Finite Group Databasebeta

← #82886

Subgroups — #82886

Order 384.

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

normal subgroups only

This group has 512 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×#123C211yesmaximal, characteristic
2C2×#936C211yesmaximal, characteristic
3#5220C211yesmaximal, characteristic
4#5210C212yesmaximal, complemented
5#4979C212yesmaximal, complemented
6#4978C212yesmaximal, complemented
7#4952C212yesmaximal, complemented
8#4952C212yesmaximal, complemented
9#4942C212yesmaximal, complemented
10#2717—31nomaximal, Sylow
11C22×C3×D4C2211yescharacteristic
12C23×C3×C4C2211yescharacteristic
13C23×Dic3C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C3×#123—24no—
16C2×C3×C4⋊C4C2212yes—
17C2×C3×C22⋊C4C2212yes—
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×#217C2212yes—
20C2×#217C2212yes—
21C2×#217C2212yes—
22C2×#217C2211yescharacteristic
23#936—24no—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28C3⋊#95C2212yes—
29C3⋊#95C2212yes—
30C3⋊#95C2212yes—
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C2×#124—31no—
35C2×#123D311yescharacteristic
36#397—32no—
37#397—32no—
38#395—32no—
39#387—32no—
40#386—31no—
41#385—32no—
42#380—32no—
43C24×C3C2311yescharacteristic
44C24×C3C2311yescharacteristic
45C2×C3×D4—41no—
46C2×C3×D4—41no—
47C2×C3×D4D414yes—
48C22×C3×C4—22no—
49C22×C3×C4—41no—
50C22×C3×C4—41no—
51C22×C3×C4C2312yes—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×Dic3—24no—
55C22×Dic3—41no—
56C22×Dic3C2312yes—
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C22×Dic3C2312yes—
60C2×Dic6—22no—
61C2×Dic6—41no—
62C2×Dic6—41no—
63C3×C4⋊C4—24no—
64C3×C22⋊C4—22no—
65C3×C22⋊C4—24no—
66#217—24no—
67#217—24no—
68#217—24no—
69#217—41no—
70Dic3⋊C4—24no—
71Dic3⋊C4—42no—
72Dic3⋊C4—42no—
73C4×Dic3—22no—
74C22×Q8—31no—
75C22×D4C2×D311yescharacteristic
76C23×C4—31no—
77C23×C4C2×D311yescharacteristic
78#124—64no—
79#123—24no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4C2×D312yes—
84C2×C22⋊C4—32no—
85C2×C22⋊C4—32no—
86C2×C22⋊C4—32no—
87C2×C22⋊C4—31no—
88C2×C22⋊C4C2×D312yes—
89C2×C22⋊C4C2×D311yescharacteristic
90C2×C42—31no—
91#95—32no—
92#95—32no—
93#95—32no—
94#95—32no—
95#95—32no—
96#95—32no—
97C23×C3—22no—
98C23×C3—22no—
99C23×C3—21no—
100C23×C3—21no—
101C23×C3—21no—
102C23×C3—21no—
103C23×C3—22no—
104C23×C3—22no—
105C23×C3C2×D412yes—
106C23×C3C2×D412yes—
107C23×C3C2411yescharacteristic
108C3×D4—44no—
109C2×C3×C4—21no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—44no—
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—44no—
131C2×Dic3#4114yes—
132Dic6—44no—
133C24C22×D311yescharacteristic
134C24C22×D311yescharacteristic
135C2×Q8—62no—
136C2×Q8—121no—
137C2×Q8—121no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4#6714yes—
141C22×C4—22no—
142C22×C4—32no—
143C22×C4—32no—
144C22×C4—32no—
145C22×C4—32no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—64no—
149C22×C4—121no—
150C22×C4C22×D312yes—
151C22×C4C22×D312yes—
152C22×C4C22×D311yescharacteristic
153C4⋊C4—24no—
154C4⋊C4—64no—
155C4⋊C4—122no—
156C4⋊C4—122no—
157C22⋊C4—22no—
158C22⋊C4—24no—
159C22⋊C4—64no—
160C22⋊C4—64no—
161C22⋊C4—64no—
162C22⋊C4—121no—
163C42—62no—
164C22×C3—22no—
165C22×C3—22no—
166C22×C3—24no—
167C22×C3—24no—
168C22×C3—42no—
169C22×C3—42no—
170C22×C3—41no—
171C22×C3—41no—
172C22×C3—41no—
173C22×C3—41no—
174C22×C3C22×D411yescharacteristic
175C22×C3C2×#4112yes—
176C22×C3H(5,𝔽2)11yescharacteristic
177C22×C3H(5,𝔽2)11yescharacteristic
178C22×C32−1+412yes—
179C3×C4—22no—
180C3×C4—42no—
181C3×C4—42no—
182Dic3—24no—
183Dic3—42no—
184Dic3—42no—
185Dic3—42no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23—22no—
194C23C2×#6712yes—
195C23C2×#6712yes—
196C23C23×D311yescharacteristic
197Q8—124no—
198D4—44no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—34no—
208C2×C4—44no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—124no—
221C2×C4C2×#6712yes—
222C2×C3—42no—
223C2×C3—42no—
224C2×C3#53411yescharacteristic
225C2×C3#54212yes—
226C2×C3#54712yes—
227C2×C3#55111yescharacteristic
228C2×C3#56311yescharacteristic
229C22—22no—
230C22—22no—
231C22—24no—
232C22—24no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22—42no—
239C22#100411yescharacteristic
240C22#100411yescharacteristic
241C22D3×#4112yes—
242C22#101012yes—
243C22C22×#6711yescharacteristic
244C4—22no—
245C4—42no—
246C4—42no—
247C4—64no—
248C4—122no—
249C4—122no—
250C4—122no—
251C3#271711yescharacteristic, complemented, Sylow
252C2—42no—
253C2—42no—
254C2#565111yescharacteristic
255C2#565911yescharacteristic
256C2#566212yes—
257C2#580411yescharacteristic
258C2#582412yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#123C2
prime radicalC22×C3×D4C22
hypercenterC2×#123D3
hypocenterC3#2717
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
4. #5210C2semidirect
5. #4979C2semidirect
6. #4978C2semidirect
7. #4952C2semidirect
8. #4952C2semidirect
9. #4942C2semidirect
251. C3#2717semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×#9362
#52202
#52102
#49792
#49782
#49522
#49422
#27173