Finite Group Databasebeta

← #78504

Subgroups — #78504

Order 384.

366 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 129 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#5479C211yesmaximal, characteristic
2#5470C211yesmaximal, characteristic, complemented
3C2×C3×#97C211yesmaximal, characteristic, complemented
4#5116C212yesmaximal, complemented
5#5114C212yesmaximal
6#4716C212yesmaximal, complemented
7#4715C211yesmaximal, characteristic, complemented
8#4714C211yesmaximal, characteristic, complemented
9#4710C212yesmaximal, complemented
10#4703C211yesmaximal, characteristic, complemented
11#4702C211yesmaximal, characteristic, complemented
12#3181—31nomaximal, Sylow
13C22×C3×C8C2212yes—
14C2×C3×C42C2211yescharacteristic
15#930C2211yescharacteristic
16#929—21no—
17C4×#67—21no—
18#926—21no—
19#924—21no—
20#924C2212yes—
21#924C2211yescharacteristic
22C2×Dic12C2212yes—
23C2×#204C2212yes—
24#876—21no—
25#875C2211yescharacteristic
26C4×D12C2211yescharacteristic
27#872—21no—
28#870—21no—
29#870C2211yescharacteristic
30#869C2211yescharacteristic
31#869C2211yescharacteristic
32C4×Dic6—21no—
33C4×Dic6C2212yes—
34C4×Dic6C2211yescharacteristic
35C3×#97C2211yescharacteristic
36C3×#97C2211yescharacteristic
37C3×#97C2212yes—
38#821C2212yes—
39#818C2212yes—
40#817C2212yes—
41#816C2212yes—
42#816C2212yes—
43#816C2212yes—
44#533—31no—
45#529—31no—
46#501—31no—
47#497—31no—
48#491—31no—
49#489—31no—
50#467—32no—
51#465—32no—
52#441—32no—
53#440—32no—
54C2×#97D311yescharacteristic, complemented
55C22×C3×C4D411yescharacteristic
56C22×C3×C4D411yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C2×#67—21no—
59C2×D12C2311yescharacteristic
60C2×C4×D3—21no—
61C2×Dic6—21no—
62C2×Dic6C2312yes—
63C2×Dic6C2311yescharacteristic
64C2×C3×C8—24no—
65C2×C3×C8C2312yes—
66C2×C3×C8C2312yes—
67C3×C42D411yescharacteristic
68C3×C42D411yescharacteristic
69C3×C42C2311yescharacteristic
70C3×C42C2311yescharacteristic
71#217—22no—
72#217—21no—
73(C2×D3)⋊C4—21no—
74(C2×D3)⋊C4—21no—
75(C2×D3)⋊C4—21no—
76(C2×D3)⋊C4—21no—
77C4⋊Dic3—21no—
78C4⋊Dic3—21no—
79C4⋊Dic3—21no—
80C4⋊Dic3C2311yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82C4⋊Dic3C2312yes—
83Dic3⋊C4—22no—
84Dic3⋊C4—22no—
85Dic3⋊C4—21no—
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88Dic3⋊C4—21no—
89C4×Dic3—22no—
90C4×Dic3—21no—
91Dic12—24no—
92#204—24no—
93C2×Q16—32no—
94C2×SD16—32no—
95C22×C8C2×D312yes—
96C4⋊Q8—31no—
97C4⋊Q8—31no—
98#126—61no—
99#125—31no—
100#125—61no—
101#124—31no—
102#123—61no—
103C22⋊Q8—31no—
104C22⋊Q8—32no—
105C22⋊Q8—61no—
106#121—31no—
107C4×Q8—31no—
108C4×Q8—32no—
109C4×Q8—61no—
110C4×D4—31no—
111C4×D4—61no—
112#117—61no—
113#117—61no—
114C2×C42C2×D311yescharacteristic
115#107—32no—
116#106—32no—
117#103—32no—
118#103—32no—
119#103—32no—
120#102—32no—
121#97C2×D312yes—
122#97C2×D311yescharacteristic
123#97C2×D311yescharacteristic
124C23×C3C2×D411yescharacteristic
125C22×D3—21no—
126C2×C3×C4—22no—
127C2×C3×C4—22no—
128C2×C3×C4—41no—
