Finite Group Databasebeta

← #73612

Subgroups — #73612

Order 384.

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

normal subgroups only

This group has 498 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×#929C211yesmaximal, characteristic
2C2×#924C211yesmaximal, characteristic, complemented
3C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
4#4961C212yesmaximal
5#4953C212yesmaximal, complemented
6#4940C212yesmaximal, complemented
7#4928C212yesmaximal, complemented
8#4677C211yesmaximal, characteristic, complemented
9#4673C211yesmaximal, characteristic, complemented
10#4659C211yesmaximal, characteristic, complemented
11#4656C211yesmaximal, characteristic, complemented
12#2663—31nomaximal, Sylow
13C23×C3×C4C2211yescharacteristic
14C22×#67C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C42C2212yes—
18C2×#217C2211yescharacteristic
19C2×#217C2212yes—
20#929—24no—
21C2×(C2×D3)⋊C4C2211yescharacteristic
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2212yes—
25C2×C4⋊Dic3C2211yescharacteristic
26C2×C4⋊Dic3C2212yes—
27C2×C4⋊Dic3C2212yes—
28#924—24no—
29C2×Dic3⋊C4C2211yescharacteristic
30C2×Dic3⋊C4C2211yescharacteristic
31C2×Dic3⋊C4C2212yes—
32C3×#95C2211yescharacteristic
33C3×#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#123—31no—
39C2×C22⋊Q8—31no—
40#400—31no—
41#399—31no—
42#397—32no—
43#395—31no—
44#394—32no—
45#393—31no—
46#389—32no—
47#388—32no—
48C4×C22⋊C4D311yescharacteristic, complemented
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4—42no—
52C22×C3×C4D414yes—
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2312yes—
57C2×#67—22no—
58C2×#67—42no—
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2312yes—
62C22×Dic3C2311yescharacteristic
63C22×Dic3C2312yes—
64C2×Dic6—22no—
65C2×Dic6—42no—
66C3×C22⋊C4—24no—
67C3×C42—42no—
68#217—24no—
69#217—22no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—24no—
73(C2×D3)⋊C4—42no—
74C4⋊Dic3—22no—
75C4⋊Dic3—24no—
76C4⋊Dic3—42no—
77Dic3⋊C4—24no—
78Dic3⋊C4—22no—
79Dic3⋊C4—22no—
80C22×Q8—31no—
81C22×D4—31no—
82C23×C4C2×D311yescharacteristic
83#123—64no—
84C22⋊Q8—64no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—31no—
88C2×C4⋊C4—31no—
89C2×C4⋊C4—32no—
90C2×C4⋊C4—32no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—31no—
93C2×C22⋊C4—31no—
94C2×C22⋊C4—31no—
95C2×C22⋊C4—32no—
96C2×C22⋊C4—32no—
97C2×C22⋊C4C2×D312yes—
98C2×C42C2×D312yes—
99#95—32no—
100#95—32no—
101#95—32no—
102#95—32no—
103#95C2×D311yescharacteristic
104#95C2×D311yescharacteristic
105C23×C3—22no—
106C23×C3—22no—
107C23×C3—21no—
108C23×C3—21no—
109C23×C3C2×D412yes—
110C23×C3C2411yescharacteristic
111C22×D3—21no—
112C22×D3—22no—
113C22×D3—22no—
114C22×D3—21no—
115C22×D3—21no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—22no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—44no—
124C2×C3×C4C2×D412yes—
125C2×C3×C4C2×D412yes—
126C2×C3×C4#4114yes—
127#67—44no—
128C2×Dic3—21no—
129C2×Dic3—22no—
130C2×Dic3—21no—
131C2×Dic3—21no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—21no—
140C2×Dic3—21no—
141Dic6—44no—
142C24—31no—
143C24C22×D311yescharacteristic
144C2×Q8—62no—
145C2×Q8—122no—
146C2×D4—62no—
147C2×D4—122no—
148C22×C4—31no—
149C22×C4—31no—
150C22×C4—32no—
151C22×C4—32no—
152C22×C4—31no—
153C22×C4—42no—
154C22×C4D1214yes—
155C22×C4C22×D312yes—
156C22×C4C22×D311yescharacteristic
157C22×C4C22×D311yescharacteristic
158C22×C4C22×D312yes—
159C4⋊C4—64no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—64no—
164C4⋊C4—122no—
165C22⋊C4—24no—
166C22⋊C4—64no—
167C22⋊C4—62no—
168C22⋊C4—62no—
169C22⋊C4—62no—
170C22⋊C4—64no—
171C22⋊C4—122no—
172C42—42no—
173C22×C3—22no—
174C22×C3—22no—
175C22×C3—22no—
176C22×C3—42no—
177C22×C3—42no—
178C22×C3C22×D411yescharacteristic
179C22×C3C2×#4112yes—
180C22×C3H(5,𝔽2)12yes—
181C22×C32−1+411yescharacteristic
182C22×C32−1+411yescharacteristic
183C2×D3—41no—
184C2×D3—41no—
185C2×D3—41no—
186C2×D3—42no—
187C2×D3—42no—
188C3×C4—22no—
189C3×C4—22no—
190C3×C4—24no—
191C3×C4—42no—
192Dic3—42no—
193Dic3—41no—
194Dic3—42no—
195Dic3—41no—
196Dic3—41no—
197C23—22no—
198C23—22no—
199C23—21no—
200C23—21no—
201C23—61no—
202C23—62no—
203C23—62no—
204C23—61no—
205C23—61no—
206C23C2×D1212yes—
207C23C23×D311yescharacteristic
208Q8—124no—
209D4—124no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—44no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—61no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—61no—
230C2×C4—61no—
231C2×C4C2×D1212yes—
232C2×C4C2×D1212yes—
233C2×C4#23514yes—
234C2×C3—42no—
235C2×C3#53211yescharacteristic
236C2×C3#53411yescharacteristic
237C2×C3#53611yescharacteristic
238C2×C3#53912yes—
239C2×C3#54212yes—
240D3—81no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—42no—
245C22—42no—
246C22—121no—
247C22—121no—
248C22—121no—
249C22—122no—
250C22—122no—
251C22C22×D1211yescharacteristic
252C22C2×#23512yes—
253C22#100412yes—
254C22#101011yescharacteristic
255C22#101011yescharacteristic
256C4—22no—
257C4—22no—
258C4—24no—
259C4—42no—
260C4—121no—
261C4—122no—
262C4—121no—
263C4—122no—
264C4—121no—
265C3#266311yescharacteristic, complemented, Sylow
266C2—42no—
267C2—241no—
268C2#547711yescharacteristic
269C2#549311yescharacteristic
270C2#552611yescharacteristic
271C2#556112yes—
272C2#556412yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×C4×C22⋊C4C2
prime radicalC22×#67C22
hypercenterC4×C22⋊C4D3
hypocenterC3#2663
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#924C2semidirect
3. C3×C4×C22⋊C4C2semidirect
5. #4953C2semidirect
6. #4940C2semidirect
7. #4928C2semidirect
8. #4677C2semidirect
9. #4673C2semidirect
10. #4659C2semidirect
11. #4656C2semidirect
48. C4×C22⋊C4D3semidirect
265. C3#2663semidirect

Maximal subgroups

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

subgroupindex
C2×#9292
C2×#9242
C3×C4×C22⋊C42
#49612
#49532
#49402
#49282
#46772
#46732
#46592
#46562
#26633