Finite Group Databasebeta

← #73506

Subgroups — #73506

Order 384.

560 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#930C211yesmaximal, characteristic
2C2×#870C211yesmaximal, characteristic, complemented
3C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
4#5212C211yesmaximal, characteristic, complemented
5#4959C212yesmaximal, complemented
6#4957C212yesmaximal, complemented
7#4939C211yesmaximal, characteristic, complemented
8#4939C211yesmaximal, characteristic, complemented
9C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
10#4675C212yesmaximal, complemented
11#4669C212yesmaximal, complemented
12#2490—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2212yes—
15C22×D12C2211yescharacteristic
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×C3×C22⋊C4C2211yescharacteristic
18C2×C3×C42C2211yescharacteristic
19C2×C3×C42C2211yescharacteristic
20C2×#217C2212yes—
21#930—24no—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2212yes—
32C2×C4×Dic3C2212yes—
33#870—24no—
34C3×#95C2211yescharacteristic
35C3×#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#125—31no—
39C2×#121—31no—
40#397—32no—
41#394—32no—
42#390—32no—
43#390—31no—
44#390—31no—
45#388—32no—
46#380—31no—
47C4×C4⋊C4—31no—
48C4×C22⋊C4D311yescharacteristic, complemented
49C24×C3C2311yescharacteristic
50C23×D3C2312yes—
51C22×C3×C4—22no—
52C22×C3×C4—22no—
53C22×C3×C4—22no—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C2×#67—24no—
61C2×#67—24no—
62C2×#67—42no—
63C22×Dic3C2312yes—
64C22×Dic3C2312yes—
65C22×Dic3C2312yes—
66C2×D12—22no—
67C2×D12—22no—
68C2×D12—22no—
69C3×C22⋊C4—22no—
70C3×C22⋊C4—22no—
71C3×C42—22no—
72C3×C42—22no—
73#217—42no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—24no—
76(C2×D3)⋊C4—42no—
77(C2×D3)⋊C4—42no—
78(C2×D3)⋊C4—42no—
79C4⋊Dic3—22no—
80C4⋊Dic3—22no—
81C4⋊Dic3D414yes—
82Dic3⋊C4—24no—
83Dic3⋊C4—24no—
84C4×Dic3—42no—
85C22×D4—32no—
86C22×D4—31no—
87C23×C4C2×D311yescharacteristic
88#125—64no—
89#121—64no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4—31no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4C2×D311yescharacteristic
102C2×C22⋊C4C2×D311yescharacteristic
103C2×C42—32no—
104C2×C42C2×D311yescharacteristic
105C2×C42C2×D311yescharacteristic
106#95—32no—
107#95—32no—
108#95C2×D311yescharacteristic
109#95C2×D311yescharacteristic
110C23×C3—21no—
111C23×C3—21no—
112C23×C3—21no—
113C23×C3—21no—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3#4112yes—
117C23×C3C2411yescharacteristic
118C22×D3—22no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—22no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—44no—
140C2×C3×C4C2×D412yes—
141C2×C3×C4#4112yes—
142C2×C3×C4#4112yes—
143C2×C3×C4#4112yes—
144#67—48no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3C2×D414yes—
154D12—44no—
155C24—32no—
156C24C22×D311yescharacteristic
157C2×D4—64no—
158C2×D4—64no—
159C2×D4—62no—
160C2×D4—62no—
161C2×D4—62no—
162C2×D4—122no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—32no—
167C22×C4—32no—
168C22×C4—32no—
169C22×C4C22×D311yescharacteristic
170C22×C4C22×D311yescharacteristic
171C22×C4C22×D311yescharacteristic
172C22×C4C22×D311yescharacteristic
173C22×C4C22×D311yescharacteristic
174C22×C4C22×D311yescharacteristic
175C4⋊C4—34no—
176C4⋊C4—64no—
177C4⋊C4—64no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—64no—
183C22⋊C4—64no—
184C22⋊C4—122no—
185C22⋊C4—122no—
186C22⋊C4—122no—
187C22⋊C4—122no—
188C42—22no—
189C42—22no—
190C42—122no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—41no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3C22×D411yescharacteristic
199C22×C3C2×#4111yescharacteristic
200C22×C3C2×#4111yescharacteristic
201C22×C3C2×#4111yescharacteristic
202C22×C3C2×#4111yescharacteristic
203C22×C3H(5,𝔽2)11yescharacteristic
204C22×C3H(5,𝔽2)11yescharacteristic
205C2×D3—42no—
206C2×D3—42no—
207C2×D3—42no—
208C2×D3—42no—
209C2×D3—42no—
210C2×D3—42no—
211C2×D3—42no—
212C3×C4—22no—
213C3×C4—41no—
214C3×C4—41no—
215C3×C4—41no—
216C3×C4—41no—
217C3×C4#12414yes—
218Dic3—24no—
219Dic3—42no—
220Dic3—42no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—62no—
228C23—62no—
229C23—62no—
230C23—62no—
231C23—62no—
232C23—62no—
233C23—62no—
234C23#23512yes—
235C23C23×D311yescharacteristic
236D4—128no—
237D4—124no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—34no—
253C2×C4—44no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4#23512yes—
263C2×C4#23512yes—
264C2×C4D3×D412yes—
265C2×C4#23712yes—
266C2×C3—42no—
267C2×C3C2×#12411yescharacteristic
268C2×C3#52911yescharacteristic
269C2×C3#54011yescharacteristic
270C2×C3#54611yescharacteristic
271C2×C3#55011yescharacteristic
272C2×C3#55311yescharacteristic
273C2×C3#55311yescharacteristic
274D3—82no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—122no—
283C22—122no—
284C22—122no—
285C22—122no—
286C22—122no—
287C22—122no—
288C22—122no—
289C22C2×#23511yescharacteristic
290C22C2×#23511yescharacteristic
291C22C2×#23511yescharacteristic
292C22C2×D3×D411yescharacteristic
293C22C2×#23711yescharacteristic
294C22#100911yescharacteristic
295C22#100911yescharacteristic
296C4—22no—
297C4—41no—
298C4—41no—
299C4—41no—
300C4—41no—
301C4—64no—
302C4—122no—
303C4—122no—
304C4#88514yes—
305C3#249011yescharacteristic, complemented, Sylow
306C2—42no—
307C2—242no—
308C2#547911yescharacteristic
309C2C2×#88511yescharacteristic
310C2#553511yescharacteristic
311C2#555011yescharacteristic
312C2#555311yescharacteristic
313C2#556511yescharacteristic
314C2#556511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×C4×C22⋊C4C2
prime radicalC2×#930C2
hypercenterC4×C22⋊C4D3
hypocenterC3#2490
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#870C2semidirect
3. C3×C4×C22⋊C4C2semidirect
4. #5212C2semidirect
5. #4959C2semidirect
6. #4957C2semidirect
7. #4939C2semidirect
8. #4939C2semidirect
9. C4×C4⋊Dic3C2semidirect
10. #4675C2semidirect
11. #4669C2semidirect
48. C4×C22⋊C4D3semidirect
305. C3#2490semidirect

Maximal subgroups

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

subgroupindex
C2×#9302
C2×#8702
C3×C4×C22⋊C42
#52122
#49592
#49572
#49392
C4×C4⋊Dic32
#46752
#46692
#24903