Finite Group Databasebeta

← #68334

Subgroups — #68334

Order 384.

334 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 65 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#5627C211yesmaximal, characteristic
2#4726C211yesmaximal, characteristic, complemented
3D3×#95C211yesmaximal, characteristic, complemented
4C3×#328C211yesmaximal, characteristic, complemented
5#4538C211yesmaximal, characteristic, complemented
6#4470C211yesmaximal, characteristic, complemented
7#4451C211yesmaximal, characteristic, complemented
8#1504—31nomaximal, Sylow
9C2×#239C411yescharacteristic, complemented
10C22×C4×D3—21no—
11C22×C4×D3C2211yescharacteristic
12C3×C22⋊Q8C2211yescharacteristic
13#946C411yescharacteristic, complemented
14#930C2211yescharacteristic
15C2×ΓL(1,𝔽25)—21no—
16#895C411yescharacteristic, complemented
17#894—21no—
18#893—21no—
19#892C411yescharacteristic, complemented
20#882—21no—
21#67⋊C4—21no—
22C3×C22⋊C8C2211yescharacteristic
23C3×#95C2211yescharacteristic
24C3⋊#95—21no—
25C3⋊#95C2211yescharacteristic
26#830C2211yescharacteristic
27(C2×D3)⋊C8—21no—
28#525—31no—
29#407—31no—
30C2×#95—31no—
31#328—31no—
32#328D311yescharacteristic, complemented
33#327—31no—
34#327—31no—
35C23×D3D411yescharacteristic
36C2×C3×Q8C2×C411yescharacteristic, complemented
37C22×C3×C4—21no—
38C22×C3×C4C2311yescharacteristic
39C2×#67—21no—
40C22×Dic3—21no—
41C22×Dic3D411yescharacteristic
42#239—42no—
43C2×D12—21no—
44C2×D12—21no—
45C2×D12C2×C411yescharacteristic, complemented
46C2×C4×D3—24no—
47C2×C4×D3—21no—
48C2×C4×D3—42no—
49C2×C4×D3—42no—
50C2×C4×D3—42no—
51C2×C4×D3C2×C411yescharacteristic
52C2×C4×D3C2×C411yescharacteristic
53C2×C4×D3D411yescharacteristic
54C2×C4×D3D411yescharacteristic
55C2×C3×C8—21no—
56C3×C4⋊C4—21no—
57C3×C4⋊C4C2×C411yescharacteristic, complemented
58C3×C22⋊C4—21no—
59(C2×D3)⋊C4—21no—
60(C2×D3)⋊C4—21no—
61(C2×D3)⋊C4—21no—
62(C2×D3)⋊C4—21no—
63C4⋊Dic3C2×C411yescharacteristic, complemented
64Dic3⋊C4—21no—
65C4×Dic3—21no—
66C2×C3⋊C8—21no—
67ΓL(1,𝔽25)—42no—
68C2×#41—31no—
69C23×C4—31no—
70C23×C4—61no—
71C2×M16—61no—
72#123—61no—
73C22⋊Q8—31no—
74C22⋊Q8C2×D311yescharacteristic
75#121—31no—
76#121—31no—
77#120—61no—
78C4×D4—61no—
79C4×D4—61no—
80#117—31no—
81C22⋊C8—31no—
82C22⋊C8—61no—
83C22⋊C8C2×D311yescharacteristic
84#95—31no—
85#95—61no—
86#95C2×D311yescharacteristic
87C23×C3C2×D411yescharacteristic
88C22×D3—21no—
89C22×D3—21no—
90C22×D3—22no—
91C22×D3—22no—
92C22×D3—21no—
93C22×D3C22⋊C411yescharacteristic
94C22×D3C22⋊C411yescharacteristic
95C3×Q8—41no—
96C2×C3×C4—22no—
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×C3×C4—41no—
102C2×C3×C4C22×C411yescharacteristic
103C2×C3×C4C2×D411yescharacteristic
104#67—41no—
105#67—41no—
106C2×Dic3—22no—
107C2×Dic3—21no—
108C2×Dic3—41no—
109C2×Dic3—41no—
110C2×Dic3—41no—
111C2×Dic3C22⋊C411yescharacteristic
112C2×Dic3C22⋊C411yescharacteristic
113D12—41no—
114D12—42no—
115D12—42no—
116C4×D3—22no—
117C4×D3—22no—
118C4×D3—41no—
119C4×D3—41no—
120C4×D3—42no—
121C4×D3—48no—
122C3×C8—41no—
123C3⋊C8—41no—
124C24—31no—
125#41—122no—
