Finite Group Databasebeta

← #79600

Subgroups — #79600

Order 384.

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

normal subgroups only

This group has 362 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#5561C212yesmaximal, complemented
2C3×C4×D8C211yesmaximal, characteristic, complemented
3#5157C211yesmaximal, characteristic
4#5155C211yesmaximal, characteristic, complemented
5#5018C212yesmaximal, complemented
6#4887C211yesmaximal, characteristic, complemented
7#4874C211yesmaximal, characteristic, complemented
8#4780C212yesmaximal, complemented
9#4765C212yesmaximal, complemented
10C4×Dic12C211yesmaximal, characteristic, complemented
11#4689C211yesmaximal, characteristic, complemented
12#3175—31nomaximal, Sylow
13C2×C3×D8C2211yescharacteristic
14C3×C4×D4C2212yes—
15#938C2212yes—
16#936C2212yes—
17C2×#214C2212yes—
18#929—22no—
19#924—22no—
20C2×Dic12C2211yescharacteristic
21C2×ΓL(1,𝔽25)C2211yescharacteristic
22C2×D3×C8C2211yescharacteristic
23#897C2212yes—
24#888C2212yes—
25#885—22no—
26#879—22no—
27#872C2211yescharacteristic
28C4×Dic6C2212yes—
29C3×#107C2211yescharacteristic
30C3×#102C2212yes—
31C3×C4×C8C2211yescharacteristic
32#832C2212yes—
33#818C2211yescharacteristic
34#816C2212yes—
35#815C2211yescharacteristic
36C8×Dic3C2211yescharacteristic
37#810C2212yes—
38#807C2212yes—
39#803C2211yescharacteristic
40#542—32no—
41#500—31no—
42#495—31no—
43#489—32no—
44#468—32no—
45#467—31no—
46#466—31no—
47#440—32no—
48C4×Q16—31no—
49C4×D8D311yescharacteristic, complemented
50#405—31no—
51C2×C3×D4C2312yes—
52C22×C3×C4—22no—
53C2×#67—22no—
54C2×C4×D3C2311yescharacteristic
55C2×Dic6—22no—
56C2×Dic6C2312yes—
57C3×D8—22no—
58C2×C3×C8C2311yescharacteristic
59C2×C3×C8C2311yescharacteristic
60C3×C4⋊C4C2312yes—
61C3×C22⋊C4—22no—
62C3×C42C2311yescharacteristic
63#217—22no—
64#217—22no—
65#217—22no—
66#214—24no—
67(C2×D3)⋊C4—22no—
68(C2×D3)⋊C4D411yescharacteristic
69(C2×D3)⋊C4D411yescharacteristic
70C4⋊Dic3—22no—
71C4⋊Dic3C2312yes—
72Dic3⋊C4—22no—
73Dic3⋊C4—22no—
74Dic3⋊C4D411yescharacteristic
75Dic3⋊C4D411yescharacteristic
76C4×Dic3—22no—
77C4×Dic3C2311yescharacteristic
78C2×C3⋊C8C2311yescharacteristic
79C2×C3⋊C8C2311yescharacteristic
80Dic12—22no—
81ΓL(1,𝔽25)—22no—
82D3×C8—22no—
83C2×Q16—31no—
84C2×SD16—32no—
85C2×D8C2×D311yescharacteristic
86C2×M16—31no—
87C22×C8—31no—
88C4⋊Q8—32no—
89#126—62no—
90#124—32no—
91#124—62no—
92#123—62no—
93C22⋊Q8—32no—
94C22⋊Q8—62no—
95#121—32no—
96C4×Q8—32no—
97C4×D4C2×D312yes—
98#117—31no—
99#107—31no—
100#107—32no—
101#107C2×D311yescharacteristic
102#103—32no—
103#103—32no—
104#102—32no—
105#102C2×D312yes—
106#97—31no—
107#97—31no—
108C4×C8—31no—
109C4×C8C2×D311yescharacteristic
110C23×C3—22no—
111C22×D3C2×D411yescharacteristic
112C3×D4—22no—
