Finite Group Databasebeta

← #79615

Subgroups — #79615

Order 384.

366 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#5558C212yesmaximal, complemented
2C3×C4×D8C211yesmaximal, characteristic, complemented
3#5157C211yesmaximal, characteristic, complemented
4#5157C211yesmaximal, characteristic, complemented
5#5009C212yesmaximal, complemented
6#4880C211yesmaximal, characteristic, complemented
7D3×#107C211yesmaximal, characteristic, complemented
8#4779C212yesmaximal, complemented
9#4777C212yesmaximal, complemented
10C8×D12C211yesmaximal, characteristic, complemented
11#4683C211yesmaximal, characteristic, complemented
12#3339—31nomaximal, Sylow
13C2×C3×D8C2211yescharacteristic
14C3×C4×D4C2212yes—
15#938C2212yes—
16#938C2212yes—
17#929—22no—
18C4×#67—22no—
19C2×D3×C8C2211yescharacteristic
20C2×D3×C8C2211yescharacteristic
21#892C2212yes—
22D3×C4⋊C4C2212yes—
23#884—22no—
24#879—22no—
25C4×D12C2211yescharacteristic
26#870C2212yes—
27C3×#107C2211yescharacteristic
28C3×#102C2212yes—
29C3×C4×C8C2211yescharacteristic
30#832C2212yes—
31#832C2212yes—
32(C2×D3)⋊C8C2211yescharacteristic
33(C2×D3)⋊C8C2211yescharacteristic
34#818C2211yescharacteristic
35#818C2211yescharacteristic
36#817C2212yes—
37#808C2212yes—
38#807C2212yes—
39C4⋊(C3⋊C8)C2211yescharacteristic
40#553—32no—
41#499—31no—
42#482—32no—
43#481—32no—
44#478—32no—
45#466—31no—
46#466—31no—
47C4×D8D311yescharacteristic, complemented
48C8×D4—31no—
49#427—31no—
50C2×#107—31no—
51C2×C3×D4C2312yes—
52C22×C3×C4—22no—
53C2×#67—22no—
54C2×#67—22no—
55C2×D12D411yescharacteristic
56C2×C4×D3—22no—
57C2×C4×D3C2311yescharacteristic
58C2×C4×D3C2311yescharacteristic
59C3×D8—22no—
60C2×C3×C8C2311yescharacteristic
61C2×C3×C8C2311yescharacteristic
62C3×C4⋊C4C2312yes—
63C3×C22⋊C4—22no—
64C3×C42C2311yescharacteristic
65#217—22no—
66#217—22no—
67(C2×D3)⋊C4—22no—
68(C2×D3)⋊C4D411yescharacteristic
69(C2×D3)⋊C4D411yescharacteristic
70C4⋊Dic3D411yescharacteristic
71C4⋊Dic3C2312yes—
72C4⋊Dic3C2312yes—
73Dic3⋊C4—22no—
74Dic3⋊C4—22no—
75Dic3⋊C4—22no—
76C4×Dic3—22no—
77C2×C3⋊C8C2311yescharacteristic
78C2×C3⋊C8C2311yescharacteristic
79D3×C8—22no—
80D3×C8—22no—
81C2×D8C2×D311yescharacteristic
82C22×C8—31no—
83C22×C8—31no—
84#126—62no—
85#125—32no—
86#123—62no—
87#121—32no—
88#121—32no—
89#121—62no—
90C4×D4—31no—
91C4×D4—62no—
92C4×D4C2×D312yes—
93#117—32no—
94C2×C4⋊C4—32no—
95#107—31no—
96#107—31no—
97#107—32no—
98#107C2×D311yescharacteristic
99#106—32no—
100#106—32no—
101C4⋊C8—31no—
102#102—32no—
103#102—32no—
104#102C2×D312yes—
105C22⋊C8—31no—
106C22⋊C8—31no—
107C4×C8C2×D311yescharacteristic
108C23×C3—22no—
109C22×D3C2×D411yescharacteristic
110C22×D3C2×D411yescharacteristic
111C3×D4—22no—
112C3×D4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—82no—
115C2×C3×C4C2×D411yescharacteristic
116C2×C3×C4C2×D411yescharacteristic
