Finite Group Databasebeta

← #81720

Subgroups — #81720

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#5715C211yesmaximal, characteristic
2#5698C211yesmaximal, characteristic, complemented
3C3×#490C211yesmaximal, characteristic, complemented
4#5076C211yesmaximal, characteristic, complemented
5#5068C211yesmaximal, characteristic, complemented
6#4783C212yesmaximal, complemented
7#4776C212yesmaximal, complemented
8#4767C212yesmaximal, complemented
9#4762C212yesmaximal, complemented
10#4715C211yesmaximal, characteristic, complemented
11#4709C211yesmaximal, characteristic, complemented
12#3222—31nomaximal, Sylow
13C2×#237C2211yescharacteristic
14C2×D3×D4C2211yescharacteristic
15C3×#127C2211yescharacteristic
16C3×#125C2211yescharacteristic, complemented
17#940C2211yescharacteristic
18#939—22no—
19#937—22no—
20#936C2211yescharacteristic
21C2×#214C2212yes—
22C2×#214C2212yescomplemented
23C2×Dic12C2212yescomplemented
24C2×#204C2212yes—
25#898—22no—
26#897C2212yes—
27#890—22no—
28#887C2212yescomplemented
29#875C2211yescharacteristic
30#872C2211yescharacteristic
31#869C2211yescharacteristic, complemented
32C3×#102C2212yes—
33C3×#102C2212yescomplemented
34C3×#97C2211yescharacteristic, complemented
35(C2×D3)⋊C8C2212yes—
36Dic3⋊C8C2212yescomplemented
37#810C2212yescomplemented
38#809C2212yes—
39C4×C3⋊C8C2211yescharacteristic, complemented
40#541—31no—
41#535—31no—
42#497—31no—
43#492—31no—
44#490D311yescharacteristic, complemented
45#488—31no—
46#464—32no—
47#460—32no—
48#452—32no—
49#450—32no—
50#433—31no—
51C23×D3—21no—
52C2×C3×D4—22no—
53C2×C3×D4C2311yescharacteristic
54C2×C3×D4C2311yescharacteristic
55C2×#67—21no—
56C2×#67—21no—
57C2×#67—22no—
58C22×Dic3—21no—
59#237—42no—
60D3×D4—42no—
61C2×D12C2311yescharacteristic
62C2×C4×D3C2311yescharacteristic
63C2×Dic6C2311yescharacteristic
64C2×Dic6C2312yes—
65C2×C3×C8C2312yes—
66C3×C4⋊C4—22no—
67C3×C4⋊C4C2312yes—
68C3×C42C2311yescharacteristic
69#217—22no—
70#214—42no—
71#214—42no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4D412yes—
75C4⋊Dic3—22no—
76C4⋊Dic3—22no—
77Dic3⋊C4—22no—
78Dic3⋊C4D412yes—
79C4×Dic3—22no—
80C4×Dic3C2311yescharacteristic
81C2×C3⋊C8C2312yes—
82Dic12—42no—
83#204—42no—
84C2×#41—31no—
85C22×D4—31no—
86C2×Q16—32no—
87C2×SD16—32no—
88C2×SD16—32no—
89C2×SD16—32no—
90C4⋊Q8—31no—
91#127—31no—
92#127C2×D311yescharacteristic
93#126—62no—
94#125—62no—
95#125C2×D311yescharacteristic, complemented
96#124—31no—
97#124—31no—
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#102C2×D312yes—
107#102C2×D312yescomplemented
108C22⋊C8—32no—
109#97C2×D311yescharacteristic, complemented
110C4×C8—31no—
111C23×C3—21no—
112C23×C3—21no—
113C22×D3—21no—
114C22×D3—42no—
115C22×D3—82no—
116C22×D3C2×D411yescharacteristic
117C3×D4—22no—
118C3×D4—22no—
119C3×D4—82no—
120C2×C3×C4—22no—
121C2×C3×C4C2×D412yes—
122C2×C3×C4C2411yescharacteristic
