Finite Group Databasebeta

← #79701

Subgroups — #79701

Order 384.

348 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 117 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#5585C211yesmaximal, characteristic
2#5570C211yesmaximal, characteristic, complemented
3C3×C4×Q16C211yesmaximal, characteristic, complemented
4#5185C212yesmaximal, complemented
5#5027C211yesmaximal, characteristic
6#5020C211yesmaximal, characteristic, complemented
7#4891C212yesmaximal, complemented
8#4800C212yesmaximal, complemented
9#4788C212yesmaximal, complemented
10#4694C211yesmaximal, characteristic
11C8×Dic6C211yesmaximal, characteristic, complemented
12#3365—31nomaximal, Sylow
13C2×C3×Q16C2211yescharacteristic
14C3×C4×Q8C2211yescharacteristic
15C3×C4×Q8C2211yescharacteristic
16#947C2212yes—
17Q8×Dic3C2212yes—
18C2×#215C2212yes—
19#894—22no—
20#893C2212yes—
21#890—22no—
22#890C2212yes—
23#875—21no—
24#874C2211yescharacteristic
25C4×D12—21no—
26#869—21no—
27#869C2211yescharacteristic
28C4×Dic6—21no—
29C4×Dic6C2211yescharacteristic
30C3×#107C2211yescharacteristic
31C3×#103C2211yescharacteristic
32C3×#103C2211yescharacteristic
33C3×C4×C8C2211yescharacteristic
34#835C2212yes—
35#821C2212yes—
36#821C2212yes—
37Dic3⋊C8C2212yes—
38C8×Dic3C2212yes—
39#809C2212yes—
40#808C2212yes—
41C4⋊(C3⋊C8)C2211yescharacteristic
42#559—31no—
43#557—31no—
44#488—32no—
45#486—31no—
46#478—32no—
47#475—31no—
48#464—32no—
49#460—31no—
50C8×Q8—31no—
51C4×Q16D311yescharacteristic, complemented
52C4×SD16—32no—
53C2×C3×Q8C2311yescharacteristic
54C2×C3×Q8C2311yescharacteristic
55C2×D12—21no—
56C2×D12—21no—
57C2×D12C2312yes—
58C2×C4×D3—22no—
59C2×Dic6—21no—
60C2×Dic6D411yescharacteristic
61C3×Q16—22no—
62C2×C3×C8C2311yescharacteristic
63C2×C3×C8C2311yescharacteristic
64C3×C4⋊C4—21no—
65C3×C4⋊C4—21no—
66C3×C4⋊C4C2311yescharacteristic
67C3×C4⋊C4C2311yescharacteristic
68C3×C42—21no—
69C3×C42—21no—
70C3×C42C2311yescharacteristic
71#215—24no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—22no—
75C4⋊Dic3—21no—
76C4⋊Dic3—22no—
77C4⋊Dic3—22no—
78C4⋊Dic3D411yescharacteristic
79C4⋊Dic3C2312yes—
80Dic3⋊C4—22no—
81Dic3⋊C4D412yes—
82C4×Dic3—22no—
83C4×Dic3C2312yes—
84C2×C3⋊C8C2312yes—
85C2×Q16C2×D311yescharacteristic
86C2×SD16—32no—
87C4⋊Q8—31no—
88C4⋊Q8—61no—
89#127—31no—
90#125—32no—
91#125—62no—
92#124—32no—
93#124—61no—
94#123—62no—
95C4×Q8—31no—
96C4×Q8—32no—
97C4×Q8—61no—
98C4×Q8C2×D311yescharacteristic
99C4×Q8C2×D311yescharacteristic
100C4×D4—32no—
101C4×D4—61no—
102#107C2×D311yescharacteristic
103#106—32no—
104C4⋊C8—32no—
105C4⋊C8—31no—
106#103—32no—
107#103C2×D311yescharacteristic
108#103C2×D311yescharacteristic
109#102—32no—
110#102—32no—
111#102—32no—
112C4×C8—32no—
113C4×C8C2×D311yescharacteristic
114C22×D3—22no—
115C3×Q8—21no—
116C3×Q8—21no—
117C3×Q8—21no—
