Finite Group Databasebeta

← #83708

Subgroups — #83708

Order 384.

350 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C3×Q82C211yesmaximal, characteristic, complemented
2#5585C211yesmaximal, characteristic
3#5091C211yesmaximal, characteristic, complemented
4#5089C212yesmaximal, complemented
5#5088C211yesmaximal, characteristic
6#5087C212yesmaximal, complemented
7C4×#216C211yesmaximal, characteristic, complemented
8#5028C212yesmaximal, complemented
9#5027C211yesmaximal, characteristic
10#5026C212yesmaximal, complemented
11#5024C211yesmaximal, characteristic, complemented
12#3375—31nomaximal, Sylow
13C3×C4⋊Q8—21no—
14C3×C4⋊Q8—22no—
15C3×C4⋊Q8C2212yes—
16C3×C4⋊Q8C2211yescharacteristic
17C3×C4×Q8—21no—
18C3×C4×Q8C2212yes—
19C3×C4×Q8C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21#947—21no—
22C2×#216C2211yescharacteristic
23C2×#215C2212yes—
24#894—22no—
25#893—21no—
26#875C2212yes—
27#874C2211yescharacteristic
28C4×D12C2212yes—
29C4×Dic6C2211yescharacteristic
30#835C2211yescharacteristic
31#810C2212yes—
32#810C2211yescharacteristic
33#809C2212yes—
34#809C2212yes—
35#809C2212yes—
36#807C2212yes—
37#807C2211yescharacteristic
38C4⋊(C3⋊C8)C2211yescharacteristic
39C4⋊(C3⋊C8)C2212yes—
40#803C2212yes—
41C4×C3⋊C8C2211yescharacteristic
42#559—31no—
43Q82D311yescharacteristic, complemented
44#489—32no—
45#486—31no—
46#477—31no—
47#474—32no—
48#464—32no—
49#460—31no—
50#446—31no—
51#440—32no—
52C4×Q16—31no—
53C2×C3×Q8—21no—
54C2×C3×Q8—22no—
55C2×C3×Q8D411yescharacteristic
56C2×C3×Q8C2311yescharacteristic
57C2×D12—21no—
58C2×D12—21no—
59C2×D12C2312yes—
60C2×C4×D3—22no—
61C2×Dic6C2311yescharacteristic
62C3×C4⋊C4—22no—
63C3×C4⋊C4—21no—
64C3×C4⋊C4—21no—
65C3×C4⋊C4—22no—
66C3×C4⋊C4D411yescharacteristic
67C3×C4⋊C4D412yes—
68C3×C4⋊C4C2312yes—
69C3×C4⋊C4C2311yescharacteristic
70C3×C42—22no—
71C3×C42—21no—
72C3×C42C2311yescharacteristic
73C3×C42C2312yes—
74#216—22no—
75#215—24no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4—22no—
79C4⋊Dic3C2311yescharacteristic
80Dic3⋊C4—21no—
81C4×Dic3—21no—
82C2×C3⋊C8C2311yescharacteristic
83C2×C3⋊C8C2311yescharacteristic
84C2×C3⋊C8C2312yes—
85C2×Q16—31no—
86C2×SD16—32no—
87C4⋊Q8—21no—
88C4⋊Q8—22no—
89C4⋊Q8C2×D312yes—
90C4⋊Q8C2×D311yescharacteristic
91#127—31no—
92#124—32no—
93#124—61no—
94#123—62no—
95C4×Q8—21no—
96C4×Q8—31no—
97C4×Q8C2×D311yescharacteristic
98C4×Q8C2×D312yes—
99C4×Q8C2×D311yescharacteristic
100C4×D4—32no—
101C4×D4—61no—
102#107—31no—
103#107—32no—
104C4⋊C8—31no—
105C4⋊C8—32no—
106#103—31no—
107#103—31no—
108#103—32no—
109#102—32no—
110#102—32no—
111#102—32no—
112#97—32no—
113C4×C8—31no—
114C22×D3—22no—
115C3×Q8—22no—
116C3×Q8—21no—
117C3×Q8—82no—
118C3×Q8—81no—
119C3×Q8#4112yes—
120C2×C3×C4—21no—
121C2×C3×C4—22no—
122C2×C3×C4—21no—
