Finite Group Databasebeta

← #82335

Subgroups — #82335

Order 384.

660 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic
2C2×#926C211yesmaximal, characteristic, complemented
3C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#5220C212yesmaximal
5#5212C211yesmaximal, characteristic, complemented
6#5212C211yesmaximal, characteristic, complemented
7#4980C212yesmaximal, complemented
8#4951C212yesmaximal, complemented
9Dic3×C22⋊C4C212yesmaximal, complemented
10#4936C211yesmaximal, characteristic, complemented
11C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
12#2352—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C23×Dic3C2212yes—
17C3×C4×D4C418yescomplemented
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C42C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25#926—24no—
26C2×C4⋊Dic3—24no—
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2212yes—
31C2×C4⋊Dic3C2211yescharacteristic
32C2×C4×Dic3C2212yes—
33C2×C4×Dic3C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×C4×D4D311yescharacteristic
40C2×#117—31no—
41C22×C4⋊C4—31no—
42#388—32no—
43#387—32no—
44#386—32no—
45#382—31no—
46#380—31no—
47#380—31no—
48C4×C4⋊C4—31no—
49C4×C22⋊C4—32no—
50C24×C3C2311yescharacteristic
51C24×C3C2311yescharacteristic
52C2×C3×D4—22no—
53C2×C3×D4—22no—
54C2×C3×D4C2×C414yes—
55C22×C3×C4—22no—
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4C2×C414yes—
60C22×C3×C4C2×C414yescomplemented
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2311yescharacteristic
64C22×C3×C4C2311yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C22×Dic3—24no—
67C22×Dic3—24no—
68C22×Dic3—42no—
69C22×Dic3C2312yes—
70C22×Dic3C2312yes—
71C22×Dic3C2312yes—
72C22×Dic3C2312yes—
73C3×C4⋊C4C2×C414yescomplemented
74C3×C22⋊C4C2×C414yes—
75C3×C22⋊C4C2×C414yescomplemented
76C3×C42C2×C414yescomplemented
77#217—24no—
78#217—24no—
79#217—42no—
80C4⋊Dic3—22no—
81C4⋊Dic3—22no—
82C4⋊Dic3—22no—
83C4⋊Dic3—24no—
84C4⋊Dic3D414yescomplemented
85C4×Dic3—24no—
86C4×Dic3—42no—
87C22×D4C2×D311yescharacteristic
88C23×C4—32no—
89C23×C4C2×D311yescharacteristic
90C23×C4C2×D311yescharacteristic
91C4×D4Dic318yescomplemented
92#117—64no—
93C2×C4⋊C4—31no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—32no—
96C2×C4⋊C4—31no—
97C2×C4⋊C4—31no—
98C2×C4⋊C4—64no—
99C2×C4⋊C4C2×D311yescharacteristic
100C2×C22⋊C4—32no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4C2×D311yescharacteristic
104C2×C22⋊C4C2×D311yescharacteristic
105C2×C42—32no—
106C2×C42—32no—
107C2×C42C2×D311yescharacteristic
108#95—32no—
109#95—32no—
110#95—32no—
111#95—32no—
112#95—32no—
113C23×C3—21no—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3—21no—
123C23×C3C22×C412yes—
124C23×C3C22×C412yes—
125C23×C3#4112yes—
126C23×C3#4112yes—
127C23×C3C2411yescharacteristic
128C3×D4—28no—
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—28no—
138C2×C3×C4—28no—
139C2×C3×C4C22×C412yes—
140C2×C3×C4C22×C412yes—
141C2×C3×C4C22×C412yes—
142C2×C3×C4C22×C412yes—
143C2×C3×C4C22×C412yes—
144C2×C3×C4C2×D412yes—
145C2×C3×C4#4112yes—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—48no—
158C2×Dic3C2×D414yes—
159C24C22×D311yescharacteristic
160C24C22×D311yescharacteristic
161C2×D4—22no—
162C2×D4—22no—
163C2×D4C2×Dic314yes—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—32no—
169C22×C4—32no—
170C22×C4—32no—
171C22×C4—32no—
172C22×C4—64no—
173C22×C4—64no—
174C22×C4—122no—
175C22×C4C2×Dic314yes—
176C22×C4C2×Dic314yescomplemented
