Finite Group Databasebeta

← #82198

Subgroups — #82198

Order 384.

584 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×C4×#67C211yesmaximal, characteristic
3#5213C211yesmaximal, characteristic
4#5213C211yesmaximal, characteristic
5#5212C211yesmaximal, characteristic, complemented
6#5210C211yesmaximal, characteristic, complemented
7#5210C211yesmaximal, characteristic, complemented
8#4994C212yesmaximal, complemented
9#4989C212yesmaximal, complemented
10#4982C212yesmaximal, complemented
11#4978C212yesmaximal, complemented
12#2579—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×#67C2211yescharacteristic
17C22×C4×D3C2211yescharacteristic
18C2×C3×C4⋊C4—24no—
19C2×C3×C4⋊C4C2212yes—
20C2×C3×C4⋊C4C2212yes—
21C2×#217C2211yescharacteristic
22C2×#217C2211yescharacteristic
23C2×#217C2211yescharacteristic
24C4×#67—24no—
25C2×(C2×D3)⋊C4C2212yes—
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×C4⋊Dic3C2212yes—
29C2×Dic3⋊C4C2212yes—
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C2×C4×Dic3C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4×D4—31no—
39C22×C4⋊C4D311yescharacteristic, complemented
40#398—32no—
41#396—32no—
42#387—32no—
43#386—31no—
44#386—31no—
45#385—31no—
46#385—31no—
47#385—32no—
48#380—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4—22no—
52C22×C3×C4—24no—
53C22×C3×C4—22no—
54C22×C3×C4—42no—
55C22×C3×C4—42no—
56C22×C3×C4D414yes—
57C22×C3×C4C2312yes—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2312yes—
61C2×#67—22no—
62C2×#67—22no—
63C2×#67Q814yes—
64C22×Dic3C2312yes—
65C22×Dic3C2312yes—
66C22×Dic3C2311yescharacteristic
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2311yescharacteristic
69C2×C4×D3—22no—
70C2×C4×D3—42no—
71C3×C4⋊C4—24no—
72C3×C4⋊C4—24no—
73#217—22no—
74#217—22no—
75#217—22no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—24no—
78(C2×D3)⋊C4—42no—
79C4⋊Dic3—42no—
80Dic3⋊C4—22no—
81Dic3⋊C4—24no—
82Dic3⋊C4—42no—
83C4×Dic3—22no—
84C22×D4—31no—
85C23×C4—31no—
86C23×C4C2×D311yescharacteristic
87C23×C4C2×D311yescharacteristic
88C23×C4C2×D311yescharacteristic
89C4×D4—64no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4—32no—
93C2×C4⋊C4—32no—
94C2×C4⋊C4—32no—
95C2×C4⋊C4C2×D312yes—
96C2×C4⋊C4C2×D312yes—
97C2×C22⋊C4—31no—
98C2×C22⋊C4—31no—
99C2×C22⋊C4—31no—
100C2×C22⋊C4—31no—
101C2×C22⋊C4—32no—
102C2×C22⋊C4—32no—
103C2×C42—31no—
104#95—32no—
105#95—32no—
106#95—32no—
107#95—32no—
108#95—32no—
109C23×C3—22no—
110C23×C3—22no—
111C23×C3C2×D412yes—
112C23×C3C2×Q812yes—
113C23×C3#4112yes—
114C23×C3C2411yescharacteristic
115C22×D3—21no—
116C22×D3—22no—
117C22×D3—21no—
118C22×D3—22no—
119C22×D3C2×Q812yes—
120C2×C3×C4—28no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—44no—
129C2×C3×C4—44no—
130C2×C3×C4C2×D412yes—
131C2×C3×C4C2×D412yes—
132C2×C3×C4#4114yes—
133#67—28no—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—21no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143C2×Dic3—22no—
144C2×Dic3—22no—
145C2×Dic3—22no—
146C2×Dic3C2×Q812yes—
147C4×D3—44no—
148C24—31no—
149C24C22×D311yescharacteristic
150C2×D4—34no—
151C2×D4—62no—
152C2×D4—62no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—22no—
156C22×C4—31no—
157C22×C4—32no—
158C22×C4—32no—
159C22×C4—31no—
160C22×C4—31no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—62no—
164C22×C4—122no—
165C22×C4#6714yes—
166C22×C4C22×D312yes—
167C22×C4C22×D312yes—
168C22×C4C22×D311yescharacteristic
169C22×C4C22×D311yescharacteristic
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—64no—
173C4⋊C4—62no—
174C4⋊C4—122no—
175C4⋊C4—122no—
176C22⋊C4—62no—
177C22⋊C4—62no—
178C22⋊C4—62no—
179C22⋊C4—64no—
180C22⋊C4—62no—
181C22⋊C4—122no—
182C42—62no—
183C22×C3—22no—
184C22×C3—22no—
185C22×C3—24no—
186C22×C3—24no—
187C22×C3C22⋊Q814yes—
188C22×C3C22×D411yescharacteristic
189C22×C3C22×Q811yescharacteristic
190C22×C3C2×#4112yes—
191C22×C3C2×#4111yescharacteristic
192C22×C3H(5,𝔽2)12yes—
193C2×D3—22no—
194C2×D3—24no—
195C2×D3—41no—
196C2×D3—41no—
197C2×D3—42no—
198C3×C4—24no—
199C3×C4—22no—
200C3×C4—22no—
201C3×C4—42no—
202Dic3—22no—
203Dic3—41no—
204Dic3—41no—
205Dic3—42no—
206Dic3—42no—
207C23—22no—
208C23—22no—
209C23—32no—
210C23—62no—
211C23—62no—
212C23—61no—
213C23—61no—
214C23D3×Q812yes—
215C23#23912yes—
216C23C2×#6712yes—
217C23C23×D311yescharacteristic
218D4—68no—
219C2×C4—28no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—32no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—61no—
231C2×C4—61no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4—62no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—124no—
243C2×C4#23514yes—
244C2×C4C2×#6712yes—
245C2×C4C2×#6712yes—
246C2×C3—24no—
247C2×C3C2×C22⋊Q811yescharacteristic
248C2×C3#52511yescharacteristic
249C2×C3#53411yescharacteristic
250C2×C3#55312yes—
251C2×C3#55412yes—
252D3—42no—
253C22—24no—
254C22—24no—
255C22—22no—
256C22—22no—
257C22—64no—
258C22—62no—
259C22—122no—
260C22—121no—
261C22—121no—
262C22#94614yes—
263C22C2×#23512yes—
264C22#100412yes—
265C22C2×D3×Q811yescharacteristic
266C22C2×#23911yescharacteristic
267C22C22×#6711yescharacteristic
268C4—24no—
269C4—22no—
270C4—22no—
271C4—42no—
272C4—62no—
273C4—122no—
274C4—122no—
275C4—121no—
276C4—121no—
277C3#257911yescharacteristic, complemented, Sylow
278C2—24no—
279C2—122no—
280C2#551112yes—
281C2#551412yes—
282C2#579111yescharacteristic
283C2#580411yescharacteristic
284C2C2×#94611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC22×C3×C4⋊C4C2
prime radicalC22×#67C22
hypercenterC22×C4⋊C4D3
hypocenterC3#2579
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊C4C2semidirect
5. #5212C2semidirect
6. #5210C2semidirect
7. #5210C2semidirect
8. #4994C2semidirect
9. #4989C2semidirect
10. #4982C2semidirect
11. #4978C2semidirect
39. C22×C4⋊C4D3semidirect
277. C3#2579semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊C42
C2×C4×#672
#52132
#52122
#52102
#49942
#49892
#49822
#49782
#25793