Finite Group Databasebeta

← #79745

Subgroups — #79745

Order 384.

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

normal subgroups only

This group has 342 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#5584C211yesmaximal, characteristic
2#5579C211yesmaximal, characteristic
3C3×C4×Q16C211yesmaximal, characteristic, complemented
4#5189C212yesmaximal, complemented
5#5021C211yesmaximal, characteristic, complemented
6#5020C211yesmaximal, characteristic, complemented
7#4881C212yesmaximal, complemented
8#4815C212yesmaximal, complemented
9#4805C212yesmaximal, complemented
10#4688C211yesmaximal, characteristic
11#4680C211yesmaximal, characteristic, complemented
12#3356—31nomaximal, Sylow
13C2×C3×Q16C2211yescharacteristic
14C3×C4×Q8C2211yescharacteristic
15C3×C4×Q8C2211yescharacteristic
16#946C2212yes—
17#946C2212yes—
18C2×ΓL(1,𝔽25)C2212yes—
19#898—22no—
20#896—22no—
21#892C2212yes—
22D3×C4⋊C4C2212yes—
23#875—21no—
24C4×D12—21no—
25C4×D12C2211yescharacteristic
26#870—21no—
27#869C2211yescharacteristic
28#869C2211yescharacteristic
29C4×Dic6—21no—
30C3×#107C2211yescharacteristic
31C3×#103C2211yescharacteristic
32C3×#103C2211yescharacteristic
33C3×C4×C8C2211yescharacteristic
34#835C2212yes—
35#835C2212yes—
36(C2×D3)⋊C8C2212yes—
37#817C2212yes—
38#817C2212yes—
39#808C2212yes—
40#807C2212yes—
41C4⋊(C3⋊C8)C2211yescharacteristic
42#556—31no—
43#554—31no—
44#498—31no—
45#485—32no—
46#483—32no—
47#477—31no—
48#475—31no—
49#470—32no—
50C4×Q16D311yescharacteristic, complemented
51#436—31no—
52#428—32no—
53C2×C3×Q8C2311yescharacteristic
54C2×C3×Q8C2311yescharacteristic
55C2×D12D411yescharacteristic
56C2×C4×D3—22no—
57C2×C4×D3C2312yes—
58C2×Dic6—21no—
59C2×Dic6—21no—
60C3×Q16—22no—
61C2×C3×C8C2311yescharacteristic
62C2×C3×C8C2311yescharacteristic
63C3×C4⋊C4—21no—
64C3×C4⋊C4—21no—
65C3×C4⋊C4C2311yescharacteristic
66C3×C4⋊C4C2311yescharacteristic
67C3×C42—21no—
68C3×C42—21no—
69C3×C42C2311yescharacteristic
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4D412yes—
74C4⋊Dic3—21no—
75C4⋊Dic3—21no—
76C4⋊Dic3D411yescharacteristic
77C4⋊Dic3C2312yes—
78C4⋊Dic3C2312yes—
79Dic3⋊C4—22no—
80Dic3⋊C4—22no—
81Dic3⋊C4—22no—
82C4×Dic3—22no—
83C2×C3⋊C8C2312yes—
84ΓL(1,𝔽25)—24no—
85C2×Q16C2×D311yescharacteristic
86C2×M16—32no—
87C4⋊Q8—31no—
88C4⋊Q8—31no—
89#126—62no—
90#125—61no—
91#124—61no—
92C22⋊Q8—32no—
93C22⋊Q8—32no—
94C22⋊Q8—62no—
95C4×Q8—61no—
96C4×Q8C2×D311yescharacteristic
97C4×Q8C2×D311yescharacteristic
98C4×D4—31no—
99C4×D4—61no—
100#117—32no—
101C2×C4⋊C4—32no—
102#107—32no—
103#107C2×D311yescharacteristic
104#106—32no—
105#106—32no—
106#106—32no—
107C4⋊C8—31no—
108#103—32no—
109#103—32no—
110#103C2×D311yescharacteristic
111#103C2×D311yescharacteristic
112C22⋊C8—32no—
