Finite Group Databasebeta

← #81878

Subgroups — #81878

Order 384.

396 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 109 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#5731C211yesmaximal, characteristic, complemented
2#5674C211yesmaximal, characteristic
3C3×#497C211yesmaximal, characteristic, complemented
4#5187C212yesmaximal, complemented
5#5185C212yesmaximal, complemented
6#5169C212yesmaximal, complemented
7#5163C212yesmaximal, complemented
8#5085C211yesmaximal, characteristic, complemented
9#5056C211yesmaximal, characteristic, complemented
10#4713C211yesmaximal, characteristic, complemented
11#4709C211yesmaximal, characteristic, complemented
12#3222—31nomaximal, Sylow
13C2×#239C2211yescharacteristic
14C2×D3×D4C2211yescharacteristic
15C2×C3×Q16C2212yescomplemented
16C2×C3×SD16C2212yes—
17C3×C4⋊Q8C2211yescharacteristic, complemented
18C3×#124C2211yescharacteristic
19#947C2211yescharacteristic
20#946C2212yes—
21Q8×Dic3C2212yescomplemented
22C2×#215C2212yes—
23C2×#215C2212yescomplemented
24#940C2211yescharacteristic
25#898—22no—
26#889—22no—
27#884—22no—
28#882—22no—
29#874C2211yescharacteristic
30#872C2211yescharacteristic
31#870C2211yescharacteristic, complemented
32C3×#97C2211yescharacteristic, complemented
33#835C2212yescomplemented
34#832C2212yes—
35#821C2212yes—
36#821C2212yescomplemented
37(C2×D3)⋊C8C2212yes—
38Dic3⋊C8C2212yescomplemented
39C4×C3⋊C8C2211yescharacteristic, complemented
40#541—31no—
41#535—31no—
42#497D311yescharacteristic, complemented
43#492—31no—
44#490—31no—
45#488—31no—
46#464—32no—
47#460—32no—
48#452—32no—
49#450—32no—
50#433—31no—
51C23×D3—21no—
52C2×C3×Q8C2311yescharacteristic
53C2×C3×Q8C2312yes—
54C2×C3×D4C2311yescharacteristic
55C2×#67—21no—
56C2×#67—22no—
57#239—42no—
58D3×D4—42no—
59C2×D12—21no—
60C2×D12—22no—
61C2×D12C2311yescharacteristic
62C2×D12C2311yescharacteristic
63C2×C4×D3—21no—
64C2×C4×D3C2311yescharacteristic
65C3×Q16—42no—
66C3×SD16—42no—
67C2×C3×C8C2312yes—
68C3×C4⋊C4—22no—
69C3×C22⋊C4—22no—
70C3×C42C2311yescharacteristic
71#215—42no—
72#215—42no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4D412yes—
76C4⋊Dic3—22no—
77C4⋊Dic3C2312yes—
78Dic3⋊C4—22no—
79Dic3⋊C4—22no—
80Dic3⋊C4D412yes—
81C4×Dic3—22no—
82C4×Dic3C2311yescharacteristic
83C2×C3⋊C8C2312yes—
84C2×#41—31no—
85C22×D4—31no—
86C2×Q16C2×D312yescomplemented
87C2×SD16—32no—
88C2×SD16—32no—
89C2×SD16C2×D312yes—
90C4⋊Q8C2×D311yescharacteristic, complemented
91#127—31no—
92#127—31no—
93#126—62no—
94#125—31no—
95#125—62no—
96#124—31no—
97#124C2×D311yescharacteristic
98C22⋊Q8—32no—
99#121—62no—
100#120—62no—
101C4×Q8—32no—
102#117—31no—
103C4⋊C8—32no—
104#103—32no—
105#102—32no—
106#102—32no—
107#102—32no—
108C22⋊C8—32no—
109#97C2×D311yescharacteristic, complemented
110C4×C8—31no—
111C23×C3—21no—
112C22×D3—21no—
113C22×D3—21no—
114C22×D3—42no—
115C22×D3—82no—
116C22×D3C2×D411yescharacteristic
117C3×Q8—22no—
118C3×Q8—22no—
119C3×Q8—22no—
120C3×D4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4C2×D412yes—