113C3×D4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—82no—
116C2×C3×C4C2×D411yescharacteristic
117C2×C3×C4C2×D411yescharacteristic
118C2×C3×C4C2411yescharacteristic
119#67—82no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3C2×D411yescharacteristic
123C2×Dic3C2×D411yescharacteristic
124C2×Dic3C2×D411yescharacteristic
125C4×D3—21no—
126C4×D3—21no—
127Dic6—22no—
128Dic6—22no—
129Dic6—82no—
130C3×C8—21no—
131C3×C8#4112yes—
132C3⋊C8—21no—
133C3⋊C8#4112yes—
134C2×Q8—32no—
135C2×Q8—62no—
136C2×D4—62no—
137C2×D4C22×D312yes—
138C22×C4—22no—
139C22×C4—31no—
140Q16—62no—
141SD16—64no—
142D8—22no—
143M16—62no—
144C2×C8—31no—
145C2×C8—31no—
146C2×C8—62no—
147C2×C8C22×D311yescharacteristic
148C2×C8C22×D311yescharacteristic
149C4⋊C4—31no—
150C4⋊C4—31no—
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—
162C22⋊C4—62no—
163C42—31no—
164C42—62no—
165C42C22×D311yescharacteristic
166C22×C3—42no—
167C22×C3—42no—
168C22×C3—42no—
169C22×C3C22×D411yescharacteristic
170C2×D3—21no—
171C2×D3—21no—
172C2×D3—21no—
173C3×C4—21no—
174C3×C4—21no—
175C3×C4—42no—
176C3×C4C2×#4111yescharacteristic
177C3×C4C2×#4111yescharacteristic
178Dic3—21no—
179Dic3—21no—
180Dic3—21no—
181Dic3—42no—
182Dic3—42no—
183C23—22no—
184C23—31no—
185Q8—62no—
186Q8—62no—
187Q8—242no—
188D4—22no—
189D4—22no—
190D4—242no—
191C2×C4—22no—
192C2×C4—31no—
193C2×C4—31no—
194C2×C4—31no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—62no—
198C2×C4—62no—
199C2×C4—82no—
200C2×C4D3×D411yescharacteristic
201C2×C4D3×D411yescharacteristic
202C2×C4C23×D311yescharacteristic
203C8—21no—
204C8—32no—
205C8—61no—
206C8#23512yes—
207C2×C3—82no—
208C2×C3#53211yescharacteristic
209C2×C3#57611yescharacteristic
210C2×C3#57811yescharacteristic
211D3—41no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—61no—
216C22—61no—
217C22—61no—
218C22C2×D3×D411yescharacteristic
219C4—21no—
220C4—21no—
221C4—42no—
222C4—61no—
223C4—61no—
224C4—61no—
225C4—122no—
226C4—122no—
227C4C2×#23511yescharacteristic
228C4D3×#4111yescharacteristic
229C3#317511yescharacteristic, complemented, Sylow
230C2—82no—
231C2—121no—
232C2#554811yescharacteristic
233C2#575711yescharacteristic
234C2#577111yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×D8C2
socleC22×C3C22×D4
prime radical#5157C2
hypercenterC4×D8D3
hypocenterC3#3175
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. #5561C2semidirect
2. C3×C4×D8C2semidirect
4. #5155C2semidirect
5. #5018C2semidirect
6. #4887C2semidirect
7. #4874C2semidirect
8. #4780C2semidirect
9. #4765C2semidirect
10. C4×Dic12C2semidirect
11. #4689C2semidirect
49. C4×D8D3semidirect
229. C3#3175semidirect

Maximal subgroups

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

subgroupindex
#55612
C3×C4×D82
#51572
#51552
#50182
#48872
#48742
#47802
#47652
C4×Dic122
#46892
#31753