123#67—42no—
124#67—42no—
125#67—82no—
126C2×Dic3—22no—
127C2×Dic3—21no—
128C2×Dic3—42no—
129C2×Dic3C2×D412yes—
130C2×Dic3C2×D411yescharacteristic
131D12—22no—
132C4×D3—22no—
133Dic6—22no—
134Dic6—22no—
135Dic6—22no—
136C3×C8—22no—
137C3⋊C8—22no—
138C24—61no—
139#41—122no—
140C2×Q8—31no—
141C2×Q8—32no—
142C2×D4—22no—
143C2×D4—31no—
144C2×D4—61no—
145C2×D4—62no—
146C2×D4—61no—
147C2×D4—122no—
148C2×D4C22×D311yescharacteristic
149C2×D4C22×D311yescharacteristic
150C22×C4—31no—
151C22×C4—61no—
152Q16—122no—
153SD16—122no—
154SD16—122no—
155SD16—122no—
156C2×C8—32no—
157C2×C8C22×D312yes—
158C4⋊C4—22no—
159C4⋊C4—32no—
160C4⋊C4—62no—
161C4⋊C4—62no—
162C4⋊C4—62no—
163C4⋊C4C22×D312yes—
164C22⋊C4—32no—
165C22⋊C4—62no—
166C22⋊C4—62no—
167C22⋊C4—62no—
168C42—31no—
169C42—62no—
170C42C22×D311yescharacteristic
171C22×C3—41no—
172C22×C3—41no—
173C22×C3—42no—
174C22×C3—42no—
175C22×C3C22×D411yescharacteristic
176C2×D3—21no—
177C2×D3—22no—
178C2×D3—41no—
179C2×D3—42no—
180C2×D3—84no—
181C3×C4—22no—
182C3×C4—42no—
183C3×C4H(5,𝔽2)12yes—
184Dic3—21no—
185Dic3—22no—
186Dic3—42no—
187Dic3—41no—
188C23—21no—
189C23—21no—
190C23—31no—
191C23—61no—
192C23—122no—
193C23—242no—
194Q8—62no—
195Q8—62no—
196Q8—62no—
197D4—22no—
198D4—22no—
199D4—62no—
200D4—82no—
201D4—122no—
202D4—122no—
203D4—242no—
204C2×C4—22no—
205C2×C4—32no—
206C2×C4—31no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—61no—
210C2×C4—122no—
211C2×C4D3×D412yes—
212C2×C4C23×D311yescharacteristic
213C8—22no—
214C8—62no—
215C2×C3—81no—
216C2×C3—81no—
217C2×C3#53511yescharacteristic
218C2×C3#57712yes—
219D3—41no—
220D3—81no—
221C22—41no—
222C22—41no—
223C22—42no—
224C22—42no—
225C22—61no—
226C22—62no—
227C22—121no—
228C22—122no—
229C22—244no—
230C22C2×D3×D411yescharacteristic
231C4—22no—
232C4—42no—
233C4—61no—
234C4—62no—
235C4—121no—
236C4—122no—
237C4#100912yes—
238C3#322211yescharacteristic, complemented, Sylow
239C2—81no—
240C2—81no—
241C2—121no—
242C2—241no—
243C2#567411yescharacteristic
244C2#577512yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#490C2
socleC22×C3C22×D4
prime radical#5715C2
hypercenter#490D3
hypocenterC3#3222
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. #5698C2semidirect
3. C3×#490C2semidirect
4. #5076C2semidirect
5. #5068C2semidirect
6. #4783C2semidirect
7. #4776C2semidirect
8. #4767C2semidirect
9. #4762C2semidirect
10. #4715C2semidirect
11. #4709C2semidirect
16. C3×#125C22semidirect
22. C2×#214C22semidirect
23. C2×Dic12C22semidirect
28. #887C22semidirect
31. #869C22semidirect
33. C3×#102C22semidirect
34. C3×#97C22semidirect
36. Dic3⋊C8C22semidirect
37. #810C22semidirect
39. C4×C3⋊C8C22semidirect
44. #490D3semidirect
95. #125C2×D3semidirect
107. #102C2×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
#57152
#56982
C3×#4902
#50762
#50682
#47832
#47762
#47672
#47622
#47152
#47092
#32223