3⋊C8—82no—
155C24—31no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—31no—
159C22×C4—32no—
160C22×C4—62no—
161C22×C4—62no—
162C22×C4—62no—
163C22×C4—62no—
164C22×C4—61no—
165C22×C4—61no—
166C22×C4—61no—
167C22×C4—61no—
168C22×C4—242no—
169C22×C4C2×C4×D311yescharacteristic
170C22×C4D3×D412yes—
171M16—244no—
172C2×C8—42no—
173C2×C8—122no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—64no—
177C4⋊C4—122no—
178C4⋊C4—121no—
179C4⋊C4—121no—
180C4⋊C4—242no—
181C4⋊C4—242no—
182C22⋊C4—64no—
183C22⋊C4—122no—
184C42—21no—
185C42—32no—
186C42—61no—
187C42—122no—
188C42ΓL(1,𝔽25)12yescomplemented
189C22×C3—21no—
190C22×C3—21no—
191C22×C3#40711yescharacteristic
192C22×C3C2×#9911yescharacteristic
193C22×C3C2×#10111yescharacteristic
194C2×D3—22no—
195C2×D3—22no—
196C2×D3—22no—
197C2×D3—42no—
198C2×D3—42no—
199C3×C4—21no—
200C3×C4—21no—
201C3×C4—42no—
202C3×C4—42no—
203C3×C4—81no—
204Dic3—22no—
205Dic3—42no—
206Dic3—42no—
207Dic3—81no—
208C23—32no—
209C23—61no—
210C23—61no—
211C23—121no—
212C23—121no—
213C23D3×C22⋊C411yescharacteristic
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—32no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—64no—
225C2×C4—64no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—124no—
231C2×C4—121no—
232C2×C4—121no—
233C2×C4—122no—
234C2×C4—122no—
235C2×C4—121no—
236C2×C4—242no—
237C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
238C2×C4D3×M1611yescharacteristic
239C8—82no—
240C8—242no—
241C2×C3—21no—
242C2×C3—21no—
243C2×C3#146311yescharacteristic
244C2×C3#212711yescharacteristic
245C2×C3#213311yescharacteristic
246D3—42no—
247C22—21no—
248C22—21no—
249C22—62no—
250C22—62no—
251C22—62no—
252C22—122no—
253C22—122no—
254C22#472611yescharacteristic
255C22D3×#9911yescharacteristic
256C22D3×#10111yescharacteristic
257C4—21no—
258C4—21no—
259C4—42no—
260C4—42no—
261C4—62no—
262C4—81no—
263C4—122no—
264C4—122no—
265C4—241no—
266C3#1151411yescharacteristic, complemented, Sylow
267C2—21no—
268C2—21no—
269C2—122no—
270C2#6814411yescharacteristic
271C2#6915711yescharacteristic
272C2#6916411yescharacteristic
subgroupG/H
centerC22D3×#99
commutatorC22×C3×C4C22×C4
FrattiniC2×C4⋊C4C22×D3
FittingC3×#1324C2
socleC22×C3C2×#99
prime radicalC23×D3C22⋊C4
cosocleC2×C3×C4⋊C4C23
hypercenter#1324D3
hypocenterC3#11514
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
3. C3×#1324C2semidirect
4. #67632C2semidirect
5. #67453C2semidirect
24. #1324D3semidirect
83. C3×C42M16semidirect
94. C4×Dic3M16semidirect
188. C42ΓL(1,𝔽25)semidirect
266. C3#11514semidirect

Maximal subgroups

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

subgroupindex
#742252
#681442
C3×#13242
#676322
#674532
#115143