Finite Group Databasebeta

← #80906

Subgroups — #80906

Order 384.

484 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#919C211yesmaximal, characteristic, complemented
2C2×#907C211yesmaximal, characteristic, complemented
3#5629C212yesmaximal, complemented
4C3×#470C211yesmaximal, characteristic, complemented
5#5189C211yesmaximal, characteristic, complemented
6#5135C211yesmaximal, characteristic, complemented
7#5048C212yesmaximal, complemented
8#4865C211yesmaximal, characteristic, complemented
9D3×#106C211yesmaximal, characteristic, complemented
10#4807C212yesmaximal, complemented
11#4803C212yesmaximal, complemented
12#3060—31nomaximal, Sylow
13C2×#239C2212yes—
14C2×#235C2211yescharacteristic
15C2×C3×Q16C2211yescharacteristic
16C2×C3×M16C2211yescharacteristic
17C3×C22⋊Q8C2212yes—
18#946C2212yes—
19C2×#215C2212yes—
20#930C2212yes—
21C22×C3⋊C8C2211yescharacteristic
22#919—24no—
23#907—24no—
24C2×D24C2211yescharacteristic
25C2×ΓL(1,𝔽25)C2211yescharacteristic
26C2×ΓL(1,𝔽25)C2211yescharacteristic
27C2×D3×C8C2211yescharacteristic
28C2×D3×C8C2211yescharacteristic
29#896—22no—
30#895—22no—
31#895C2212yes—
32D3×C4⋊C4C2212yes—
33#884—22no—
34#884—22no—
35C3×#106C2211yescharacteristic
36C3×#103C2212yes—
37#835C2212yes—
38#821C2212yes—
39#817C2211yescharacteristic
40#809C2212yes—
41#808C2212yes—
42C2×#135—31no—
43C2×#131—31no—
44#537—32no—
45#470—31no—
46#470D311yescharacteristic, complemented
47#469—31no—
48#469—31no—
49#465—32no—
50#465—32no—
51C2×#106—31no—
52#417—32no—
53C2×C3×Q8C2312yes—
54C22×C3×C4C2311yescharacteristic
55C2×#67—22no—
56C2×#67D411yescharacteristic
57C2×#67D411yescharacteristic
58#239—24no—
59#239—24no—
60#235—22no—
61#235—22no—
62C2×D12—22no—
63C2×D12—22no—
64C2×D12D411yescharacteristic
65C2×D12C2312yes—
66C2×C4×D3—22no—
67C2×C4×D3—22no—
68C2×C4×D3C2311yescharacteristic
69C2×C4×D3C2311yescharacteristic
70C2×Dic6D411yescharacteristic
71C3×Q16—22no—
72C3×M16—22no—
73C2×C3×C8C2311yescharacteristic
74C2×C3×C8C2311yescharacteristic
75C3×C4⋊C4—22no—
76C3×C4⋊C4C2312yes—
77C3×C22⋊C4—22no—
78#215—24no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—22no—
82C4⋊Dic3C2312yes—
83Dic3⋊C4—22no—
84Dic3⋊C4—22no—
85C2×C3⋊C8—22no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88D24—22no—
89ΓL(1,𝔽25)—22no—
90ΓL(1,𝔽25)—22no—
91D3×C8—22no—
92D3×C8D414yescomplemented
93C2×#41—31no—
94C2×#41—32no—
95#135—64no—
96C2×Q16C2×D311yescharacteristic
97C2×SD16—32no—
98C2×D8—31no—
99#131—64no—
100C2×M16—31no—
101C2×M16—31no—
102C2×M16C2×D311yescharacteristic
103C22×C8—31no—
104C22×C8—31no—
105C22×C8—31no—
106C22⋊Q8—32no—
107C22⋊Q8—62no—
108C22⋊Q8C2×D312yes—
109#121—32no—
110#121—32no—
111#121—62no—
112#121—62no—
113#121—62no—
114C2×C4⋊C4—32no—
115#106—31no—
116#106—32no—
117#106C2×D311yescharacteristic
118#103—32no—
119#103C2×D312yes—
120#102—32no—
121#102—32no—
122C23×C3C2×D411yescharacteristic
123C22×D3—22no—
124C22×D3C2×D411yescharacteristic
125C22×D3C2×D411yescharacteristic
126C3×Q8—22no—
127C3×Q8—22no—
128C2×C3×C4—21no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2411yescharacteristic
134#67—22no—