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D412yes—
125C2×C3×C4C2×D412yes—
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2411yescharacteristic
128C2×Dic3—21no—
129C2×Dic3—21no—
130D12—22no—
131D12—22no—
132D12—42no—
133D12—81no—
134C4×D3—82no—
135Dic6—21no—
136Dic6—21no—
137C3⋊C8—22no—
138C3⋊C8—21no—
139C3⋊C8—21no—
140C2×Q8—22no—
141C2×Q8—21no—
142C2×Q8—31no—
143C2×Q8#6711yescharacteristic
144C2×Q8C22×D311yescharacteristic
145C2×D4—32no—
146C2×D4—61no—
147C2×D4—61no—
148C22×C4—62no—
149Q16—62no—
150SD16—64no—
151C2×C8—31no—
152C2×C8—31no—
153C2×C8—32no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—31no—
159C4⋊C4—61no—
160C4⋊C4#6711yescharacteristic
161C4⋊C4#6712yes—
162C4⋊C4C22×D312yes—
163C4⋊C4C22×D311yescharacteristic
164C22⋊C4—62no—
165C22⋊C4—62no—
166C22⋊C4—62no—
167C42—22no—
168C42—21no—
169C42—61no—
170C42C22×D311yescharacteristic
171C42C22×D312yes—
172C22×C3C22×D411yescharacteristic
173C2×D3—42no—
174C2×D3—42no—
175C2×D3—42no—
176C3×C4—22no—
177C3×C4—22no—
178C3×C4—22no—
179C3×C4—21no—
180C3×C4—41no—
181C3×C4—42no—
182C3×C4#13712yes—
183C3×C4C2×#4111yescharacteristic
184C3×C42−1+411yescharacteristic
185Dic3—41no—
186Dic3—41no—
187C23—62no—
188Q8—22no—
189Q8—21no—
190Q8—61no—
191Q8—61no—
192Q8—82no—
193Q8—81no—
194Q8#23912yes—
195D4—62no—
196D4—62no—
197D4—122no—
198D4—241no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—242no—
205C2×C4C2×#6712yes—
206C2×C4C2×#6712yes—
207C2×C4C2×#6711yescharacteristic
208C2×C4C2×#6711yescharacteristic
209C2×C4C23×D311yescharacteristic
210C8—62no—
211C8—61no—
212C8—61no—
213C2×C3#54811yescharacteristic
214C2×C3C2×#13711yescharacteristic
215C2×C3#57611yescharacteristic
216D3—82no—
217C22—122no—
218C22—122no—
219C22—122no—
220C22C22×#6711yescharacteristic
221C4—22no—
222C4—22no—
223C4—22no—
224C4—21no—
225C4—41no—
226C4—42no—
227C4—121no—
228C4—121no—
229C4#94212yes—
230C4C2×#23911yescharacteristic
231C4#100711yescharacteristic
232C3#337511yescharacteristic, complemented, Sylow
233C2—242no—
234C2C2×#94211yescharacteristic
235C2#581711yescharacteristic
236C2#583511yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×Q82C2
socleC22×C3C22×D4
prime radical#874C22
hypercenterQ82D3
hypocenterC3#3375
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×Q8 ◁ Q82

Complements

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

NH ≅ G/Ntype
1. C3×Q82C2semidirect
3. #5091C2semidirect
4. #5089C2semidirect
6. #5087C2semidirect
7. C4×#216C2semidirect
8. #5028C2semidirect
10. #5026C2semidirect
11. #5024C2semidirect
43. Q82D3semidirect
232. C3#3375semidirect

Maximal subgroups

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

subgroupindex
C3×Q822
#55852
#50912
#50892
#50882
#50872
C4×#2162
#50282
#50272
#50262
#50242
#33753