Finite Group Databasebeta

← #82399

Subgroups — #82399

Order 384.

788 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×#938C211yesmaximal, characteristic
3C2×C4×#67C212yesmaximal, complemented
4C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
5#5221C211yesmaximal, characteristic, complemented
6#5210C212yesmaximal, complemented
7Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
8#4956C212yesmaximal, complemented
9#4945C212yesmaximal, complemented
10C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
11#4928C211yesmaximal, characteristic, complemented
12#2361—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2212yes—
15C22×#67C2212yes—
16C22×C4×D3C2212yes—
17C22×C4×D3C2211yescharacteristic
18C3×C4×D4—24no—
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2212yes—
21C2×C3×C42C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25#938C418yescomplemented
26C4×#67—28no—
27C2×(C2×D3)⋊C4C2211yescharacteristic
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×Dic3⋊C4C2212yes—
33C2×C4×Dic3C2211yescharacteristic
34C2×C4×Dic3C2212yes—
35D3×C4⋊C4—24no—
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2211yescharacteristic
39C3⋊#95C2212yes—
40C2×#121—31no—
41C2×C4×D4—32no—
42C2×C4×D4D311yescharacteristic, complemented
43C22×C4⋊C4—31no—
44#390—31no—
45#389—31no—
46#388—32no—
47#386—32no—
48#385—32no—
49C4×C4⋊C4—31no—
50C4×C22⋊C4—31no—
51C24×C3C2312yes—
52C23×D3C2311yescharacteristic
53C2×C3×D4—22no—
54C2×C3×D4—22no—
55C2×C3×D4C2×C414yescomplemented
56C22×C3×C4—24no—
57C22×C3×C4—24no—
58C22×C3×C4—42no—
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C2×#67—24no—
64C2×#67—24no—
65C2×#67C2×C418yescomplemented
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2312yes—
68C22×Dic3C2312yes—
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2311yescharacteristic
71C2×C4×D3—22no—
72C2×C4×D3—22no—
73C2×C4×D3—24no—
74C2×C4×D3—24no—
75C2×C4×D3—42no—
76C2×C4×D3C2×C414yescomplemented
77C3×C4⋊C4D414yescomplemented
78C3×C22⋊C4—24no—
79C3×C42—22no—
80#217—24no—
81#217—42no—
82#217C2×C418yescomplemented
83(C2×D3)⋊C4—22no—
84(C2×D3)⋊C4D414yescomplemented
85C4⋊Dic3—22no—
86C4⋊Dic3C2×C414yescomplemented
87Dic3⋊C4—24no—
88Dic3⋊C4—24no—
89C4×Dic3—22no—
90C4×Dic3—24no—
91C22×D4—32no—
92C22×D4C2×D311yescharacteristic
93C23×C4—31no—
94C23×C4—32no—
95C23×C4C2×D312yes—
96#121—38no—
97C4×D4—24no—
98C4×D4—68no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4—32no—
103C2×C4⋊C4—64no—
104C2×C4⋊C4C2×D311yescharacteristic
105C2×C22⋊C4—32no—
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—31no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4C2×D312yes—
111C2×C42—32no—
112C2×C42—31no—
113C2×C42C2×D311yescharacteristic
114#95—31no—
115#95—31no—
116#95—32no—
117#95—32no—
118C23×C3—22no—
119C23×C3—22no—
120C23×C3—22no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3—22no—
124C23×C3C22×C414yes—
125C23×C3C2411yescharacteristic
126C22×D3—21no—
127C22×D3—21no—
128C22×D3—21no—
129C22×D3—21no—
130C22×D3C22×C412yes—
131C22×D3C2×D412yes—
132C22×D3#4112yes—
133C3×D4—28no—
134C2×C3×C4—22no—
135C2×C3×C4—21no—
136C2×C3×C4—22no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—48no—
142C2×C3×C4C22×C412yes—
143C2×C3×C4C2×D412yes—
144C2×C3×C4C2×D414yes—
145C2×C3×C4C2×D412yes—
146C2×C3×C4#4112yes—
147#67—216no—
148C2×Dic3—22no—
149C2×Dic3—21no—
150C2×Dic3—21no—
151C2×Dic3—21no—
152C2×Dic3—22no—
153C2×Dic3—21no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—21no—
157C2×Dic3—21no—
158C2×Dic3—21no—
159C2×Dic3—22no—
160C2×Dic3C22×C414yes—
161C2×Dic3C22×C412yes—
162C2×Dic3C2×D412yes—
163C4×D3—28no—
164C4×D3—48no—
165C24—31no—
166C24C22×D312yes—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—38no—
170C2×D4—64no—
171C2×D4—64no—
172C2×D4C4×D314yescomplemented
