Finite Group Databasebeta

← #82161

Subgroups — #82161

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
1C22×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C4×#67C211yesmaximal, characteristic
3C2×#892C211yesmaximal, characteristic, complemented
4#5213C212yesmaximal
5#5212C211yesmaximal, characteristic, complemented
6C4×#217C212yesmaximal, complemented
7#4990C212yesmaximal, complemented
8#4987C211yesmaximal, characteristic, complemented
9#4978C212yesmaximal, complemented
10#4977C211yesmaximal, characteristic, complemented
11Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
12#2352—31nomaximal, Sylow
13C23×C3×C4C2212yes—
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C22×C4×D3C2211yescharacteristic
17C2×C3×C4⋊C4—24no—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C4⋊C4C2212yes—
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×#217C2212yes—
22C2×#217C2211yescharacteristic
23C4×#67C418yescomplemented
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×Dic3⋊C4C2211yescharacteristic
30C2×C4×Dic3C2212yes—
31C2×C4×Dic3C2212yes—
32C2×C4×Dic3C2211yescharacteristic
33#892—24no—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C3⋊#95C2212yes—
39C2×C4×D4—31no—
40C2×#117—31no—
41C22×C4⋊C4D311yescharacteristic, complemented
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
51C23×D3C2311yescharacteristic
52C22×C3×C4—22no—
53C22×C3×C4—22no—
54C22×C3×C4—24no—
55C22×C3×C4—24no—
56C22×C3×C4—42no—
57C22×C3×C4C2×C414yes—
58C22×C3×C4C2312yes—
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C2×#67—22no—
63C2×#67—22no—
64C2×#67C2×C414yes—
65C22×Dic3C2312yes—
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C2×C4×D3—22no—
71C2×C4×D3—22no—
72C2×C4×D3C2×C414yescomplemented
73C3×C4⋊C4—24no—
74C3×C4⋊C4—22no—
75C3×C4⋊C4D414yescomplemented
76#217—24no—
77#217C2×C414yes—
78(C2×D3)⋊C4—24no—
79(C2×D3)⋊C4—42no—
80(C2×D3)⋊C4C2×C414yescomplemented
81C4⋊Dic3—22no—
82C4⋊Dic3—22no—
83Dic3⋊C4C2×C414yescomplemented
84C4×Dic3—24no—
85C4×Dic3—42no—
86C4×Dic3C2×C414yescomplemented
87C22×D4—31no—
88C23×C4—31no—
89C23×C4C2×D312yes—
90C23×C4C2×D311yescharacteristic
91C4×D4—38no—
92#117—64no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—31no—
95C2×C4⋊C4—31no—
96C2×C4⋊C4—31no—
97C2×C4⋊C4C2×D311yescharacteristic
98C2×C4⋊C4C2×D312yes—
99C2×C4⋊C4C2×D311yescharacteristic
100C2×C22⋊C4—31no—
101C2×C22⋊C4—31no—
102C2×C22⋊C4—32no—
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C42—31no—
106C2×C42—32no—
107C2×C42—32no—
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×C3C22×C412yes—
118C23×C3#4112yes—
119C23×C3#4112yes—
120C23×C3C2411yescharacteristic
121C22×D3—21no—
122C22×D3—21no—
123C22×D3—21no—
124C22×D3—21no—
125C22×D3—21no—
126C22×D3—21no—
127C22×D3C22×C412yes—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—28no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—48no—
137C2×C3×C4C22×C412yes—
138C2×C3×C4C22×C412yes—
139C2×C3×C4C2×D414yes—
140C2×C3×C4C2×D412yes—
141C2×C3×C4#4112yes—
142#67—28no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—21no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3C22×C412yes—
156C2×Dic3C22×C412yes—
157C2×Dic3C22×C412yes—
158C4×D3—28no—
159C24—31no—
160C24C22×D311yescharacteristic
161C2×D4—34no—
162C2×D4—62no—
163C2×D4—62no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—34no—
172C22×C4—32no—
173C22×C4—32no—
174C22×C4—42no—
175C22×C4—62no—
176C22×C4—62no—
177C22×C4C4×D314yes—
178C22×C4C22×D312yes—
179C22×C4C22×D311yescharacteristic
