Finite Group Databasebeta

← #83018

Subgroups — #83018

Order 384.

334 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 121 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×#125C211yesmaximal, characteristic, complemented
2#5479C211yesmaximal, characteristic
3#5068C212yesmaximal, complemented
4#5066C212yesmaximal, complemented
5#5065C212yesmaximal, complemented
6#5063C212yesmaximal, complemented
7#4971C211yesmaximal, characteristic, complemented
8#4970C211yesmaximal, characteristic, complemented
9#4968C211yesmaximal, characteristic
10#4967C211yesmaximal, characteristic
11#4925C211yesmaximal, characteristic, complemented
12#3232—31nomaximal, Sylow
13C3×#125—21no—
14C3×#125—21no—
15C3×#125C2212yes—
16C3×#125C2212yes—
17C2×C3×C4⋊C4—21no—
18C2×C3×C4⋊C4—21no—
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2211yescharacteristic
21C2×C3×C42C2211yescharacteristic
22#930C2211yescharacteristic
23#929—21no—
24C4×#67—21no—
25#924C2211yescharacteristic
26#876—21no—
27#875C2211yescharacteristic
28C4×D12C2211yescharacteristic
29#872—21no—
30#870C2211yescharacteristic
31C4×Dic6C2211yescharacteristic
32#830C2211yescharacteristic
33#830C2211yescharacteristic
34#810C2212yes—
35#810C2212yes—
36#809C2212yes—
37#809C2212yes—
38#808C2212yes—
39#808C2212yes—
40#807C2212yes—
41#807C2212yes—
42C4⋊(C3⋊C8)C2212yes—
43C4×C3⋊C8C2212yes—
44#529—31no—
45C2×#125D311yescharacteristic, complemented
46#499—32no—
47#488—32no—
48#484—31no—
49#483—31no—
50#481—31no—
51#480—31no—
52#479—32no—
53#478—32no—
54#431—31no—
55C22×C3×C4—21no—
56C22×C3×C4—21no—
57C22×C3×C4D411yescharacteristic
58C22×C3×C4D411yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C2×#67—21no—
61C2×D12C2311yescharacteristic
62C2×C4×D3—21no—
63C2×Dic6C2311yescharacteristic
64C3×C4⋊C4—22no—
65C3×C4⋊C4—22no—
66C3×C4⋊C4—22no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—21no—
69C3×C4⋊C4—21no—
70C3×C4⋊C4C2312yes—
71C3×C4⋊C4C2312yes—
72C3×C42D411yescharacteristic
73C3×C42D411yescharacteristic
74C3×C42C2311yescharacteristic
75C3×C42C2311yescharacteristic
76#217—21no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4—21no—
79(C2×D3)⋊C4—21no—
80(C2×D3)⋊C4—21no—
81C4⋊Dic3C2311yescharacteristic
82C4⋊Dic3C2311yescharacteristic
83Dic3⋊C4—21no—
84Dic3⋊C4—21no—
85Dic3⋊C4—21no—
86Dic3⋊C4—21no—
87C4×Dic3—21no—
88C2×C3⋊C8C2312yes—
89C2×C3⋊C8C2312yes—
90#126—61no—
91#125—21no—
92#125—21no—
93#125—31no—
94#125C2×D312yes—
95#125C2×D312yes—
96#124—31no—
97#123—61no—
98C22⋊Q8—31no—
99#121—31no—
100C4×Q8—31no—
101C4×D4—31no—
102C4×D4—61no—
103#117—61no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4C2×D311yescharacteristic
107C2×C4⋊C4C2×D311yescharacteristic
108C2×C42C2×D311yescharacteristic
109#107—32no—
110#107—32no—
111#106—32no—
112#106—32no—
113C4⋊C8—32no—
114#103—32no—
115#103—32no—
116#102—32no—
117#102—32no—
118C22⋊C8—31no—
119C22⋊C8—31no—
120C4×C8—32no—
121C23×C3C2×D411yescharacteristic
122C22×D3—21no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4—22no—
127C2×C3×C4—41no—
128C2×C3×C4—42no—
129C2×C3×C4—42no—
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2×D411yescharacteristic
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2411yescharacteristic
136#67—42no—
137C2×Dic3—21no—
138C2×Dic3—21no—
139C2×Dic3—21no—
140D12—22no—
141C4×D3—42no—
142Dic6—22no—
143C3⋊C8—22no—
144C3⋊C8—22no—
145C2×Q8—31no—
146C2×D4—31no—
147C2×D4—61no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—61no—
151C22×C4#6711yescharacteristic
152C22×C4#6711yescharacteristic
153C22×C4C22×D311yescharacteristic
154C2×C8—32no—
155C2×C8—32no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—31no—
163C4⋊C4—31no—
164C4⋊C4—61no—
165C4⋊C4—61no—
166C4⋊C4—61no—
167C4⋊C4—61no—
168C4⋊C4C22×D312yes—
169C4⋊C4C22×D312yes—
170C22⋊C4—61no—
171C22⋊C4—61no—
172C22⋊C4—61no—
173C22⋊C4—61no—
174C22⋊C4—61no—
175C42—61no—
176C42#6711yescharacteristic
177C42#6711yescharacteristic
178C42C22×D311yescharacteristic
179C42C22×D311yescharacteristic
180C22×C3—21no—
181C22×C3—21no—
182C22×C3#13512yes—
183C22×C3C22×D411yescharacteristic
184C2×D3—41no—
185C2×D3—41no—
186C2×D3—41no—
187C3×C4—21no—
188C3×C4—21no—
189C3×C4—21no—
190C3×C4—21no—
191C3×C4—21no—
192C3×C4—42no—
193C3×C4—42no—
194C3×C42−1+412yes—
195Dic3—41no—
196Dic3—41no—
197Dic3—41no—
198C23—61no—
199C23C2×#6711yescharacteristic
200Q8—62no—
201D4—62no—
202D4—122no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—61no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—122no—
214C2×C4C2×#6711yescharacteristic
215C2×C4C2×#6711yescharacteristic
216C2×C4C2×#6711yescharacteristic
217C2×C4C2×#6711yescharacteristic
218C2×C4C2×#6711yescharacteristic
219C2×C4C23×D311yescharacteristic
220C8—62no—
221C8—62no—
222C2×C3—22no—
223C2×C3#53611yescharacteristic
224C2×C3C2×#13511yescharacteristic
225C2×C3#57511yescharacteristic
226D3—81no—
227C22—21no—
228C22—21no—
229C22—121no—
230C22—121no—
231C22—121no—
232C22#95012yes—
233C22C22×#6711yescharacteristic
234C4—21no—
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—42no—
240C4—42no—
241C4—121no—
242C4—121no—
243C4—121no—
244C4#100712yes—
245C3#323211yescharacteristic, complemented, Sylow
246C2—22no—
247C2—241no—
248C2#581611yescharacteristic
249C2C2×#95011yescharacteristic
250C2#582211yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#125C2
socleC22×C3C22×D4
prime radical#930C22
hypercenterC2×#125D3
hypocenterC3#3232
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
1. C2×C3×#125C2semidirect
3. #5068C2semidirect
4. #5066C2semidirect
5. #5065C2semidirect
6. #5063C2semidirect
7. #4971C2semidirect
8. #4970C2semidirect
11. #4925C2semidirect
45. C2×#125D3semidirect
245. C3#3232semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1252
#54792
#50682
#50662
#50652
#50632
#49712
#49702
#49682
#49672
#49252
#32323