Finite Group Databasebeta

← #73763

Subgroups — #73763

Order 384.

668 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#895C211yesmaximal, characteristic
2C2×C4×D12C211yesmaximal, characteristic, complemented
3C2×#872C211yesmaximal, characteristic, complemented
4C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5#4988C211yesmaximal, characteristic, complemented
6#4987C211yesmaximal, characteristic, complemented
7#4984C211yesmaximal, characteristic, complemented
8#4940C212yesmaximal, complemented
9#4939C212yesmaximal, complemented
10#4668C212yesmaximal, complemented
11#4665C212yesmaximal, complemented
12#2324—31nomaximal, Sylow
13C22×D12C2211yescharacteristic
14C22×D12C2212yes—
15C22×C4×D3C2211yescharacteristic
16C22×C4×D3C2212yes—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C42C2212yes—
20C2×C3×C42C2211yescharacteristic
21C2×(C2×D3)⋊C4C2211yescharacteristic
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⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2212yes—
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2211yescharacteristic
32C2×C4×Dic3C2211yescharacteristic
33#895C418yescomplemented
34C4×D12—24no—
35#872—24no—
36C3×#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×#121—31no—
40C2×C4×D4—31no—
41C2×#117—31no—
42#390—31no—
43#390—32no—
44#389—31no—
45#388—32no—
46#386—32no—
47#385—32no—
48#380—31no—
49C4×C4⋊C4D311yescharacteristic, complemented
50C23×D3C2311yescharacteristic
51C23×D3C2312yes—
52C22×C3×C4C2312yes—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C2×D12—22no—
62C2×D12—22no—
63C2×D12—42no—
64C2×D12—42no—
65C2×D12C2×C414yes—
66C2×D12C2×C418yescomplemented
67C2×C4×D3—22no—
68C2×C4×D3—22no—
69C2×C4×D3—24no—
70C2×C4×D3—42no—
71C2×C4×D3—42no—
72C2×C4×D3C2×C414yescomplemented
73C3×C4⋊C4—22no—
74C3×C4⋊C4C2×C414yescomplemented
75C3×C42—42no—
76C3×C42D414yescomplemented
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—24no—
80(C2×D3)⋊C4—42no—
81(C2×D3)⋊C4—42no—
82(C2×D3)⋊C4—42no—
83(C2×D3)⋊C4C2×C418yescomplemented
84C4⋊Dic3—22no—
85C4⋊Dic3—42no—
86Dic3⋊C4—22no—
87Dic3⋊C4—22no—
88C4×Dic3—22no—
89C22×D4—31no—
90C22×D4—32no—
91C23×C4—31no—
92C23×C4—32no—
93#121—38no—
94C4×D4—64no—
95#117—64no—
96C2×C4⋊C4—31no—
97C2×C4⋊C4—31no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—32no—
100C2×C4⋊C4C2×D311yescharacteristic
101C2×C4⋊C4C2×D311yescharacteristic
102C2×C22⋊C4—32no—
103C2×C22⋊C4—31no—
104C2×C22⋊C4—31no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C42—31no—
110C2×C42C2×D312yes—
111C2×C42C2×D311yescharacteristic
112#95—32no—
113#95—32no—
114#95C2×D312yes—
115C23×C3C2411yescharacteristic
116C22×D3—21no—
117C22×D3—22no—
118C22×D3—21no—
119C22×D3—22no—
120C22×D3—21no—
121C22×D3—22no—
122C22×D3—22no—
123C22×D3—22no—
124C22×D3—21no—
125C22×D3—21no—
126C22×D3—21no—
127C22×D3—22no—
128C22×D3C22×C414yes—
129C22×D3C22×C412yes—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4C22×C414yes—
140C2×C3×C4C22×C412yes—
141C2×C3×C4C2×D412yes—
142C2×C3×C4C2×D412yes—
143C2×C3×C4C2×D412yes—
144C2×C3×C4#4112yes—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—21no—
151C2×Dic3—22no—
152C2×Dic3—21no—
153C2×Dic3—21no—
154C2×Dic3—22no—
155C2×Dic3—21no—
156C2×Dic3C22×C412yes—
157D12—28no—
158D12—48no—
159C4×D3—28no—
160C4×D3—48no—
161C24—32no—
162C24—31no—
163C2×D4—34no—
164C2×D4—38no—
165C2×D4—62no—
166C2×D4—62no—
167C2×D4—122no—
168C2×D4—122no—
169C22×C4—31no—
170C22×C4—34no—
171C22×C4—31no—
172C22×C4—32no—
173C22×C4—31no—
174C22×C4—62no—
175C22×C4—62no—
176C22×C4—64no—
177C22×C4—122no—
178C22×C4—122no—
179C22×C4C22×D312yes—
