Finite Group Databasebeta

← #82275

Subgroups — #82275

Order 384.

786 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#117C211yesmaximal, characteristic, complemented
2C2×#929C211yesmaximal, characteristic, complemented
3C22×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
4C2×#882C211yesmaximal, characteristic
5#4994C212yesmaximal, complemented
6#4991C212yesmaximal, complemented
7#4958C211yesmaximal, characteristic, complemented
8#4955C211yesmaximal, characteristic, complemented
9#4950C211yesmaximal, characteristic, complemented
10#4940C212yesmaximal, complemented
11#4939C212yesmaximal, complemented
12#2411—31nomaximal, Sylow
13C24×D3C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C22×#67C2211yescharacteristic
16C23×Dic3C2211yescharacteristic
17C22×D12C2212yes—
18C3×#117—24no—
19C2×C3×C4⋊C4C2212yescomplemented
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C22⋊C4C2211yescharacteristic
22C2×C3×C42C2212yescomplemented
23C2×#217C2211yescharacteristic
24#929—24no—
25C2×(C2×D3)⋊C4—24no—
26C2×(C2×D3)⋊C4C2212yes—
27C2×(C2×D3)⋊C4C2212yes—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2212yes—
31C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
32C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
33C2×(C2×D3)⋊C4C2212yescomplemented
34C2×C4⋊Dic3C2212yescomplemented
35C2×Dic3⋊C4C2211yescharacteristic, complemented
36C2×Dic3⋊C4C2211yescharacteristic, complemented
37#882—24no—
38C3⋊#95C2212yescomplemented
39C3⋊#95C2212yescomplemented
40C2×#123—31no—
41C2×#120—31no—
42C2×#117D311yescharacteristic, complemented
43C22×C22⋊C4—31no—
44#396—32no—
45#394—32no—
46#390—32no—
47#388—32no—
48#386—31no—
49#385—31no—
50#379—31no—
51C24×C3C2311yescharacteristic
52C23×D3—22no—
53C23×D3—24no—
54C23×D3—22no—
55C23×D3—22no—
56C23×D3—42no—
57C23×D3C2311yescharacteristic
58C23×D3C2312yes—
59C22×C3×C4—22no—
60C22×C3×C4—22no—
61C22×C3×C4D414yescomplemented
62C22×C3×C4C2312yes—
63C22×C3×C4C2312yes—
64C22×C3×C4C2311yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C2×#67—22no—
67C2×#67—22no—
68C2×#67—22no—
69C22×Dic3—22no—
70C22×Dic3—42no—
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2311yescharacteristic
73C22×Dic3C2312yes—
74C22×Dic3C2311yescharacteristic
75C2×D12—24no—
76C2×D12—24no—
77C2×D12—42no—
78C3×C4⋊C4—24no—
79C3×C22⋊C4—22no—
80C3×C22⋊C4D414yes—
81C3×C42—24no—
82#217—22no—
83(C2×D3)⋊C4—24no—
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—24no—
86(C2×D3)⋊C4—24no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—22no—
89(C2×D3)⋊C4—42no—
90(C2×D3)⋊C4—42no—
91C4⋊Dic3—42no—
92Dic3⋊C4—22no—
93Dic3⋊C4—22no—
94C25—31no—
95C22×D4—31no—
96C22×D4—32no—
97C23×C4—31no—
98C23×C4C2×D311yescharacteristic
99#123—64no—
100#120—64no—
101#117—24no—
102C2×C4⋊C4—31no—
103C2×C4⋊C4—31no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4C2×D312yescomplemented
106C2×C22⋊C4—31no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4—32no—
109C2×C22⋊C4—32no—
110C2×C22⋊C4—31no—
111C2×C22⋊C4—31no—
112C2×C22⋊C4—32no—
113C2×C22⋊C4—32no—
114C2×C22⋊C4—32no—
115C2×C22⋊C4—64no—
116C2×C22⋊C4C2×D311yescharacteristic
117C2×C22⋊C4C2×D311yescharacteristic
118C2×C42C2×D312yescomplemented
119#95—32no—
120#95—32no—
121C23×C3—22no—
122C23×C3—22no—
123C23×C3C2×D412yes—
124C23×C3C2×D412yes—
125C23×C3#4112yes—
126C23×C3C2411yescharacteristic
127C22×D3—28no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—22no—
133C22×D3—22no—
134C22×D3—21no—
135C22×D3—21no—
136C22×D3—22no—
137C22×D3—21no—
138C22×D3—22no—
139C22×D3—48no—
140C22×D3—48no—
141C22×D3—44no—
142C22×D3—44no—
143C22×D3#4114yes—
144C2×C3×C4—28no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—22no—
148C2×C3×C4—21no—
149C2×C3×C4—21no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4C2×D412yes—
155C2×C3×C4C2×D412yes—
156C2×C3×C4C2×D414yes—
157#67—44no—
158C2×Dic3—21no—
159C2×Dic3—21no—
160C2×Dic3—21no—
161C2×Dic3—22no—
162C2×Dic3—21no—
163C2×Dic3—22no—
164C2×Dic3—21no—
