Finite Group Databasebeta

← #82372

Subgroups — #82372

Order 384.

540 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic
2C2×#926C211yesmaximal, characteristic, complemented
3#5222C212yesmaximal
4#5212C211yesmaximal, characteristic, complemented
5#5212C211yesmaximal, characteristic, complemented
6#5212C211yesmaximal, characteristic, complemented
7#4978C212yesmaximal, complemented
8#4952C212yesmaximal, complemented
9#4944C212yesmaximal, complemented
10#4928C211yesmaximal, characteristic, complemented
11#4928C211yesmaximal, characteristic, complemented
12#2485—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C3×C4×D4—24no—
17C2×C3×C4⋊C4C2211yescharacteristic
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C42C2211yescharacteristic
21C2×#217C2212yes—
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25#926—24no—
26C2×C4⋊Dic3C2211yescharacteristic
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×Dic3⋊C4C2212yes—
31C2×Dic3⋊C4C2212yes—
32C2×C4×Dic3C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×D4D311yescharacteristic
39C2×#117—31no—
40#397—32no—
41#394—32no—
42#389—31no—
43#389—31no—
44#388—32no—
45#387—32no—
46#380—31no—
47#380—31no—
48#380—31no—
49C24×C3C2311yescharacteristic
50C24×C3C2311yescharacteristic
51C2×C3×D4—22no—
52C2×C3×D4—22no—
53C2×C3×D4—22no—
54C22×C3×C4—22no—
55C22×C3×C4—22no—
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4D414yes—
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C22×C3×C4C2311yescharacteristic
64C22×C3×C4C2311yescharacteristic
65C22×Dic3C2312yes—
66C22×Dic3C2312yes—
67C22×Dic3C2312yes—
68C22×Dic3C2312yes—
69C3×C4⋊C4—22no—
70C3×C22⋊C4—22no—
71C3×C22⋊C4—22no—
72C3×C42—22no—
73#217—24no—
74#217—24no—
75#217—42no—
76#217—42no—
77C4⋊Dic3—22no—
78C4⋊Dic3—22no—
79C4⋊Dic3—22no—
80C4⋊Dic3—22no—
81Dic3⋊C4—24no—
82Dic3⋊C4—42no—
83C4×Dic3—24no—
84C22×D4C2×D311yescharacteristic
85C23×C4C2×D311yescharacteristic
86C23×C4C2×D311yescharacteristic
87C4×D4—24no—
88#117—64no—
89C2×C4⋊C4—31no—
90C2×C4⋊C4—31no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—31no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4C2×D311yescharacteristic
96C2×C22⋊C4—32no—
97C2×C22⋊C4—32no—
98C2×C22⋊C4—32no—
99C2×C22⋊C4—32no—
100C2×C22⋊C4C2×D311yescharacteristic
101C2×C22⋊C4C2×D311yescharacteristic
102C2×C42—32no—
103C2×C42C2×D311yescharacteristic
104#95—32no—
105#95—32no—
106#95—32no—
107#95—32no—
108#95—32no—
109C23×C3—21no—
110C23×C3—21no—
111C23×C3—21no—
112C23×C3—21no—
113C23×C3—21no—
114C23×C3—21no—
115C23×C3—21no—
116C23×C3—21no—
117C23×C3—21no—
118C23×C3—21no—
119C23×C3—21no—
120C23×C3—21no—
121C23×C3C2×D412yes—
122C23×C3#4112yes—
123C23×C3C2411yescharacteristic
124C3×D4—28no—
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—28no—
130C2×C3×C4—21no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4—44no—
137C2×C3×C4C2×D412yes—
138C2×C3×C4C2×D412yes—
139C2×C3×C4#4112yes—
140C2×C3×C4#4112yes—
141C2×C3×C4#4112yes—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3—22no—
147C2×Dic3—22no—
148C2×Dic3—22no—
149C2×Dic3—22no—
150C2×Dic3—22no—
151C2×Dic3—22no—
152C2×Dic3—22no—
153C2×Dic3—22no—
154C24C22×D311yescharacteristic
155C24C22×D311yescharacteristic
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—32no—
165C22×C4—32no—
166C22×C4—32no—
167C22×C4—32no—
168C22×C4#6714yes—
169C22×C4C22×D311yescharacteristic
170C22×C4C22×D311yescharacteristic
171C22×C4C22×D311yescharacteristic
172C22×C4C22×D311yescharacteristic
173C22×C4C22×D311yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—64no—
176C4⋊C4—62no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C4⋊C4—62no—
180C4⋊C4—122no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—64no—
184C22⋊C4—64no—
185C22⋊C4—122no—
186C22⋊C4—122no—
187C42—22no—
188C42—64no—
189C22×C3—22no—
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—22no—
194C22×C3—22no—
195C22×C3—41no—
196C22×C3—41no—
197C22×C3—41no—
198C22×C3—41no—
199C22×C3—41no—
200C22×C3—41no—
201C22×C3—41no—
202C22×C3—41no—
203C22×C3C22×D411yescharacteristic
204C22×C3C2×#4111yescharacteristic
205C22×C3C2×#4111yescharacteristic
206C22×C3C2×#4111yescharacteristic
207C22×C3C2×#4111yescharacteristic
208C22×C3H(5,𝔽2)11yescharacteristic
209C22×C32−1+411yescharacteristic
210C3×C4—22no—
211C3×C4—22no—
212C3×C4—41no—
213C3×C4—41no—
214C3×C4#12314yes—
215Dic3—42no—
216Dic3—42no—
217Dic3—42no—
218Dic3—42no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23#23512yes—
232C23C2×#6712yes—
233C23C23×D311yescharacteristic
234D4—28no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—28no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—44no—
247C2×C4—62no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—62no—
259C2×C4#23512yes—
260C2×C4#23712yes—
261C2×C4#23712yes—
262C2×C4C2×#6712yes—
263C2×C4C2×#6712yes—
264C2×C3—42no—
265C2×C3—42no—
266C2×C3C2×#12311yescharacteristic
267C2×C3#52511yescharacteristic
268C2×C3#53711yescharacteristic
269C2×C3#55211yescharacteristic
270C2×C3#55311yescharacteristic
271C2×C3#55511yescharacteristic
272C2×C3#55611yescharacteristic
273C22—22no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22C2×#23511yescharacteristic
288C22C2×#23511yescharacteristic
289C22C2×#23711yescharacteristic
290C22C2×#23711yescharacteristic
291C22#100911yescharacteristic
292C22#101011yescharacteristic
293C22C22×#6711yescharacteristic
294C4—22no—
295C4—22no—
296C4—41no—
297C4—41no—
298C4—122no—
299C4—122no—
300C4—122no—
301C4—122no—
302C4#93514yes—
303C3#248511yescharacteristic, complemented, Sylow
304C2—42no—
305C2—42no—
306C2#554111yescharacteristic
307C2#554211yescharacteristic
308C2#556311yescharacteristic
309C2#556511yescharacteristic
310C2#579111yescharacteristic
311C2C2×#93511yescharacteristic
312C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
prime radicalC22×C3×D4C22
hypercenterC2×C4×D4D3
hypocenterC3#2485
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#926C2semidirect
4. #5212C2semidirect
5. #5212C2semidirect
6. #5212C2semidirect
7. #4978C2semidirect
8. #4952C2semidirect
9. #4944C2semidirect
10. #4928C2semidirect
11. #4928C2semidirect
303. C3#2485semidirect

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×#9262
#52222
#52122
#49782
#49522
#49442
#49282
#24853