177C22×C4C22×D311yescharacteristic
178C22×C4C22×D311yescharacteristic
179C22×C4C22×D311yescharacteristic
180C22×C4C22×D311yescharacteristic
181C22×C4C22×D311yescharacteristic
182C4⋊C4—34no—
183C4⋊C4—64no—
184C4⋊C4—62no—
185C4⋊C4—62no—
186C4⋊C4—62no—
187C4⋊C4C2×Dic314yescomplemented
188C22⋊C4—64no—
189C22⋊C4—64no—
190C22⋊C4—122no—
191C22⋊C4C2×Dic314yes—
192C22⋊C4C2×Dic314yescomplemented
193C42—64no—
194C42—122no—
195C42C2×Dic314yescomplemented
196C22×C3—22no—
197C22×C3—22no—
198C22×C3—22no—
199C22×C3—22no—
200C22×C3—22no—
201C22×C3—22no—
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3#11714yescomplemented
210C22×C3C23×C411yescharacteristic
211C22×C3C22×D411yescharacteristic
212C22×C3C2×#4111yescharacteristic
213C22×C3C2×#4111yescharacteristic
214C22×C3C2×#4111yescharacteristic
215C22×C3H(5,𝔽2)11yescharacteristic
216C22×C32−1+411yescharacteristic
217C3×C4—22no—
218C3×C4—22no—
219C3×C4—22no—
220C3×C4—22no—
221C3×C4C4×D414yes—
222Dic3—24no—
223Dic3—42no—
224Dic3—42no—
225Dic3—42no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23#23512yes—
237C23#23512yes—
238C23C22×Dic312yes—
239C23C22×Dic312yes—
240C23C23×D311yescharacteristic
241D4—28no—
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—28no—
251C2×C4—28no—
252C2×C4—34no—
253C2×C4—62no—
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—62no—
263C2×C4—62no—
264C2×C4—128no—
265C2×C4D3×D412yes—
266C2×C4#23712yes—
267C2×C4C22×Dic312yes—
268C2×C4C22×Dic312yes—
269C2×C4C22×Dic312yes—
270C2×C4C22×Dic312yes—
271C2×C4C22×Dic312yes—
272C2×C3—24no—
273C2×C3—42no—
274C2×C3C2×#11711yescharacteristic
275C2×C3C2×C4×D411yescharacteristic
276C2×C3#52011yescharacteristic
277C2×C3#54611yescharacteristic
278C2×C3#54811yescharacteristic
279C2×C3#55211yescharacteristic
280C2×C3#55311yescharacteristic
281C22—22no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22#92614yescomplemented
295C22C2×#23511yescharacteristic
296C22C2×#23511yescharacteristic
297C22C2×D3×D411yescharacteristic
298C22C2×#23711yescharacteristic
299C22#100911yescharacteristic
300C22#101011yescharacteristic
301C22C23×Dic311yescharacteristic
302C4—22no—
303C4—22no—
304C4—22no—
305C4—22no—
306C4—64no—
307C4—122no—
308C4—122no—
309C4—122no—
310C4D4×Dic314yes—
311C3#235211yescharacteristic, complemented, Sylow
312C2—24no—
313C2—42no—
314C2#554111yescharacteristic
315C2#555011yescharacteristic
316C2#555211yescharacteristic
317C2#556511yescharacteristic
318C2C2×#92611yescharacteristic
319C2C2×D4×Dic311yescharacteristic
320C2#582711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC2×C3×C4×D4C2
socle, cosocleC23×C3C24
prime radicalC22×C3×D4C22
hypercenterC2×C4×D4D3
hypocenterC3#2352
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#926C2semidirect
3. C22×C4⋊Dic3C2semidirect
5. #5212C2semidirect
6. #5212C2semidirect
7. #4980C2semidirect
8. #4951C2semidirect
9. Dic3×C22⋊C4C2semidirect
10. #4936C2semidirect
11. C4×C4⋊Dic3C2semidirect
17. C3×C4×D4C4semidirect
60. C22×C3×C4C2×C4semidirect
73. C3×C4⋊C4C2×C4semidirect
75. C3×C22⋊C4C2×C4semidirect
76. C3×C42C2×C4semidirect
84. C4⋊Dic3D4semidirect
91. C4×D4Dic3semidirect
176. C22×C4C2×Dic3semidirect
187. C4⋊C4C2×Dic3semidirect
192. C22⋊C4C2×Dic3semidirect
195. C42C2×Dic3semidirect
209. C22×C3#117semidirect
294. C22#926semidirect
311. C3#2352semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D42
C2×#9262
C22×C4⋊Dic32
#52202
#52122
#49802
#49512
Dic3×C22⋊C42
#49362
C4×C4⋊Dic32
#23523