124C2×C3×C4C2411yescharacteristic
125#67—42no—
126#67—82no—
127C2×Dic3—22no—
128C2×Dic3C2×D412yes—
129C2×Dic3C2×D411yescharacteristic
130D12—22no—
131D12—22no—
132D12—42no—
133D12—82no—
134C4×D3—22no—
135C4×D3—42no—
136C3×C8—22no—
137C3⋊C8—22no—
138C24—61no—
139#41—122no—
140C2×Q8C22×D311yescharacteristic
141C2×Q8C22×D312yes—
142C2×D4—31no—
143C2×D4—31no—
144C2×D4—61no—
145C2×D4—61no—
146C2×D4—62no—
147C2×D4—62no—
148C2×D4—122no—
149C2×D4C22×D311yescharacteristic
150C22×C4—31no—
151C22×C4—61no—
152Q16—42no—
153SD16—42no—
154SD16—122no—
155SD16—122no—
156C2×C8—32no—
157C2×C8C22×D312yes—
158C4⋊C4—22no—
159C4⋊C4—32no—
160C4⋊C4—32no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4—62no—
164C22⋊C4—22no—
165C22⋊C4—32no—
166C22⋊C4—62no—
167C22⋊C4—62no—
168C42—31no—
169C42—62no—
170C42C22×D311yescharacteristic
171C22×C3—41no—
172C22×C3—42no—
173C22×C3C22×D411yescharacteristic
174C2×D3—21no—
175C2×D3—22no—
176C2×D3—41no—
177C2×D3—41no—
178C2×D3—42no—
179C2×D3—42no—
180C2×D3—84no—
181C3×C4—22no—
182C3×C4—41no—
183C3×C4—42no—
184C3×C4H(5,𝔽2)12yes—
185Dic3—21no—
186Dic3—22no—
187Dic3—42no—
188C23—21no—
189C23—31no—
190C23—61no—
191C23—61no—
192C23—122no—
193C23—242no—
194Q8—22no—
195Q8—22no—
196Q8—22no—
197D4—22no—
198D4—62no—
199D4—62no—
200D4—122no—
201D4—122no—
202D4—242no—
203D4—242no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—32no—
207C2×C4—31no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—122no—
211C2×C4D3×D412yes—
212C2×C4C23×D311yescharacteristic
213C8—22no—
214C8—62no—
215C2×C3—81no—
216C2×C3#53511yescharacteristic
217C2×C3#57712yes—
218D3—41no—
219D3—81no—
220D3—81no—
221C22—41no—
222C22—42no—
223C22—61no—
224C22—62no—
225C22—121no—
226C22—121no—
227C22—122no—
228C22—122no—
229C22—244no—
230C22C2×D3×D411yescharacteristic
231C4—22no—
232C4—41no—
233C4—42no—
234C4—61no—
235C4—62no—
236C4—122no—
237C4#100412yes—
238C3#322211yescharacteristic, complemented, Sylow
239C2—81no—
240C2—121no—
241C2—241no—
242C2—241no—
243C2#571511yescharacteristic
244C2#577812yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#497C2
socleC22×C3C22×D4
prime radical#5674C2
hypercenter#497D3
hypocenterC3#3222
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #5731C2semidirect
3. C3×#497C2semidirect
4. #5187C2semidirect
5. #5185C2semidirect
6. #5169C2semidirect
7. #5163C2semidirect
8. #5085C2semidirect
9. #5056C2semidirect
10. #4713C2semidirect
11. #4709C2semidirect
15. C2×C3×Q16C22semidirect
17. C3×C4⋊Q8C22semidirect
21. Q8×Dic3C22semidirect
23. C2×#215C22semidirect
31. #870C22semidirect
32. C3×#97C22semidirect
33. #835C22semidirect
36. #821C22semidirect
38. Dic3⋊C8C22semidirect
39. C4×C3⋊C8C22semidirect
42. #497D3semidirect
86. C2×Q16C2×D3semidirect
90. C4⋊Q8C2×D3semidirect
109. #97C2×D3semidirect
238. C3#3222semidirect

Maximal subgroups

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

subgroupindex
#57312
#56742
C3×#4972
#51872
#51852
#51692
#51632
#50852
#50562
#47132
#47092
#32223