117C2×C3×C4C2411yescharacteristic
118#67—44no—
119#67—82no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3C2×D411yescharacteristic
123C2×Dic3C2×D411yescharacteristic
124D12—22no—
125C4×D3—21no—
126C4×D3—21no—
127C4×D3—21no—
128C4×D3—21no—
129C4×D3—82no—
130C3×C8—21no—
131C3×C8#4112yes—
132C3⋊C8—21no—
133C3⋊C8—21no—
134C2×D4—31no—
135C2×D4—62no—
136C2×D4—62no—
137C2×D4C22×D312yes—
138C22×C4—22no—
139C22×C4—31no—
140C22×C4—31no—
141C22×C4—62no—
142D8—22no—
143C2×C8—31no—
144C2×C8—31no—
145C2×C8—62no—
146C2×C8—62no—
147C2×C8C22×D311yescharacteristic
148C2×C8C22×D311yescharacteristic
149C4⋊C4—31no—
150C4⋊C4—32no—
151C4⋊C4—32no—
152C4⋊C4—62no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4C22×D312yes—
156C22⋊C4—22no—
157C22⋊C4—31no—
158C22⋊C4—31no—
159C22⋊C4—62no—
160C22⋊C4—62no—
161C22⋊C4—62no—
162C42—62no—
163C42C22×D311yescharacteristic
164C22×C3—42no—
165C22×C3—42no—
166C22×C3—42no—
167C22×C3C22×D411yescharacteristic
168C2×D3—21no—
169C2×D3—21no—
170C2×D3—21no—
171C2×D3—21no—
172C2×D3—21no—
173C2×D3#13512yes—
174C3×C4—21no—
175C3×C4—21no—
176C3×C4—42no—
177C3×C4C2×#4111yescharacteristic
178C3×C42−1+411yescharacteristic
179Dic3—21no—
180Dic3—21no—
181Dic3—42no—
182Dic3—42no—
183C23—22no—
184C23—31no—
185C23—31no—
186D4—22no—
187D4—22no—
188D4—62no—
189D4—124no—
190D4—242no—
191C2×C4—22no—
192C2×C4—31no—
193C2×C4—31no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—82no—
201C2×C4—242no—
202C2×C4D3×D411yescharacteristic
203C2×C4D3×D411yescharacteristic
204C2×C4C23×D311yescharacteristic
205C8—21no—
206C8—61no—
207C8—61no—
208C8#23512yes—
209C2×C3—82no—
210C2×C3#54711yescharacteristic
211C2×C3C2×#13511yescharacteristic
212C2×C3#57611yescharacteristic
213D3—22no—
214D3—41no—
215C22—32no—
216C22—42no—
217C22—42no—
218C22—42no—
219C22—61no—
220C22—61no—
221C22—61no—
222C22—61no—
223C22—61no—
224C22C2×D3×D411yescharacteristic
225C4—21no—
226C4—21no—
227C4—42no—
228C4—61no—
229C4—61no—
230C4—122no—
231C4—122no—
232C4C2×#23511yescharacteristic
233C4#101011yescharacteristic
234C3#333911yescharacteristic, complemented, Sylow
235C2—62no—
236C2—82no—
237C2—121no—
238C2#555111yescharacteristic
239C2#575711yescharacteristic
240C2D3×#13511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
hypercenterC4×D8D3
hypocenterC3#3339
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. #5558C2semidirect
2. C3×C4×D8C2semidirect
3. #5157C2semidirect
4. #5157C2semidirect
5. #5009C2semidirect
6. #4880C2semidirect
7. D3×#107C2semidirect
8. #4779C2semidirect
9. #4777C2semidirect
10. C8×D12C2semidirect
11. #4683C2semidirect
47. C4×D8D3semidirect
234. C3#3339semidirect

Maximal subgroups

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

subgroupindex
#55582
C3×C4×D82
#51572
#50092
#48802
D3×#1072
#47792
#47772
C8×D122
#46832
#33393