Finite Group Databasebeta

← #73787

Subgroups — #73787

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×#895C211yesmaximal, characteristic
2C2×#890C211yesmaximal, characteristic, complemented
3C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
4#4988C211yesmaximal, characteristic, complemented
5#4987C211yesmaximal, characteristic, complemented
6#4940C212yesmaximal, complemented
7#4939C212yesmaximal, complemented
8#4939C211yesmaximal, characteristic, complemented
9C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
10#4674C212yesmaximal, complemented
11#4672C212yesmaximal, complemented
12#2490—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×D12C2212yes—
15C22×C4×D3C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C42C2211yescharacteristic
19C2×C3×C42C2212yes—
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2211yescharacteristic
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×C4⋊Dic3C2212yes—
29C2×C4⋊Dic3C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×C4×Dic3C2211yescharacteristic
32#895—24no—
33#890—24no—
34C3×#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×#125—31no—
39C2×#121—31no—
40#397—32no—
41#394—32no—
42#390—31no—
43#390—31no—
44#390—32no—
45#388—32no—
46#380—31no—
47C4×C4⋊C4D311yescharacteristic, complemented
48C4×C22⋊C4—31no—
49C23×D3C2312yes—
50C23×D3C2311yescharacteristic
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2312yes—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2312yes—
60C2×D12—24no—
61C2×D12—24no—
62C2×D12—22no—
63C2×D12—22no—
64C2×D12—22no—
65C2×D12—42no—
66C2×C4×D3—22no—
67C2×C4×D3—22no—
68C2×C4×D3—22no—
69C3×C4⋊C4—22no—
70C3×C4⋊C4D414yes—
71C3×C42—22no—
72C3×C42—42no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—24no—
77(C2×D3)⋊C4—42no—
78(C2×D3)⋊C4—42no—
79(C2×D3)⋊C4—42no—
80(C2×D3)⋊C4—42no—
81C4⋊Dic3—24no—
82C4⋊Dic3—22no—
83Dic3⋊C4—24no—
84C4×Dic3—22no—
85C22×D4—31no—
86C22×D4—32no—
87C23×C4—31no—
88#125—64no—
89#121—64no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—32no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4C2×D311yescharacteristic
94C2×C4⋊C4C2×D311yescharacteristic
95C2×C22⋊C4—32no—
96C2×C22⋊C4—31no—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C42—31no—
104C2×C42C2×D311yescharacteristic
105C2×C42C2×D312yes—
106#95—31no—
107#95—31no—
108#95—32no—
109#95C2×D312yes—
110C23×C3C2411yescharacteristic
111C22×D3—21no—
112C22×D3—21no—
113C22×D3—21no—
114C22×D3—21no—
115C22×D3—21no—
116C22×D3—21no—
117C22×D3—22no—
118C22×D3—22no—
119C22×D3—22no—
120C22×D3—22no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3#4112yes—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—21no—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—22no—
135C2×C3×C4C2×D414yes—
136C2×C3×C4C2×D412yes—
137C2×C3×C4#4112yes—
138C2×C3×C4#4112yes—
139C2×C3×C4#4112yes—
140C2×Dic3—21no—
141C2×Dic3—21no—
142C2×Dic3—22no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—22no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152D12—48no—
153D12—44no—
154C4×D3—44no—
155C24—31no—
156C24—32no—
157C2×D4—64no—
158C2×D4—64no—
159C2×D4—62no—
160C2×D4—62no—
161C2×D4—62no—
162C2×D4—122no—
163C22×C4—31no—
164C22×C4—31no—
165C22×C4—31no—
166C22×C4—32no—
167C22×C4—62no—