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—32no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—34no—
179C22×C4—31no—
180C22×C4—32no—
181C22×C4—42no—
182C22×C4—64no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4—64no—
186C22×C4—122no—
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D312yes—
190C22×C4C22×D311yescharacteristic
191C4⋊C4—34no—
192C4⋊C4—64no—
193C4⋊C4—64no—
194C4⋊C4—62no—
195C4⋊C4#6714yescomplemented
196C22⋊C4—24no—
197C22⋊C4—38no—
198C22⋊C4—34no—
199C22⋊C4—62no—
200C22⋊C4—64no—
201C22⋊C4—122no—
202C42—22no—
203C42—64no—
204C42—62no—
205C22×C3—24no—
206C22×C3—24no—
207C22×C3—24no—
208C22×C3—42no—
209C22×C3—42no—
210C22×C3—42no—
211C22×C3—42no—
212C22×C3C23×C411yescharacteristic
213C22×C3C22×D411yescharacteristic
214C22×C3C22×D411yescharacteristic
215C22×C3C2×#4111yescharacteristic
216C22×C3C2×#4111yescharacteristic
217C22×C3H(5,𝔽2)11yescharacteristic
218C22×C32−1+411yescharacteristic
219C2×D3—22no—
220C2×D3—22no—
221C2×D3—22no—
222C2×D3—22no—
223C2×D3—22no—
224C2×D3—22no—
225C2×D3C4×D414yescomplemented
226C3×C4—24no—
227C3×C4—22no—
228C3×C4—41no—
229C3×C4C4×D414yescomplemented
230Dic3—22no—
231Dic3—24no—
232Dic3—22no—
233Dic3—42no—
234Dic3—41no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—32no—
242C23—32no—
243C23—32no—
244C23—61no—
245C23—61no—
246C23—61no—
247C23—61no—
248C23C2×C4×D314yes—
249C23C23×D311yescharacteristic
250D4—28no—
251D4—616no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—32no—
260C2×C4—34no—
261C2×C4—32no—
262C2×C4—48no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—61no—
268C2×C4—61no—
269C2×C4—68no—
270C2×C4—61no—
271C2×C4—62no—
272C2×C4—61no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—62no—
276C2×C4—128no—
277C2×C4C2×C4×D312yes—
278C2×C4#23512yes—
279C2×C4D3×D412yes—
280C2×C4C2×#6714yes—
281C2×C4C2×#6712yes—
282C2×C3—44no—
283C2×C3C2×C4×D411yescharacteristic
284C2×C3C2×C4×D411yescharacteristic
285C2×C3#52011yescharacteristic
286C2×C3D4211yescharacteristic
287C2×C3#54711yescharacteristic
288C2×C3#54811yescharacteristic
289C2×C3#55311yescharacteristic
290D3—24no—
291C22—24no—
292C22—24no—
293C22—24no—
294C22—34no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—62no—
300C22—62no—
301C22—62no—
302C22—62no—
303C22—62no—
304C22—62no—
305C22C22×C4×D311yescharacteristic
306C22C2×#23511yescharacteristic
307C22C2×D3×D411yescharacteristic
308C22#100411yescharacteristic
309C22D3×#4111yescharacteristic
310C22#101011yescharacteristic
311C22C22×#6711yescharacteristic
312C4—22no—
313C4—24no—
314C4—41no—
315C4—62no—
316C4—64no—
317C4—62no—
318C4—122no—
319C4—121no—
320C4C4×#6714yescomplemented
321C3#236111yescharacteristic, complemented, Sylow
322C2—44no—
323C2—64no—
324C2C4×D3×D411yescharacteristic
325C2#554611yescharacteristic
326C2#555111yescharacteristic
327C2#555811yescharacteristic
328C2C2×C4×#6711yescharacteristic
329C2D4×#6711yescharacteristic
330C2#583111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC2×C3×C4×D4C2
socle, cosocleC23×C3C24
prime radicalC2×#938C2
hypercenterC2×C4×D4D3
hypocenterC3#2361
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
3. C2×C4×#67C2semidirect
4. C2×D3×C4⋊C4C2semidirect
5. #5221C2semidirect
6. #5210C2semidirect
7. Dic3×C4⋊C4C2semidirect
8. #4956C2semidirect
9. #4945C2semidirect
10. C4×(C2×D3)⋊C4C2semidirect
11. #4928C2semidirect
25. #938C4semidirect
42. C2×C4×D4D3semidirect
55. C2×C3×D4C2×C4semidirect
65. C2×#67C2×C4semidirect
76. C2×C4×D3C2×C4semidirect
77. C3×C4⋊C4D4semidirect
82. #217C2×C4semidirect
84. (C2×D3)⋊C4D4semidirect
86. C4⋊Dic3C2×C4semidirect
172. C2×D4C4×D3semidirect
195. C4⋊C4#67semidirect
225. C2×D3C4×D4semidirect
229. C3×C4C4×D4semidirect
320. C4C4×#67semidirect
321. C3#2361semidirect

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×#9382
C2×C4×#672
C2×D3×C4⋊C42
#52212
#52102
Dic3×C4⋊C42
#49562
#49452
C4×(C2×D3)⋊C42
#49282
#23613