Finite Group Databasebeta

← #82236

Subgroups — #82236

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×C3×#117C211yesmaximal, characteristic, complemented
2C2×#930C211yesmaximal, characteristic
3C2×C4×D12C211yesmaximal, characteristic, complemented
4#5212C211yesmaximal, characteristic, complemented
5#4990C212yesmaximal, complemented
6#4989C212yesmaximal, complemented
7#4957C212yesmaximal, complemented
8#4956C212yesmaximal, complemented
9#4939C211yesmaximal, characteristic, complemented
10C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
11#4928C211yesmaximal, characteristic, complemented
12#2324—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2212yes—
15C22×D12C2211yescharacteristic
16C22×C4×D3C2212yes—
17C3×#117—24no—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C42C2211yescharacteristic
23C2×C3×C42C2211yescharacteristic
24C2×#217C2212yes—
25#930C418yescomplemented
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic
32C2×C4⋊Dic3C2211yescharacteristic
33C2×Dic3⋊C4C2212yes—
34C2×C4×Dic3C2212yes—
35C4×D12—24no—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×#121—31no—
40C2×C4×D4—31no—
41C2×#117D311yescharacteristic, complemented
42#390—32no—
43#390—31no—
44#389—31no—
45#388—32no—
46#386—32no—
47#385—32no—
48#380—31no—
49C4×C4⋊C4—31no—
50C24×C3C2311yescharacteristic
51C23×D3C2312yes—
52C22×C3×C4—22no—
53C22×C3×C4—22no—
54C22×C3×C4C2×C414yescomplemented
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C2×#67—42no—
62C2×#67—42no—
63C2×#67C2×C418yescomplemented
64C22×Dic3C2312yes—
65C22×Dic3C2312yes—
66C22×Dic3C2312yes—
67C2×D12—22no—
68C2×D12—22no—
69C2×D12C2×C414yes—
70C2×C4×D3—24no—
71C2×C4×D3—42no—
72C2×C4×D3—42no—
73C3×C4⋊C4—22no—
74C3×C4⋊C4—22no—
75C3×C22⋊C4—22no—
76C3×C22⋊C4—22no—
77C3×C42—22no—
78C3×C42D414yescomplemented
79#217—42no—
80(C2×D3)⋊C4—24no—
81(C2×D3)⋊C4—42no—
82(C2×D3)⋊C4—42no—
83(C2×D3)⋊C4C2×C418yescomplemented
84C4⋊Dic3—22no—
85C4⋊Dic3—22no—
86C4⋊Dic3C2×C414yescomplemented
87Dic3⋊C4—42no—
88C4×Dic3—42no—
89C22×D4—32no—
90C22×D4—31no—
91C23×C4—32no—
92C23×C4C2×D311yescharacteristic
93#121—38no—
94C4×D4—64no—
95#117—24no—
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—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4C2×D311yescharacteristic
108C2×C22⋊C4C2×D311yescharacteristic
109C2×C42—32no—
110C2×C42C2×D311yescharacteristic
111C2×C42C2×D311yescharacteristic
112#95—32no—
113#95—32no—
114#95—32no—
115C23×C3—21no—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3C22×C412yes—
122C23×C3C2411yescharacteristic
123C22×D3—22no—
124C22×D3—22no—
125C22×D3—22no—
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3C22×C414yes—
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—28no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4C22×C412yes—
144C2×C3×C4C22×C412yes—
145C2×C3×C4C2×D412yes—
146C2×C3×C4C2×D412yes—
147C2×C3×C4C2×D412yes—
148C2×C3×C4#4112yes—
149#67—48no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3C22×C414yes—
159D12—28no—
160C4×D3—48no—
161C24—32no—
162C24C22×D311yescharacteristic
163C2×D4—38no—
164C2×D4—34no—
165C2×D4—62no—
166C2×D4—62no—
167C2×D4—122no—
168C2×D4—122no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—32no—
172C22×C4—32no—
173C22×C4—32no—
174C22×C4—64no—
175C22×C4—122no—
176C22×C4—122no—
177C22×C4C4×D314yescomplemented
178C22×C4C22×D311yescharacteristic
179C22×C4C22×D311yescharacteristic