126C2×Q8C4×D311yescharacteristic, complemented
127C2×D4—31no—
128C2×D4—61no—
129C2×D4—61no—
130C2×D4—61no—
131C22×C4—21no—
132C22×C4—31no—
133C22×C4—31no—
134C22×C4—31no—
135C22×C4—31no—
136C22×C4—31no—
137C22×C4—61no—
138C22×C4—64no—
139C22×C4—61no—
140C22×C4—122no—
141C22×C4—122no—
142C22×C4—122no—
143C22×C4C22×D311yescharacteristic
144M16—122no—
145C2×C8—21no—
146C2×C8—61no—
147C4⋊C4—21no—
148C4⋊C4—31no—
149C4⋊C4—61no—
150C4⋊C4C4×D311yescharacteristic, complemented
151C22⋊C4—21no—
152C22⋊C4—61no—
153C22⋊C4—61no—
154C22⋊C4—61no—
155C22⋊C4—61no—
156C42—61no—
157C22×C3—21no—
158C22×C3—21no—
159C22×C3C2×C22⋊C411yescharacteristic
160C22×C3AGL(1,ℤ/8)11yescharacteristic
161C22×C3#13711yescharacteristic
162C2×D3—24no—
163C2×D3—24no—
164C2×D3—21no—
165C2×D3—21no—
166C2×D3—21no—
167C2×D3—21no—
168C2×D3—41no—
169C2×D3—41no—
170C2×D3—41no—
171C2×D3#9912yes—
172C2×D3#10412yes—
173C3×C4—21no—
174C3×C4—21no—
175C3×C4—42no—
176C3×C4—41no—
177C3×C4—41no—
178Dic3—21no—
179Dic3—21no—
180Dic3—42no—
181Dic3—41no—
182C23—31no—
183C23—31no—
184C23—62no—
185C23—62no—
186C23—61no—
187C23—61no—
188C23—61no—
189C23D3×D411yescharacteristic
190Q8—41no—
191D4—121no—
192D4—122no—
193D4—122no—
194D4—121no—
195D4—121no—
196C2×C4—22no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—61no—
208C2×C4—121no—
209C2×C4—121no—
210C2×C4—122no—
211C2×C4—128no—
212C2×C4—121no—
213C2×C4—121no—
214C2×C4—121no—
215C2×C4C2×C4×D311yescharacteristic
216C2×C4D3×D411yescharacteristic
217C8—41no—
218C8—121no—
219C2×C3—21no—
220C2×C3—21no—
221C2×C3C2×#9911yescharacteristic
222C2×C3#41711yescharacteristic
223C2×C3C2×#10411yescharacteristic
224D3—22no—
225D3—22no—
226D3—81no—
227C22—21no—
228C22—21no—
229C22—32no—
230C22—32no—
231C22—64no—
232C22—64no—
233C22—61no—
234C22—61no—
235C22—61no—
236C22—61no—
237C22—121no—
238C22—121no—
239C22—121no—
240C22D3×C22⋊C411yescharacteristic
241C22#91411yescharacteristic
242C22#91811yescharacteristic
243C4—21no—
244C4—21no—
245C4—42no—
246C4—41no—
247C4—41no—
248C4—61no—
249C4—61no—
250C4—122no—
251C4—121no—
252C3#150411yescharacteristic, complemented, Sylow
253C2—21no—
254C2—21no—
255C2—62no—
256C2—62no—
257C2—241no—
258C2D3×#9911yescharacteristic
259C2#480311yescharacteristic
260C2D3×#10411yescharacteristic
subgroupG/H
centerC22D3×C22⋊C4
commutatorC2×C3×C4C22×C4
FrattiniC22×C4C22×D3
FittingC3×#328C2
socleC22×C3C2×C22⋊C4
prime radical#5627C2
cosocleC22×C3×C4C23
hypercenter#328D3
hypocenterC3#1504
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #4726C2semidirect
3. D3×#95C2semidirect
4. C3×#328C2semidirect
5. #4538C2semidirect
6. #4470C2semidirect
7. #4451C2semidirect
9. C2×#239C4semidirect
13. #946C4semidirect
16. #895C4semidirect
19. #892C4semidirect
32. #328D3semidirect
36. C2×C3×Q8C2×C4semidirect
45. C2×D12C2×C4semidirect
57. C3×C4⋊C4C2×C4semidirect
63. C4⋊Dic3C2×C4semidirect
126. C2×Q8C4×D3semidirect
150. C4⋊C4C4×D3semidirect
252. C3#1504semidirect

Maximal subgroups

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

subgroupindex
#56272
#47262
D3×#952
C3×#3282
#45382
#44702
#44512
#15043