Finite Group Databasebeta

← #73467

Subgroups — #73467

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×#930C211yesmaximal, characteristic
2C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C2×#67⋊C4C212yesmaximal, complemented
4C2×C4×D12C211yesmaximal, characteristic, complemented
5C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
6#4958C212yesmaximal, complemented
7#4939C211yesmaximal, characteristic, complemented
8C4×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#4928C211yesmaximal, characteristic, complemented
10#4670C212yesmaximal, complemented
11#4668C212yesmaximal, complemented
12#2361—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2212yes—
15C23×Dic3C2212yes—
16C22×D12C2211yescharacteristic
17C22×C4×D3C2212yes—
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C42C2211yescharacteristic
21C2×C3×C42C2211yescharacteristic
22#930C418yescomplemented
23C2×(C2×D3)⋊C4C2212yes—
24C2×(C2×D3)⋊C4C2212yes—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2212yes—
27C2×C4⋊Dic3—24no—
28C2×C4⋊Dic3C2212yes—
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2211yescharacteristic
31C2×C4⋊Dic3C2211yescharacteristic
32C2×Dic3⋊C4C2212yes—
33C2×C4×Dic3C2212yes—
34#67⋊C4—28no—
35C4×D12—24no—
36C3×#95C2211yescharacteristic
37C3×#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C3⋊#95C2212yes—
40C2×#121—31no—
41C2×C4×D4—32no—
42C2×C4×D4—31no—
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⋊C4D311yescharacteristic, complemented
51C24×C3C2311yescharacteristic
52C23×D3C2312yes—
53C22×C3×C4—22no—
54C22×C3×C4—22no—
55C22×C3×C4C2×C414yescomplemented
56C22×C3×C4C2311yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C2×#67—24no—
63C2×#67—24no—
64C2×#67C2×C418yescomplemented
65C22×Dic3—24no—
66C22×Dic3—24no—
67C22×Dic3—42no—
68C22×Dic3C2312yes—
69C22×Dic3C2312yes—
70C22×Dic3C2312yes—
71C2×D12—22no—
72C2×D12—22no—
73C2×D12C2×C414yescomplemented
74C2×C4×D3—24no—
75C2×C4×D3—24no—
76C2×C4×D3—42no—
77C3×C22⋊C4—22no—
78C3×C22⋊C4D414yescomplemented
79C3×C42—22no—
80C3×C42—22no—
81(C2×D3)⋊C4—24no—
82(C2×D3)⋊C4—24no—
83(C2×D3)⋊C4—42no—
84(C2×D3)⋊C4C2×C418yescomplemented
85C4⋊Dic3—22no—
86C4⋊Dic3—24no—
87C4⋊Dic3C2×C414yescomplemented
88C4⋊Dic3D414yescomplemented
89Dic3⋊C4—24no—
90C4×Dic3—24no—
91C22×D4—32no—
92C22×D4—31no—
93C23×C4—32no—
94C23×C4—32no—
95C23×C4C2×D311yescharacteristic
96#121—38no—
97C4×D4—64no—
98C4×D4—68no—
99C2×C4⋊C4—31no—
100C2×C4⋊C4—31no—
101C2×C4⋊C4—32no—
102C2×C4⋊C4—31no—
103C2×C4⋊C4—32no—
104C2×C4⋊C4—64no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4C2×D311yescharacteristic
110C2×C22⋊C4C2×D311yescharacteristic
111C2×C42—32no—
112C2×C42C2×D311yescharacteristic
113C2×C42C2×D311yescharacteristic
114#95—32no—
115#95—32no—
116#95C2×D311yescharacteristic
117#95C2×D311yescharacteristic
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3—21no—
122C23×C3C22×C412yes—
123C23×C3C2×D412yes—
124C23×C3#4112yes—
125C23×C3C2411yescharacteristic
126C22×D3—22no—
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—22no—
132C22×D3C22×C414yes—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—21no—
139C2×C3×C4—28no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4C22×C412yes—
147C2×C3×C4C22×C412yes—
148C2×C3×C4C2×D412yes—
149C2×C3×C4C2×D412yes—
150C2×C3×C4C2×D412yes—
151C2×C3×C4#4112yes—
152#67—216no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—22no—
159C2×Dic3—22no—
160C2×Dic3—48no—
161C2×Dic3C22×C414yes—
162C2×Dic3C2×D414yes—
163D12—28no—
164C4×D3—48no—
165C24—32no—
166C24C22×D311yescharacteristic
167C2×D4—38no—
168C2×D4—34no—
169C2×D4—64no—
170C2×D4—64no—
171C2×D4—62no—
172C2×D4—62no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—32no—
176C22×C4—32no—