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D311yescharacteristic
182C22×C4C22×D311yescharacteristic
183C22×C4C22×D311yescharacteristic
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—34no—
187C4⋊C4—62no—
188C4⋊C4—62no—
189C4⋊C4—122no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—38no—
193C22⋊C4—64no—
194C22⋊C4—122no—
195C22⋊C4—122no—
196C22⋊C4—122no—
197C42—22no—
198C42—122no—
199C42#6714yescomplemented
200C22×C3—22no—
201C22×C3—22no—
202C22×C3—22no—
203C22×C3—41no—
204C22×C3—41no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3C23×C411yescharacteristic
208C22×C3C22×D411yescharacteristic
209C22×C3C2×#4111yescharacteristic
210C22×C3H(5,𝔽2)11yescharacteristic
211C22×C3H(5,𝔽2)11yescharacteristic
212C22×C3H(5,𝔽2)11yescharacteristic
213C22×C32−1+411yescharacteristic
214C2×D3—24no—
215C2×D3—24no—
216C2×D3—24no—
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—42no—
221C3×C4—22no—
222C3×C4—22no—
223C3×C4—22no—
224C3×C4—41no—
225C3×C4—41no—
226C3×C4C4×D414yes—
227Dic3—24no—
228Dic3—42no—
229Dic3—42no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—34no—
237C23—62no—
238C23—62no—
239C23—62no—
240C23—62no—
241C23—62no—
242C23—62no—
243C23C2×C4×D312yes—
244C23C23×D311yescharacteristic
245D4—68no—
246D4—128no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—28no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—34no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—128no—
270C2×C4C2×C4×D312yes—
271C2×C4C2×C4×D312yes—
272C2×C4#23512yes—
273C2×C4C2×#6712yes—
274C2×C4C2×#6712yes—
275C2×C4C2×#6712yes—
276C2×C3—42no—
277C2×C3C2×C4×D411yescharacteristic
278C2×C3#51811yescharacteristic
279C2×C3#52011yescharacteristic
280C2×C3#53511yescharacteristic
281C2×C3#53711yescharacteristic
282C2×C3#54011yescharacteristic
283C2×C3#54211yescharacteristic
284D3—44no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—64no—
293C22—64no—
294C22—64no—
295C22—122no—
296C22—122no—
297C22—122no—
298C22—122no—
299C22C22×C4×D311yescharacteristic
300C22C2×#23511yescharacteristic
301C22#100911yescharacteristic
302C22#100911yescharacteristic
303C22#100911yescharacteristic
304C22#101011yescharacteristic
305C22C22×#6711yescharacteristic
306C4—22no—
307C4—22no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—64no—
312C4—122no—
313C4—122no—
314C4C4×#6714yes—
315C3#232411yescharacteristic, complemented, Sylow
316C2—42no—
317C2—124no—
318C2#552111yescharacteristic
319C2#552311yescharacteristic
320C2#553411yescharacteristic
321C2#553511yescharacteristic
322C2C2×C4×#6711yescharacteristic
323C2#583011yescharacteristic
324C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC2×C3×#117C2
socle, cosocleC23×C3C24
prime radicalC2×#930C2
hypercenterC2×#117D3
hypocenterC3#2324
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C3×#117C2semidirect
3. C2×C4×D12C2semidirect
4. #5212C2semidirect
5. #4990C2semidirect
6. #4989C2semidirect
7. #4957C2semidirect
8. #4956C2semidirect
9. #4939C2semidirect
10. C4×C4⋊Dic3C2semidirect
11. #4928C2semidirect
25. #930C4semidirect
41. C2×#117D3semidirect
54. C22×C3×C4C2×C4semidirect
63. C2×#67C2×C4semidirect
78. C3×C42D4semidirect
83. (C2×D3)⋊C4C2×C4semidirect
86. C4⋊Dic3C2×C4semidirect
177. C22×C4C4×D3semidirect
199. C42#67semidirect
315. C3#2324semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1172
C2×#9302
C2×C4×D122
#52122
#49902
#49892
#49572
#49562
#49392
C4×C4⋊Dic32
#49282
#23243