165C2×Dic3—22no—
166C2×Dic3—22no—
167C2×Dic3—22no—
168C2×Dic3—44no—
169D12—48no—
170C24—32no—
171C24—31no—
172C24—64no—
173C24—62no—
174C24—62no—
175C24—62no—
176C24—122no—
177C24C22×D311yescharacteristic
178C2×D4—64no—
179C2×D4—62no—
180C2×D4—62no—
181C2×D4—62no—
182C2×D4—64no—
183C2×D4—122no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—31no—
187C22×C4—32no—
188C22×C4—31no—
189C22×C4—31no—
190C22×C4—62no—
191C22×C4—122no—
192C22×C4D1214yescomplemented
193C22×C4C22×D312yes—
194C22×C4C22×D311yescharacteristic
195C22×C4C22×D311yescharacteristic
196C22×C4C22×D312yes—
197C4⋊C4—24no—
198C4⋊C4—62no—
199C4⋊C4—62no—
200C4⋊C4—122no—
201C22⋊C4—22no—
202C22⋊C4—64no—
203C22⋊C4—62no—
204C22⋊C4—64no—
205C22⋊C4—64no—
206C22⋊C4—62no—
207C22⋊C4—62no—
208C22⋊C4—64no—
209C22⋊C4—122no—
210C22⋊C4—122no—
211C22⋊C4#6714yes—
212C42—24no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—24no—
216C22×C3—24no—
217C22×C3#12114yes—
218C22×C3C22×D411yescharacteristic
219C22×C3C22×D411yescharacteristic
220C22×C3C2×#4111yescharacteristic
221C22×C3C2×#4112yes—
222C22×C3H(5,𝔽2)12yes—
223C2×D3—24no—
224C2×D3—24no—
225C2×D3—24no—
226C2×D3—416no—
227C2×D3—42no—
228C2×D3—42no—
229C2×D3—42no—
230C2×D3—42no—
231C2×D3—41no—
232C2×D3—41no—
233C2×D3—41no—
234C2×D3—42no—
235C2×D3—42no—
236C3×C4—22no—
237C3×C4—22no—
238C3×C4—24no—
239C3×C4—42no—
240Dic3—41no—
241Dic3—41no—
242Dic3—41no—
243Dic3—42no—
244C23—22no—
245C23—22no—
246C23—34no—
247C23—68no—
248C23—62no—
249C23—62no—
250C23—62no—
251C23—62no—
252C23—62no—
253C23—62no—
254C23—61no—
255C23—61no—
256C23—62no—
257C23—62no—
258C23—61no—
259C23—128no—
260C23—128no—
261C23—124no—
262C23—124no—
263C23C2×D1212yes—
264C23#23512yes—
265C23C2×#6712yes—
266C23C23×D311yescharacteristic
267D4—128no—
268D4—124no—
269C2×C4—28no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—61no—
280C2×C4—61no—
281C2×C4—62no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—62no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4—62no—
288C2×C4—61no—
289C2×C4—124no—
290C2×C4C2×D1212yes—
291C2×C4C2×D1212yes—
292C2×C4C2×#6714yes—
293C2×C3—24no—
294C2×C3C2×#12111yescharacteristic
295C2×C3#52511yescharacteristic
296C2×C3#53511yescharacteristic
297C2×C3#54612yes—
298C2×C3#55112yes—
299D3—44no—
300D3—81no—
301C22—22no—
302C22—22no—
303C22—24no—
304C22—24no—
305C22—64no—
306C22—64no—
307C22—64no—
308C22—1216no—
309C22—122no—
310C22—122no—
311C22—122no—
312C22—122no—
313C22—121no—
314C22—121no—
315C22—122no—
316C22—121no—
317C22—122no—
318C22#93014yes—
319C22C22×D1211yescharacteristic
320C22C2×#23511yescharacteristic
321C22D3×#4112yes—
322C22#100912yes—
323C22C22×#6711yescharacteristic
324C4—22no—
325C4—22no—
326C4—24no—
327C4—42no—
328C4—121no—
329C4—122no—
330C4—121no—
331C4—121no—
332C3#241111yescharacteristic, complemented, Sylow
333C2—24no—
334C2—124no—
335C2—241no—
336C2#552411yescharacteristic
337C2#552511yescharacteristic
338C2#552712yes—
339C2C2×#93011yescharacteristic
340C2#582912yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#117C2
prime radicalC2×#882C2
hypercenterC2×#117D3
hypocenterC3#2411
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C3×#117C2semidirect
2. C2×#929C2semidirect
3. C22×(C2×D3)⋊C4C2semidirect
5. #4994C2semidirect
6. #4991C2semidirect
7. #4958C2semidirect
8. #4955C2semidirect
9. #4950C2semidirect
10. #4940C2semidirect
11. #4939C2semidirect
19. C2×C3×C4⋊C4C22semidirect
22. C2×C3×C42C22semidirect
31. C2×(C2×D3)⋊C4C22semidirect
32. C2×(C2×D3)⋊C4C22semidirect
33. C2×(C2×D3)⋊C4C22semidirect
34. C2×C4⋊Dic3C22semidirect
35. C2×Dic3⋊C4C22semidirect
36. C2×Dic3⋊C4C22semidirect
38. C3⋊#95C22semidirect
39. C3⋊#95C22semidirect
42. C2×#117D3semidirect
61. C22×C3×C4D4semidirect
105. C2×C4⋊C4C2×D3semidirect
118. C2×C42C2×D3semidirect
192. C22×C4D12semidirect
332. C3#2411semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1172
C2×#9292
C22×(C2×D3)⋊C42
C2×#8822
#49942
#49912
#49582
#49552
#49502
#49402
#49392
#24113