Finite Group Databasebeta

← #82392

Subgroups — #82392

Order 384.

520 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic
2C2×#924C211yesmaximal, characteristic, complemented
3#5222C212yesmaximal
4#5211C211yesmaximal, characteristic, complemented
5#5210C212yesmaximal, complemented
6#4981C211yesmaximal, characteristic, complemented
7#4979C211yesmaximal, characteristic, complemented
8#4953C211yesmaximal, characteristic, complemented
9#4952C212yesmaximal, complemented
10#4947C211yesmaximal, characteristic, complemented
11#4933C212yesmaximal, complemented
12#2638—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C23×C3×C4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C22×Dic6C2211yescharacteristic
17C3×C4×D4—24no—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2211yescharacteristic
20C2×C3×C22⋊C4C2211yescharacteristic
21C2×C3×C42C2211yescharacteristic
22C2×#217C2212yes—
23C2×#217C2212yes—
24C2×#217C2212yes—
25C2×#217C2212yes—
26C2×C4⋊Dic3C2211yescharacteristic
27#924—24no—
28C2×Dic3⋊C4C2212yes—
29C2×Dic3⋊C4C2212yes—
30C2×Dic3⋊C4C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2211yescharacteristic
38C2×C22⋊Q8—31no—
39C2×C4×D4D311yescharacteristic
40#398—31no—
41#397—32no—
42#397—31no—
43#395—31no—
44#394—32no—
45#393—31no—
46#387—32no—
47#385—32no—
48#381—31no—
49C24×C3C2311yescharacteristic
50C24×C3C2311yescharacteristic
51C2×C3×D4—22no—
52C2×C3×D4—22no—
53C2×C3×D4—22no—
54C22×C3×C4—22no—
55C22×C3×C4—42no—
56C22×C3×C4—42no—
57C22×C3×C4D414yes—
58C22×C3×C4C2311yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C22×C3×C4C2311yescharacteristic
61C22×C3×C4C2311yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C22×Dic3C2312yes—
64C22×Dic3C2312yes—
65C22×Dic3C2312yes—
66C22×Dic3C2312yes—
67C2×Dic6—22no—
68C2×Dic6—42no—
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—
78Dic3⋊C4—24no—
79Dic3⋊C4—24no—
80Dic3⋊C4—42no—
81C22×Q8—31no—
82C22×D4C2×D311yescharacteristic
83C23×C4C2×D311yescharacteristic
84C23×C4C2×D311yescharacteristic
85C22⋊Q8—64no—
86C4×D4—24no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C4⋊C4C2×D311yescharacteristic
92C2×C22⋊C4—32no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4C2×D311yescharacteristic
97C2×C22⋊C4C2×D311yescharacteristic
98C2×C42C2×D311yescharacteristic
99#95—32no—
100#95—31no—
101#95—31no—
102#95—32no—
103#95—32no—
104#95—32no—
105#95—32no—
106C23×C3—21no—
107C23×C3—21no—
108C23×C3—21no—
109C23×C3—21no—
110C23×C3—22no—
111C23×C3—22no—
112C23×C3—22no—
113C23×C3—22no—
114C23×C3C2×D412yes—
115C23×C3#4112yes—
116C23×C3C2411yescharacteristic
117C3×D4—28no—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—44no—
127C2×C3×C4—44no—
128C2×C3×C4C2×D412yes—
129C2×C3×C4C2×D412yes—
130C2×C3×C4#4112yes—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—22no—
141C2×Dic3—22no—
142C2×Dic3—22no—
143Dic6—44no—
144C24C22×D311yescharacteristic
145C24C22×D311yescharacteristic
146C2×Q8—62no—
147C2×Q8—122no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C22×C4—22no—
152C22×C4—32no—
153C22×C4—32no—
154C22×C4—32no—
155C22×C4—32no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4#6714yes—
159C22×C4C22×D311yescharacteristic
160C22×C4C22×D311yescharacteristic
161C22×C4C22×D311yescharacteristic
162C22×C4C22×D311yescharacteristic
163C22×C4C22×D311yescharacteristic
164C4⋊C4—22no—
165C4⋊C4—64no—
166C4⋊C4—64no—
167C4⋊C4—62no—
168C4⋊C4—122no—
169C22⋊C4—22no—
170C22⋊C4—22no—
171C22⋊C4—64no—
172C22⋊C4—64no—
173C22⋊C4—122no—
174C22⋊C4—122no—
175C42—22no—
176C22×C3—22no—
177C22×C3—22no—
178C22×C3—22no—
179C22×C3—22no—
180C22×C3—24no—
181C22×C3—41no—
182C22×C3—41no—
183C22×C3—42no—
184C22×C3—42no—
185C22×C3—42no—
186C22×C3C22×D411yescharacteristic
187C22×C3C2×#4111yescharacteristic
188C22×C3C2×#4111yescharacteristic
189C22×C3H(5,𝔽2)12yes—
190C22×C32−1+412yes—
191C3×C4—22no—
192C3×C4—22no—
193C3×C4—22no—
194C3×C4—41no—
195C3×C4—41no—
196Dic3—42no—
197Dic3—42no—
198Dic3—42no—
199Dic3—42no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—21no—
208C23#23512yes—
209C23C2×#6712yes—
210C23C23×D311yescharacteristic
211Q8—124no—
212D4—28no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—62no—
235C2×C4#23512yes—
236C2×C4C2×#6712yes—
237C2×C4C2×#6712yes—
238C2×C3—42no—
239C2×C3—42no—
240C2×C3#52511yescharacteristic
241C2×C3#53411yescharacteristic
242C2×C3#53611yescharacteristic
243C2×C3#53912yes—
244C2×C3#54212yes—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—24no—
249C22—22no—
250C22—41no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—41no—
255C22C2×#23511yescharacteristic
256C22C2×#23511yescharacteristic
257C22#100412yes—
258C22#101012yes—
259C22C22×#6711yescharacteristic
260C4—22no—
261C4—22no—
262C4—22no—
263C4—41no—
264C4—41no—
265C4—122no—
266C4—122no—
267C4—122no—
268C4—122no—
269C3#263811yescharacteristic, complemented, Sylow
270C2—42no—
271C2—42no—
272C2#556112yes—
273C2#556412yes—
274C2#579111yescharacteristic
275C2#580411yescharacteristic
276C2#582511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×C4×D4C2
prime radicalC22×C3×D4C22
hypercenterC2×C4×D4D3
hypocenterC3#2638
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×#924C2semidirect
4. #5211C2semidirect
5. #5210C2semidirect
6. #4981C2semidirect
7. #4979C2semidirect
8. #4953C2semidirect
9. #4952C2semidirect
10. #4947C2semidirect
11. #4933C2semidirect
269. C3#2638semidirect

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×#9242
#52222
#52112
#52102
#49812
#49792
#49532
#49522
#49472
#49332
#26383