Finite Group Databasebeta

← #82380

Subgroups — #82380

Order 384.

724 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×#939C212yesmaximal, complemented
3C2×#930C211yesmaximal, characteristic, complemented
4#5213C212yesmaximal, complemented
5C2×C3⋊#95C211yesmaximal, characteristic, complemented
6#4991C211yesmaximal, characteristic, complemented
7#4981C211yesmaximal, characteristic, complemented
8#4960C211yesmaximal, characteristic, complemented
9#4958C212yesmaximal, complemented
10#4953C211yesmaximal, characteristic, complemented
11#4940C212yesmaximal, complemented
12#2543—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×#67C2212yes—
17C22×#67C2212yes—
18C23×Dic3C2212yes—
19C22×D12C2211yescharacteristic
20C3×C4×D4—24no—
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C22⋊C4C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C42C2211yescharacteristic
25C2×#217C2212yes—
26#939—28no—
27#930—24no—
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2212yes—
31C2×(C2×D3)⋊C4C2212yes—
32C2×C4⋊Dic3C2211yescharacteristic
33C2×Dic3⋊C4C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2211yescharacteristic
37C3⋊#95C2211yescharacteristic
38C3⋊#95C2212yes—
39C2×#121—32no—
40C2×#121—31no—
41C2×C4×D4D311yescharacteristic, complemented
42#398—31no—
43#397—31no—
44#394—31no—
45#392—31no—
46#388—32no—
47#386—32no—
48#386—32no—
49C2×#95—31no—
50C24×C3C2311yescharacteristic
51C24×C3C2311yescharacteristic
52C23×D3C2312yes—
53C2×C3×D4—22no—
54C2×C3×D4—22no—
55C2×C3×D4D414yescomplemented
56C22×C3×C4—22no—
57C22×C3×C4—42no—
58C22×C3×C4—42no—
59C22×C3×C4D414yescomplemented
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2311yescharacteristic
64C22×C3×C4C2311yescharacteristic
65C2×#67—24no—
66C2×#67—24no—
67C2×#67—24no—
68C2×#67—24no—
69C2×#67—24no—
70C2×#67—42no—
71C22×Dic3—24no—
72C22×Dic3—24no—
73C22×Dic3—42no—
74C22×Dic3C2312yes—
75C22×Dic3C2312yes—
76C22×Dic3C2312yes—
77C2×D12—22no—
78C2×D12—42no—
79C3×C4⋊C4—22no—
80C3×C22⋊C4—22no—
81C3×C22⋊C4—22no—
82C3×C42—22no—
83#217—24no—
84(C2×D3)⋊C4—24no—
85(C2×D3)⋊C4—24no—
86(C2×D3)⋊C4—24no—
87(C2×D3)⋊C4—42no—
88C4⋊Dic3—22no—
89Dic3⋊C4—24no—
90C22×D4—32no—
91C22×D4—32no—
92C22×D4—31no—
93C22×D4C2×D311yescharacteristic
94C23×C4—32no—
95C23×C4C2×D311yescharacteristic
96C23×C4C2×D311yescharacteristic
97#121—68no—
98#121—64no—
99C4×D4—24no—
100C2×C4⋊C4—32no—
101C2×C4⋊C4—31no—
102C2×C4⋊C4C2×D311yescharacteristic
103C2×C22⋊C4—32no—
104C2×C22⋊C4—32no—
105C2×C22⋊C4—32no—
106C2×C22⋊C4—32no—
107C2×C22⋊C4—32no—
108C2×C22⋊C4C2×D311yescharacteristic
109C2×C22⋊C4C2×D311yescharacteristic
110C2×C42C2×D311yescharacteristic
111#95—32no—
112#95—31no—
113#95—31no—
114#95—32no—
115#95—32no—
116C23×C3—22no—
117C23×C3—22no—
118C23×C3—21no—
119C23×C3—22no—
120C23×C3—21no—
121C23×C3—22no—
122C23×C3C2×D412yes—
123C23×C3C2×D412yes—
124C23×C3C2×D412yes—
125C23×C3#4112yes—
126C23×C3C2411yescharacteristic
127C22×D3—22no—
128C22×D3—22no—
129C22×D3—22no—
130C22×D3—22no—
131C22×D3—22no—
132C22×D3—22no—
133C22×D3—22no—
134C3×D4—28no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4—22no—
138C2×C3×C4—22no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—44no—
144C2×C3×C4—44no—
145C2×C3×C4C2×D412yes—
146C2×C3×C4C2×D412yes—
147C2×C3×C4C2×D412yes—
148#67—48no—
149#67—48no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—22no—
158C2×Dic3—48no—
159C2×Dic3#4114yes—
160D12—44no—
161C24—32no—
162C24C22×D311yescharacteristic