113C4×C8C2×D311yescharacteristic
114C22×D3C2×D412yes—
115C3×Q8—21no—
116C3×Q8—21no—
117C3×Q8—21no—
118C3×Q8—21no—
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×Dic3—22no—
128C2×Dic3C2×D412yes—
129D12—22no—
130C4×D3—22no—
131C4×D3—22no—
132C4×D3—82no—
133Dic6—42no—
134Dic6—81no—
135C3×C8—21no—
136C3×C8#4112yes—
137C3⋊C8—22no—
138C2×Q8—61no—
139C2×Q8—61no—
140C2×Q8C22×D311yescharacteristic
141C2×Q8C22×D311yescharacteristic
142C2×D4—31no—
143C22×C4—32no—
144C22×C4—62no—
145Q16—22no—
146M16—64no—
147C2×C8—32no—
148C2×C8C22×D311yescharacteristic
149C2×C8C22×D311yescharacteristic
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—31no—
153C4⋊C4—32no—
154C4⋊C4—32no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C4⋊C4—61no—
159C4⋊C4—61no—
160C4⋊C4C22×D311yescharacteristic
161C4⋊C4C22×D311yescharacteristic
162C22⋊C4—32no—
163C22⋊C4—62no—
164C22⋊C4—62no—
165C22⋊C4—62no—
166C42—21no—
167C42—21no—
168C42—62no—
169C42C22×D311yescharacteristic
170C22×C3C22×D411yescharacteristic
171C2×D3—22no—
172C2×D3—22no—
173C2×D3—22no—
174C3×C4—21no—
175C3×C4—41no—
176C3×C4—41no—
177C3×C4—41no—
178C3×C4—41no—
179C3×C4#13712yes—
180C3×C4C2×#4111yescharacteristic
181C3×C42−1+411yescharacteristic
182Dic3—22no—
183Dic3—42no—
184Dic3—42no—
185C23—32no—
186Q8—21no—
187Q8—21no—
188Q8—21no—
189Q8—21no—
190Q8—122no—
191Q8—241no—
192D4—62no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—32no—
198C2×C4—62no—
199C2×C4—62no—
200C2×C4—62no—
201C2×C4—62no—
202C2×C4—242no—
203C2×C4D3×D411yescharacteristic
204C2×C4D3×D411yescharacteristic
205C2×C4C23×D311yescharacteristic
206C8—21no—
207C8—62no—
208C8#23512yes—
209C2×C3#54711yescharacteristic
210C2×C3C2×#13711yescharacteristic
211C2×C3#57711yescharacteristic
212D3—42no—
213C22—62no—
214C22—62no—
215C22—62no—
216C22C2×D3×D411yescharacteristic
217C4—21no—
218C4—41no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—62no—
223C4—122no—
224C4—122no—
225C4#91812yes—
226C4C2×#23511yescharacteristic
227C4#101011yescharacteristic
228C3#335611yescharacteristic, complemented, Sylow
229C2—122no—
230C2#555111yescharacteristic
231C2C2×#91811yescharacteristic
232C2#577011yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×C4×Q16C2
socleC22×C3C22×D4
prime radicalC2×D12D4
hypercenterC4×Q16D3
hypocenterC3#3356
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
3. C3×C4×Q16C2semidirect
4. #5189C2semidirect
5. #5021C2semidirect
6. #5020C2semidirect
7. #4881C2semidirect
8. #4815C2semidirect
9. #4805C2semidirect
11. #4680C2semidirect
50. C4×Q16D3semidirect
228. C3#3356semidirect

Maximal subgroups

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

subgroupindex
#55842
#55792
C3×C4×Q162
#51892
#50212
#50202
#48812
#48152
#48052
#46882
#46802
#33563