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2×D411yescharacteristic
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2411yescharacteristic
135#67—42no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3—22no—
139C2×Dic3—21no—
140C2×Dic3—21no—
141D12—22no—
142C4×D3—42no—
143Dic6—22no—
144Dic6—22no—
145Dic6—22no—
146Dic6—42no—
147C3×C8—22no—
148C3×C8#4114yes—
149C2×Q8—31no—
150C2×Q8—32no—
151C2×Q8—61no—
152C2×D4—31no—
153C2×D4—61no—
154C22×C4—61no—
155C22×C4D1211yescharacteristic
156C22×C4D1211yescharacteristic
157C22×C4C22×D311yescharacteristic
158Q16—64no—
159SD16—64no—
160C2×C8—24no—
161C2×C8C22×D312yes—
162C2×C8C22×D312yes—
163C4⋊C4—32no—
164C4⋊C4—31no—
165C4⋊C4—31no—
166C4⋊C4—61no—
167C4⋊C4—61no—
168C4⋊C4—61no—
169C4⋊C4—61no—
170C4⋊C4—61no—
171C4⋊C4—62no—
172C4⋊C4—61no—
173C4⋊C4—62no—
174C4⋊C4—61no—
175C22⋊C4—62no—
176C22⋊C4—61no—
177C22⋊C4—61no—
178C22⋊C4—61no—
179C22⋊C4—61no—
180C22⋊C4—61no—
181C42—61no—
182C42—62no—
183C42D1211yescharacteristic
184C42D1211yescharacteristic
185C42C22×D311yescharacteristic
186C42C22×D311yescharacteristic
187C22×C3—21no—
188C22×C3—21no—
189C22×C3#13712yes—
190C22×C3C22×D411yescharacteristic
191C2×D3—41no—
192C2×D3—41no—
193C2×D3—41no—
194C3×C4—21no—
195C3×C4—21no—
196C3×C4—21no—
197C3×C4—21no—
198C3×C4—21no—
199C3×C4C2×#4112yes—
200Dic3—42no—
201Dic3—41no—
202Dic3—41no—
203Dic3—42no—
204Dic3—41no—
205C23—61no—
206C23C2×D1211yescharacteristic
207Q8—62no—
208Q8—62no—
209Q8—62no—
210Q8—122no—
211D4—62no—
212D4—122no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—61no—
221C2×C4—122no—
222C2×C4C2×D1211yescharacteristic
223C2×C4C2×D1211yescharacteristic
224C2×C4C2×D1211yescharacteristic
225C2×C4C2×D1211yescharacteristic
226C2×C4C2×D1211yescharacteristic
227C2×C4C23×D311yescharacteristic
228C8—22no—
229C8#23514yes—
230C2×C3—22no—
231C2×C3#53211yescharacteristic
232C2×C3C2×#13711yescharacteristic
233C2×C3#57511yescharacteristic
234D3—81no—
235C22—21no—
236C22—21no—
237C22—121no—
238C22—121no—
239C22—121no—
240C22#90912yes—
241C22C22×D1211yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—121no—
248C4—122no—
249C4—121no—
250C4—121no—
251C4—122no—
252C4C2×#23512yes—
253C3#318111yescharacteristic, complemented, Sylow
254C2—22no—
255C2—241no—
256C2#547711yescharacteristic
257C2C2×#90911yescharacteristic
258C2#574811yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#97C2
socleC22×C3C22×D4
prime radical#930C22
hypercenterC2×#97D3
hypocenterC3#3181
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#97

Complements

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

NH ≅ G/Ntype
2. #5470C2semidirect
3. C2×C3×#97C2semidirect
4. #5116C2semidirect
6. #4716C2semidirect
7. #4715C2semidirect
8. #4714C2semidirect
9. #4710C2semidirect
10. #4703C2semidirect
11. #4702C2semidirect
54. C2×#97D3semidirect
253. C3#3181semidirect

Maximal subgroups

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

subgroupindex
#54792
#54702
C2×C3×#972
#51162
#51142
#47162
#47152
#47142
#47102
#47032
#47022
#31813