135#67—22no—
136#67—82no—
137C2×Dic3—22no—
138C2×Dic3C2×D411yescharacteristic
139C2×Dic3C2×D411yescharacteristic
140D12—22no—
141D12—22no—
142D12—22no—
143D12—44no—
144D12—82no—
145C4×D3—21no—
146C4×D3—21no—
147C4×D3—44no—
148C4×D3—82no—
149C4×D3C2×D412yes—
150C4×D3#4112yes—
151Dic6—22no—
152C3×C8—21no—
153C3×C8C2×D412yes—
154C3⋊C8—21no—
155C3⋊C8C2×D412yes—
156#41—62no—
157#41—62no—
158#41—64no—
159#41—64no—
160C2×Q8—31no—
161C2×Q8C22×D312yes—
162C2×D4—31no—
163C2×D4—31no—
164C2×D4—31no—
165C2×D4—32no—
166C2×D4—62no—
167C2×D4—62no—
168C2×D4—62no—
169C22×C4—31no—
170C22×C4—31no—
171C22×C4—62no—
172C22×C4—62no—
173C22×C4C22×D311yescharacteristic
174Q16—22no—
175SD16—64no—
176D8—62no—
177M16—22no—
178M16—62no—
179M16—62no—
180C2×C8—31no—
181C2×C8—34no—
182C2×C8—31no—
183C2×C8—62no—
184C2×C8—62no—
185C2×C8C22×D311yescharacteristic
186C2×C8C22×D311yescharacteristic
187C4⋊C4—22no—
188C4⋊C4—32no—
189C4⋊C4—62no—
190C4⋊C4—62no—
191C4⋊C4C22×D312yes—
192C22⋊C4—22no—
193C22⋊C4—62no—
194C22⋊C4—62no—
195C22⋊C4—62no—
196C22×C3—21no—
197C22×C3—22no—
198C22×C3C22×D411yescharacteristic
199C2×D3—22no—
200C2×D3—21no—
201C2×D3—22no—
202C2×D3—42no—
203C2×D3—42no—
204C2×D3—42no—
205C2×D3#12112yes—
206C3×C4—21no—
207C3×C4—42no—
208C3×C4—42no—
209C3×C4C22×D411yescharacteristic
210C3×C4C2×#4111yescharacteristic
211Dic3—21no—
212Dic3—42no—
213Dic3#12112yes—
214C23—31no—
215C23—31no—
216C23—62no—
217C23D3×D411yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—62no—
221D4—62no—
222D4—62no—
223D4—62no—
224D4—62no—
225D4—62no—
226D4—124no—
227D4—242no—
228D4—242no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—31no—
234C2×C4—32no—
235C2×C4—32no—
236C2×C4—31no—
237C2×C4—61no—
238C2×C4—61no—
239C2×C4—62no—
240C2×C4—124no—
241C2×C4—242no—
242C2×C4D3×D411yescharacteristic
243C2×C4C23×D311yescharacteristic
244C8—21no—
245C8—32no—
246C8—61no—
247C8D3×D412yes—
248C2×C3—41no—
249C2×C3C2×#12111yescharacteristic
250C2×C3#57712yes—
251D3—22no—
252D3—41no—
253D3—82no—
254C22—22no—
255C22—21no—
256C22—32no—
257C22—62no—
258C22—62no—
259C22—61no—
260C22—122no—
261C22—122no—
262C22—122no—
263C22C2×D3×D411yescharacteristic
264C4—21no—
265C4—32no—
266C4—42no—
267C4—42no—
268C4—61no—
269C4—122no—
270C4C2×D3×D411yescharacteristic
271C4D3×#4111yescharacteristic
272C3#306011yescharacteristic, complemented, Sylow
273C2—41no—
274C2—62no—
275C2—121no—
276C2—242no—
277C2D3×#12111yescharacteristic
278C2#577812yes—
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#470C2
socleC22×C3C22×D4
hypercenter#470D3
hypocenterC3#3060
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#919C2semidirect
2. C2×#907C2semidirect
3. #5629C2semidirect
4. C3×#470C2semidirect
5. #5189C2semidirect
6. #5135C2semidirect
7. #5048C2semidirect
8. #4865C2semidirect
9. D3×#106C2semidirect
10. #4807C2semidirect
11. #4803C2semidirect
46. #470D3semidirect
92. D3×C8D4semidirect
272. C3#3060semidirect

Maximal subgroups

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

subgroupindex
C2×#9192
C2×#9072
#56292
C3×#4702
#51892
#51352
#50482
#48652
D3×#1062
#48072
#48032
#30603