180C22×C4C22×D312yes—
181C22×C4C22×D311yescharacteristic
182C22×C4C22×D311yescharacteristic
183C22×C4C22×D311yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—62no—
186C4⋊C4—62no—
187C4⋊C4—62no—
188C4⋊C4—122no—
189C4⋊C4C4×D314yescomplemented
190C22⋊C4—38no—
191C22⋊C4—62no—
192C22⋊C4—62no—
193C22⋊C4—64no—
194C22⋊C4—122no—
195C22⋊C4—122no—
196C22⋊C4—122no—
197C42—42no—
198C42—62no—
199C42D1214yescomplemented
200C22×C3C23×C411yescharacteristic
201C22×C3C22×D411yescharacteristic
202C22×C3C2×#4111yescharacteristic
203C22×C3H(5,𝔽2)11yescharacteristic
204C22×C3H(5,𝔽2)11yescharacteristic
205C22×C3H(5,𝔽2)11yescharacteristic
206C22×C32−1+411yescharacteristic
207C2×D3—22no—
208C2×D3—24no—
209C2×D3—22no—
210C2×D3—24no—
211C2×D3—22no—
212C2×D3—24no—
213C2×D3—42no—
214C2×D3—41no—
215C2×D3—41no—
216C2×D3—42no—
217C2×D3—41no—
218C2×D3—42no—
219C2×D3—41no—
220C2×D3—42no—
221C3×C4—24no—
222C3×C4—22no—
223C3×C4—22no—
224C3×C4—42no—
225C3×C4C4×D414yes—
226Dic3—22no—
227Dic3—42no—
228Dic3—41no—
229Dic3—41no—
230C23—34no—
231C23—32no—
232C23—61no—
233C23—62no—
234C23—61no—
235C23—62no—
236C23—62no—
237C23—61no—
238C23—62no—
239C23—61no—
240C23—61no—
241C23—62no—
242C23—62no—
243C23—61no—
244C23C23×D311yescharacteristic
245D4—68no—
246D4—128no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—32no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—68no—
260C2×C4—61no—
261C2×C4—61no—
262C2×C4—62no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—62no—
267C2×C4—61no—
268C2×C4—62no—
269C2×C4—128no—
270C2×C4C2×C4×D314yes—
271C2×C4C2×C4×D312yes—
272C2×C4C2×D1212yes—
273C2×C4C2×D1212yes—
274C2×C4C2×D1212yes—
275C2×C4#23512yes—
276C2×C3C2×C4×D411yescharacteristic
277C2×C3#51811yescharacteristic
278C2×C3#52011yescharacteristic
279C2×C3#53511yescharacteristic
280C2×C3#53711yescharacteristic
281C2×C3#54011yescharacteristic
282C2×C3#54211yescharacteristic
283D3—42no—
284D3—44no—
285C22—62no—
286C22—64no—
287C22—62no—
288C22—64no—
289C22—64no—
290C22—62no—
291C22—121no—
292C22—122no—
293C22—122no—
294C22—121no—
295C22—121no—
296C22—122no—
297C22—121no—
298C22—122no—
299C22C22×C4×D311yescharacteristic
300C22C22×D1211yescharacteristic
301C22C2×#23511yescharacteristic
302C22#100411yescharacteristic
303C22#100711yescharacteristic
304C22#100911yescharacteristic
305C22#100911yescharacteristic
306C4—22no—
307C4—22no—
308C4—24no—
309C4—42no—
310C4—62no—
311C4—121no—
312C4—122no—
313C4—121no—
314C4C4×D1214yes—
315C3#232411yescharacteristic, complemented, Sylow
316C2—122no—
317C2—124no—
318C2C2×C4×D1211yescharacteristic
319C2#551011yescharacteristic
320C2#552511yescharacteristic
321C2#554511yescharacteristic
322C2#556211yescharacteristic
323C2#557411yescharacteristic
324C2#558211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×C4×C4⋊C4C2
socle, cosocleC23×C3C24
prime radicalC2×#895C2
hypercenterC4×C4⋊C4D3
hypocenterC3#2324
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×D12C2semidirect
3. C2×#872C2semidirect
4. C3×C4×C4⋊C4C2semidirect
5. #4988C2semidirect
6. #4987C2semidirect
7. #4984C2semidirect
8. #4940C2semidirect
9. #4939C2semidirect
10. #4668C2semidirect
11. #4665C2semidirect
33. #895C4semidirect
49. C4×C4⋊C4D3semidirect
66. C2×D12C2×C4semidirect
72. C2×C4×D3C2×C4semidirect
74. C3×C4⋊C4C2×C4semidirect
76. C3×C42D4semidirect
83. (C2×D3)⋊C4C2×C4semidirect
189. C4⋊C4C4×D3semidirect
199. C42D12semidirect
315. C3#2324semidirect

Maximal subgroups

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

subgroupindex
C2×#8952
C2×C4×D122
C2×#8722
C3×C4×C4⋊C42
#49882
#49872
#49842
#49402
#49392
#46682
#46652
#23243