118C3×Q8#4112yes—
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2411yescharacteristic
126C2×Dic3—22no—
127C2×Dic3C2×D412yes—
128C2×Dic3C2×D412yes—
129D12—22no—
130D12—22no—
131D12—42no—
132D12—81no—
133C4×D3—82no—
134Dic6—22no—
135Dic6—42no—
136C3×C8—21no—
137C3×C8—21no—
138C3⋊C8—22no—
139C2×Q8—31no—
140C2×Q8—61no—
141C2×Q8C22×D311yescharacteristic
142C2×Q8C22×D311yescharacteristic
143C2×D4—32no—
144C2×D4—61no—
145C2×D4—61no—
146C22×C4—62no—
147Q16—22no—
148SD16—64no—
149C2×C8—32no—
150C2×C8C22×D311yescharacteristic
151C2×C8C22×D311yescharacteristic
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—31no—
155C4⋊C4—32no—
156C4⋊C4—32no—
157C4⋊C4—62no—
158C4⋊C4—62no—
159C4⋊C4—62no—
160C4⋊C4—61no—
161C4⋊C4C22×D311yescharacteristic
162C4⋊C4C22×D311yescharacteristic
163C22⋊C4—62no—
164C22⋊C4—62no—
165C22⋊C4—62no—
166C42—21no—
167C42—21no—
168C42—32no—
169C42—62no—
170C42C22×D311yescharacteristic
171C22×C3C22×D411yescharacteristic
172C2×D3—42no—
173C2×D3—42no—
174C2×D3—42no—
175C3×C4—22no—
176C3×C4—21no—
177C3×C4—41no—
178C3×C4—41no—
179C3×C4—41no—
180C3×C4#13512yes—
181C3×C4C2×#4111yescharacteristic
182C3×C42−1+411yescharacteristic
183Dic3—22no—
184Dic3—22no—
185Dic3—42no—
186C23—62no—
187Q8—21no—
188Q8—21no—
189Q8—21no—
190Q8—62no—
191Q8—122no—
192Q8#23512yes—
193D4—62no—
194D4—62no—
195D4—122no—
196D4—241no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—32no—
202C2×C4—32no—
203C2×C4—62no—
204C2×C4—242no—
205C2×C4D3×D411yescharacteristic
206C2×C4D3×D411yescharacteristic
207C2×C4C23×D311yescharacteristic
208C8—21no—
209C8—21no—
210C8—62no—
211C2×C3#54811yescharacteristic
212C2×C3C2×#13511yescharacteristic
213C2×C3#57711yescharacteristic
214D3—82no—
215C22—122no—
216C22—122no—
217C22—122no—
218C22C2×D3×D411yescharacteristic
219C4—21no—
220C4—22no—
221C4—41no—
222C4—41no—
223C4—41no—
224C4—62no—
225C4—62no—
226C4—122no—
227C4#91912yes—
228C4C2×#23511yescharacteristic
229C4#101011yescharacteristic
230C3#336511yescharacteristic, complemented, Sylow
231C2—242no—
232C2#555211yescharacteristic
233C2C2×#91911yescharacteristic
234C2#577011yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×Q16C2
socleC22×C3C22×D4
prime radical#874C22
hypercenterC4×Q16D3
hypocenterC3#3365
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q16

Complements

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

NH ≅ G/Ntype
2. #5570C2semidirect
3. C3×C4×Q16C2semidirect
4. #5185C2semidirect
6. #5020C2semidirect
7. #4891C2semidirect
8. #4800C2semidirect
9. #4788C2semidirect
11. C8×Dic6C2semidirect
51. C4×Q16D3semidirect
230. C3#3365semidirect

Maximal subgroups

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

subgroupindex
#55852
#55702
C3×C4×Q162
#51852
#50272
#50202
#48912
#48002
#47882
#46942
C8×Dic62
#33653