Finite Group Databasebeta

← #82990

Subgroups — #82990

Order 384.

492 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 137 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×#124C211yesmaximal, characteristic
2C2×#944C211yesmaximal, characteristic, complemented
3#5229C211yesmaximal, characteristic, complemented
4#5222C212yesmaximal
5#4948C212yesmaximal, complemented
6#4947C212yesmaximal, complemented
7#4945C212yesmaximal, complemented
8#4944C212yesmaximal, complemented
9#4933C212yesmaximal, complemented
10#2731—31nomaximal, Sylow
11C22×C3×Q8C2211yescharacteristic
12C22×C3×D4C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C3×#124—24no—
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2212yes—
17C2×C3×C42C2211yescharacteristic
18C2×#217C2212yes—
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×#217C2212yes—
22#944—24no—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C2×C4⋊Q8—31no—
35C2×#124D311yescharacteristic
36#396—32no—
37#394—32no—
38#393—32no—
39#391—31no—
40#388—32no—
41#388—32no—
42#387—32no—
43C24×C3C2312yes—
44C2×C3×Q8—41no—
45C2×C3×Q8—41no—
46C2×C3×Q8D414yes—
47C2×C3×D4—22no—
48C2×C3×D4—41no—
49C2×C3×D4—41no—
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C22×Dic3C2312yes—
57C22×Dic3C2312yes—
58C2×Dic6—22no—
59C2×Dic6—41no—
60C2×Dic6—41no—
61C3×C22⋊C4—24no—
62C3×C22⋊C4—24no—
63C3×C42—22no—
64#217—24no—
65#217—24no—
66#217—42no—
67#217—42no—
68C4⋊Dic3—41no—
69Dic3⋊C4—24no—
70Dic3⋊C4—24no—
71Dic3⋊C4—24no—
72C4×Dic3—22no—
73C4×Dic3—41no—
74C22×Q8—31no—
75C22×Q8C2×D311yescharacteristic
76C22×D4C2×D311yescharacteristic
77C4⋊Q8—64no—
78#124—24no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—31no—
83C2×C22⋊C4—32no—
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4C2×D312yes—
88C2×C22⋊C4C2×D312yes—
89C2×C42—31no—
90C2×C42—31no—
91C2×C42C2×D311yescharacteristic
92#95—32no—
93#95—32no—
94#95—32no—
95#95—32no—
96#95—32no—
97C23×C3—22no—
98C23×C3—22no—
99C23×C3—22no—
100C23×C3—22no—
101C23×C3—22no—
102C23×C3—22no—
103C23×C3—22no—
104C23×C3C2411yescharacteristic
105C3×Q8—44no—
106C3×D4—44no—
107C2×C3×C4—21no—
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4C2×D412yes—
117C2×C3×C4C2×D412yes—
118C2×C3×C4C2×D412yes—
119C2×Dic3—22no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—22no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3#4114yes—
131Dic6—44no—
132C24C22×D312yes—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×Q8—62no—
136C2×Q8—121no—
137C2×Q8—121no—
138C2×Q8#6714yes—
139C2×D4—22no—
140C2×D4—41no—
141C2×D4—41no—
142C22×C4—32no—
143C22×C4—32no—
144C22×C4—32no—
145C22×C4—32no—
146C22×C4C22×D312yes—
147C22×C4C22×D311yescharacteristic
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D311yescharacteristic
150C4⋊C4—64no—
151C4⋊C4—64no—
152C4⋊C4—64no—
153C4⋊C4—121no—
154C22⋊C4—24no—
155C22⋊C4—24no—
156C22⋊C4—64no—
157C22⋊C4—64no—
158C22⋊C4—122no—
159C22⋊C4—122no—
160C42—22no—
161C42—62no—
162C42—121no—
163C22×C3—42no—
164C22×C3—42no—
165C22×C3—42no—
166C22×C3—42no—
167C22×C3—42no—
168C22×C3—42no—
169C22×C3—42no—
170C22×C3C22×D411yescharacteristic
171C22×C3C2×#4112yes—
172C22×C3H(5,𝔽2)11yescharacteristic
173C22×C3H(5,𝔽2)11yescharacteristic
174C22×C32−1+412yes—
175C3×C4—22no—
176C3×C4—22no—
177C3×C4—22no—
178C3×C4—42no—
179Dic3—24no—
180Dic3—42no—
181Dic3—42no—
182Dic3—42no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23C23×D311yescharacteristic
191Q8—44no—
192Q8—124no—
193D4—44no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—34no—
204C2×C4—62no—
205C2×C4—62no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4C2×#6712yes—
216C2×C4C2×#6712yes—
217C2×C4C2×#6712yes—
218C2×C3—82no—
219C2×C3#53411yescharacteristic
220C2×C3#54212yes—
221C2×C3#54812yes—
222C2×C3#55511yescharacteristic
223C2×C3#56311yescharacteristic
224C22—42no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22#100411yescharacteristic
232C22#100411yescharacteristic
233C22D3×#4112yes—
234C22#101012yes—
235C22C22×#6711yescharacteristic
236C4—22no—
237C4—22no—
238C4—22no—
239C4—42no—
240C4—64no—
241C4—122no—
242C4—122no—
243C4—122no—
244C3#273111yescharacteristic, complemented, Sylow
245C2—82no—
246C2#566912yes—
247C2#567011yescharacteristic
248C2#567211yescharacteristic
249C2#580411yescharacteristic
250C2#583112yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#124C2
prime radicalC22×C3×D4C22
hypercenterC2×#124D3
hypocenterC3#2731
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#944C2semidirect
3. #5229C2semidirect
5. #4948C2semidirect
6. #4947C2semidirect
7. #4945C2semidirect
8. #4944C2semidirect
9. #4933C2semidirect
244. C3#2731semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1242
C2×#9442
#52292
#52222
#49482
#49472
#49452
#49442
#49332
#27313