180C22×C4C22×D312yes—
181C22×C4C22×D311yescharacteristic
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—34no—
185C4⋊C4—62no—
186C4⋊C4—62no—
187C4⋊C4#6714yescomplemented
188C22⋊C4—34no—
189C22⋊C4—34no—
190C22⋊C4—64no—
191C22⋊C4—64no—
192C22⋊C4—122no—
193C42—34no—
194C42—64no—
195C42—122no—
196C22×C3—22no—
197C22×C3—22no—
198C22×C3—22no—
199C22×C3—22no—
200C22×C3—22no—
201C22×C3—22no—
202C22×C3#11714yescomplemented
203C22×C3C23×C411yescharacteristic
204C22×C3C22×D411yescharacteristic
205C22×C3C2×#4111yescharacteristic
206C22×C3C2×#4111yescharacteristic
207C22×C3C2×#4111yescharacteristic
208C22×C3H(5,𝔽2)11yescharacteristic
209C22×C32−1+411yescharacteristic
210C2×D3—22no—
211C2×D3—22no—
212C2×D3—22no—
213C2×D3—41no—
214C2×D3—41no—
215C2×D3—41no—
216C2×D3—41no—
217C3×C4—22no—
218C3×C4—24no—
219C3×C4—42no—
220C3×C4C4×D414yes—
221Dic3—22no—
222Dic3—22no—
223Dic3—22no—
224Dic3—42no—
225Dic3—42no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—32no—
231C23—61no—
232C23—61no—
233C23—61no—
234C23—61no—
235C23—61no—
236C23—61no—
237C23C2×C4×D312yes—
238C23#23712yes—
239C23#23912yes—
240C23C23×D311yescharacteristic
241D4—68no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—28no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—32no—
251C2×C4—32no—
252C2×C4—32no—
253C2×C4—48no—
254C2×C4—61no—
255C2×C4—61no—
256C2×C4—61no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—68no—
261C2×C4—62no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4C2×C4×D312yes—
268C2×C4C2×C4×D312yes—
269C2×C4#23512yes—
270C2×C4C2×#6714yes—
271C2×C4C2×#6712yes—
272C2×C3—24no—
273C2×C3C2×#11711yescharacteristic
274C2×C3C2×C4×D411yescharacteristic
275C2×C3#52011yescharacteristic
276C2×C3#54611yescharacteristic
277C2×C3#54811yescharacteristic
278C2×C3#55211yescharacteristic
279C2×C3#55311yescharacteristic
280D3—42no—
281C22—22no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—22no—
287C22—62no—
288C22—62no—
289C22—62no—
290C22—121no—
291C22—121no—
292C22—121no—
293C22—121no—
294C22#89214yescomplemented
295C22C22×C4×D311yescharacteristic
296C22C2×#23511yescharacteristic
297C22C2×#23711yescharacteristic
298C22#100411yescharacteristic
299C22C2×#23911yescharacteristic
300C22#100711yescharacteristic
301C22C22×#6711yescharacteristic
302C4—22no—
303C4—24no—
304C4—42no—
305C4—62no—
306C4—62no—
307C4—62no—
308C4—122no—
309C4—122no—
310C4C4×#6714yes—
311C3#235211yescharacteristic, complemented, Sylow
312C2—24no—
313C2—122no—
314C2C2×#89211yescharacteristic
315C2#550411yescharacteristic
316C2#551311yescharacteristic
317C2#551411yescharacteristic
318C2C2×C4×#6711yescharacteristic
319C2#580611yescharacteristic
320C2#581711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC22×C3×C4⋊C4C2
socle, cosocleC23×C3C24
prime radicalC22×#67C22
hypercenterC22×C4⋊C4D3
hypocenterC3#2352
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊C4C2semidirect
3. C2×#892C2semidirect
5. #5212C2semidirect
6. C4×#217C2semidirect
7. #4990C2semidirect
8. #4987C2semidirect
9. #4978C2semidirect
10. #4977C2semidirect
11. Dic3×C4⋊C4C2semidirect
23. C4×#67C4semidirect
41. C22×C4⋊C4D3semidirect
72. C2×C4×D3C2×C4semidirect
75. C3×C4⋊C4D4semidirect
80. (C2×D3)⋊C4C2×C4semidirect
83. Dic3⋊C4C2×C4semidirect
86. C4×Dic3C2×C4semidirect
187. C4⋊C4#67semidirect
202. C22×C3#117semidirect
294. C22#892semidirect
311. C3#2352semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊C42
C2×C4×#672
C2×#8922
#52132
#52122
C4×#2172
#49902
#49872
#49782
#49772
Dic3×C4⋊C42
#23523