168C22×C4—62no—
169C22×C4—62no—
170C22×C4C22×D311yescharacteristic
171C22×C4C22×D312yes—
172C22×C4C22×D312yes—
173C22×C4C22×D311yescharacteristic
174C22×C4C22×D311yescharacteristic
175C4⋊C4—22no—
176C4⋊C4—64no—
177C4⋊C4—64no—
178C4⋊C4—62no—
179C4⋊C4D1214yes—
180C22⋊C4—62no—
181C22⋊C4—64no—
182C22⋊C4—62no—
183C22⋊C4—64no—
184C22⋊C4—122no—
185C22⋊C4—122no—
186C22⋊C4—122no—
187C22⋊C4—122no—
188C42—22no—
189C42—42no—
190C42—62no—
191C22×C3C22×D411yescharacteristic
192C22×C3C2×#4111yescharacteristic
193C22×C3C2×#4111yescharacteristic
194C22×C3C2×#4111yescharacteristic
195C22×C3C2×#4111yescharacteristic
196C22×C3H(5,𝔽2)11yescharacteristic
197C22×C3H(5,𝔽2)11yescharacteristic
198C2×D3—22no—
199C2×D3—22no—
200C2×D3—22no—
201C2×D3—41no—
202C2×D3—41no—
203C2×D3—41no—
204C2×D3—41no—
205C2×D3—42no—
206C2×D3—42no—
207C2×D3—42no—
208C2×D3—42no—
209C2×D3—42no—
210C2×D3—42no—
211C2×D3—42no—
212C3×C4—24no—
213C3×C4—22no—
214C3×C4—42no—
215C3×C4—41no—
216C3×C4#12414yes—
217Dic3—41no—
218Dic3—41no—
219Dic3—41no—
220Dic3—42no—
221C23—32no—
222C23—61no—
223C23—61no—
224C23—61no—
225C23—61no—
226C23—61no—
227C23—61no—
228C23—62no—
229C23—62no—
230C23—62no—
231C23—62no—
232C23—62no—
233C23—62no—
234C23—62no—
235C23C23×D311yescharacteristic
236D4—128no—
237D4—124no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—61no—
249C2×C4—61no—
250C2×C4—61no—
251C2×C4—62no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—62no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—62no—
260C2×C4—124no—
261C2×C4C2×D1214yes—
262C2×C4C2×D1212yes—
263C2×C4#23512yes—
264C2×C4#23512yes—
265C2×C4#23912yes—
266C2×C3C2×#12411yescharacteristic
267C2×C3#52911yescharacteristic
268C2×C3#54011yescharacteristic
269C2×C3#54611yescharacteristic
270C2×C3#55011yescharacteristic
271C2×C3#55311yescharacteristic
272C2×C3#55311yescharacteristic
273D3—42no—
274D3—82no—
275C22—62no—
276C22—62no—
277C22—62no—
278C22—121no—
279C22—121no—
280C22—121no—
281C22—121no—
282C22—122no—
283C22—122no—
284C22—122no—
285C22—122no—
286C22—122no—
287C22—122no—
288C22—122no—
289C22C22×D1211yescharacteristic
290C22C2×#23511yescharacteristic
291C22C2×#23511yescharacteristic
292C22#100411yescharacteristic
293C22C2×#23911yescharacteristic
294C22D3×#4111yescharacteristic
295C22#100911yescharacteristic
296C4—22no—
297C4—24no—
298C4—41no—
299C4—42no—
300C4—121no—
301C4—121no—
302C4—121no—
303C4—122no—
304C4#87514yes—
305C3#249011yescharacteristic, complemented, Sylow
306C2—122no—
307C2—242no—
308C2C2×#87511yescharacteristic
309C2#551411yescharacteristic
310C2#553011yescharacteristic
311C2#555411yescharacteristic
312C2#556211yescharacteristic
313C2#557711yescharacteristic
314C2#558011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×C4×C4⋊C4C2
prime radicalC2×#895C2
hypercenterC4×C4⋊C4D3
hypocenterC3#2490
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#890C2semidirect
3. C3×C4×C4⋊C4C2semidirect
4. #4988C2semidirect
5. #4987C2semidirect
6. #4940C2semidirect
7. #4939C2semidirect
8. #4939C2semidirect
9. C4×(C2×D3)⋊C4C2semidirect
10. #4674C2semidirect
11. #4672C2semidirect
47. C4×C4⋊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×#8952
C2×#8902
C3×C4×C4⋊C42
#49882
#49872
#49402
#49392
C4×(C2×D3)⋊C42
#46742
#46722
#24903