163C24C22×D311yescharacteristic
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—64no—
167C2×D4—64no—
168C2×D4—64no—
169C2×D4—64no—
170C2×D4—64no—
171C2×D4—62no—
172C2×D4—122no—
173C2×D4—122no—
174C2×D4D1214yescomplemented
175C22×C4—22no—
176C22×C4—32no—
177C22×C4—32no—
178C22×C4—32no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—64no—
182C22×C4—64no—
183C22×C4—122no—
184C22×C4#6714yescomplemented
185C22×C4C22×D311yescharacteristic
186C22×C4C22×D311yescharacteristic
187C22×C4C22×D311yescharacteristic
188C22×C4C22×D311yescharacteristic
189C22×C4C22×D311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—64no—
192C4⋊C4—62no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—64no—
196C22⋊C4—64no—
197C22⋊C4—64no—
198C22⋊C4—64no—
199C22⋊C4—122no—
200C42—22no—
201C22×C3—22no—
202C22×C3—22no—
203C22×C3—24no—
204C22×C3—24no—
205C22×C3—22no—
206C22×C3—24no—
207C22×C3—41no—
208C22×C3—41no—
209C22×C3—42no—
210C22×C3#12114yes—
211C22×C3C22×D411yescharacteristic
212C22×C3C22×D411yescharacteristic
213C22×C3C2×#4111yescharacteristic
214C22×C3C2×#4112yes—
215C22×C3H(5,𝔽2)12yes—
216C2×D3—42no—
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—42no—
221C2×D3—42no—
222C2×D3—42no—
223C3×C4—22no—
224C3×C4—22no—
225C3×C4—22no—
226C3×C4—41no—
227C3×C4—41no—
228Dic3—24no—
229Dic3—42no—
230Dic3—42no—
231C23—22no—
232C23—22no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—21no—
237C23—62no—
238C23—62no—
239C23—62no—
240C23—62no—
241C23—62no—
242C23—62no—
243C23—62no—
244C23C2×D1212yes—
245C23C2×D1212yes—
246C23#23512yes—
247C23C2×#6712yes—
248C23C23×D311yescharacteristic
249D4—28no—
250D4—124no—
251D4—128no—
252D4—128no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—34no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—62no—
267C2×C4—62no—
268C2×C4—62no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—128no—
273C2×C4C2×D1212yes—
274C2×C4C2×#6712yes—
275C2×C4C2×#6712yes—
276C2×C3—24no—
277C2×C3—42no—
278C2×C3C2×#12111yescharacteristic
279C2×C3#53211yescharacteristic
280C2×C3#53411yescharacteristic
281C2×C3#54612yes—
282C2×C3#55312yes—
283D3—82no—
284C22—24no—
285C22—22no—
286C22—24no—
287C22—22no—
288C22—22no—
289C22—24no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—122no—
294C22—122no—
295C22—122no—
296C22—122no—
297C22—122no—
298C22—122no—
299C22—122no—
300C22#93014yes—
301C22C22×D1211yescharacteristic
302C22C2×#23511yescharacteristic
303C22#100412yes—
304C22D3×#4112yes—
305C22C22×#6711yescharacteristic
306C4—22no—
307C4—22no—
308C4—22no—
309C4—41no—
310C4—41no—
311C4—64no—
312C4—122no—
313C4—122no—
314C3#254311yescharacteristic, complemented, Sylow
315C2—24no—
316C2—42no—
317C2—242no—
318C2#554012yes—
319C2#555412yes—
320C2C2×#93011yescharacteristic
321C2#580411yescharacteristic
322C2#583211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
hypercenterC2×C4×D4D3
hypocenterC3#2543
LCSG ▷ C23×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
2. C2×#939C2semidirect
3. C2×#930C2semidirect
4. #5213C2semidirect
5. C2×C3⋊#95C2semidirect
6. #4991C2semidirect
7. #4981C2semidirect
8. #4960C2semidirect
9. #4958C2semidirect
10. #4953C2semidirect
11. #4940C2semidirect
41. C2×C4×D4D3semidirect
55. C2×C3×D4D4semidirect
59. C22×C3×C4D4semidirect
174. C2×D4D12semidirect
184. C22×C4#67semidirect
314. C3#2543semidirect

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×#9392
C2×#9302
#52132
C2×C3⋊#952
#49912
#49812
#49602
#49582
#49532
#49402
#25433