177C22×C4—32no—
178C22×C4—64no—
179C22×C4—64no—
180C22×C4—64no—
181C22×C4—64no—
182C22×C4—122no—
183C22×C4—122no—
184C22×C4C4×D314yescomplemented
185C22×C4C22×D311yescharacteristic
186C22×C4C22×D311yescharacteristic
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D311yescharacteristic
190C22×C4C22×D311yescharacteristic
191C4⋊C4—34no—
192C4⋊C4—34no—
193C4⋊C4—64no—
194C4⋊C4—64no—
195C4⋊C4—62no—
196C22⋊C4—22no—
197C22⋊C4—38no—
198C22⋊C4—64no—
199C22⋊C4—64no—
200C22⋊C4—122no—
201C22⋊C4D1214yescomplemented
202C42—22no—
203C42—22no—
204C42—64no—
205C22×C3—22no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3C4×D414yescomplemented
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—24no—
220C2×D3—24no—
221C2×D3—24no—
222C2×D3—42no—
223C2×D3—42no—
224C2×D3—42no—
225C2×D3—42no—
226C3×C4—22no—
227C3×C4—22no—
228C3×C4—22no—
229C3×C4—41no—
230C3×C4—41no—
231C3×C4C4×D414yescomplemented
232Dic3—24no—
233Dic3—24no—
234Dic3—42no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—34no—
240C23—62no—
241C23—62no—
242C23—62no—
243C23—62no—
244C23—62no—
245C23—62no—
246C23C2×C4×D312yes—
247C23C2×D1212yes—
248C23#23512yes—
249C23C23×D311yescharacteristic
250D4—616no—
251D4—68no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—28no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—34no—
266C2×C4—34no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—62no—
273C2×C4—62no—
274C2×C4—128no—
275C2×C4—128no—
276C2×C4C2×C4×D312yes—
277C2×C4C2×C4×D312yes—
278C2×C4C2×D1212yes—
279C2×C4C2×D1212yes—
280C2×C4D3×D412yes—
281C2×C4#23712yes—
282C2×C3—24no—
283C2×C3C2×C4×D411yescharacteristic
284C2×C3C2×C4×D411yescharacteristic
285C2×C3#52011yescharacteristic
286C2×C3D4211yescharacteristic
287C2×C3#54711yescharacteristic
288C2×C3#54811yescharacteristic
289C2×C3#55311yescharacteristic
290D3—44no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—64no—
298C22—64no—
299C22—64no—
300C22—122no—
301C22—122no—
302C22—122no—
303C22—122no—
304C22C4×D1214yescomplemented
305C22C22×C4×D311yescharacteristic
306C22C22×D1211yescharacteristic
307C22C2×#23511yescharacteristic
308C22C2×D3×D411yescharacteristic
309C22C2×#23711yescharacteristic
310C22#100911yescharacteristic
311C22#101011yescharacteristic
312C4—22no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—41no—
317C4—64no—
318C4—64no—
319C4—122no—
320C4#67⋊C414yescomplemented
321C3#236111yescharacteristic, complemented, Sylow
322C2—24no—
323C2—124no—
324C2C2×C4×D1211yescharacteristic
325C2C2×#67⋊C411yescharacteristic
326C2#552311yescharacteristic
327C2D4×D1211yescharacteristic
328C2#555211yescharacteristic
329C2#555511yescharacteristic
330C2#556511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×C4×C22⋊C4C2
socle, cosocleC23×C3C24
prime radicalC2×#930C2
hypercenterC4×C22⋊C4D3
hypocenterC3#2361
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊Dic3C2semidirect
3. C2×#67⋊C4C2semidirect
4. C2×C4×D12C2semidirect
5. C3×C4×C22⋊C4C2semidirect
6. #4958C2semidirect
7. #4939C2semidirect
8. C4×C4⋊Dic3C2semidirect
9. #4928C2semidirect
10. #4670C2semidirect
11. #4668C2semidirect
22. #930C4semidirect
50. C4×C22⋊C4D3semidirect
55. C22×C3×C4C2×C4semidirect
64. C2×#67C2×C4semidirect
73. C2×D12C2×C4semidirect
78. C3×C22⋊C4D4semidirect
84. (C2×D3)⋊C4C2×C4semidirect
87. C4⋊Dic3C2×C4semidirect
88. C4⋊Dic3D4semidirect
184. C22×C4C4×D3semidirect
201. C22⋊C4D12semidirect
211. C22×C3C4×D4semidirect
231. C3×C4C4×D4semidirect
304. C22C4×D12semidirect
320. C4#67⋊C4semidirect
321. C3#2361semidirect

Maximal subgroups

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

subgroupindex
C2×#9302
C22×C4⋊Dic32
C2×#67⋊C42
C2×C4×D122
C3×C4×C22⋊C42
#49582
#49392
C4×C4⋊Dic32
#49282
#46702
#46682
#23613