Finite Group Databasebeta

← #69131

Subgroups — #69131

Order 384.

270 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 41 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#5594C211yesmaximal, characteristic, complemented
2#5032C211yesmaximal, characteristic, complemented
3#4746C211yesmaximal, characteristic, complemented
4#4741C211yesmaximal, characteristic, complemented
5C3×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
6#4475C211yesmaximal, characteristic, complemented
7#4473C211yesmaximal, characteristic, complemented
8#2199—31nomaximal, Sylow
9#1004C2211yescharacteristic
10C3×#120C2211yescharacteristic
11#953—21no—
12#953—21no—
13#937—21no—
14#937C2211yescharacteristic
15#936—21no—
16#936C2211yescharacteristic
17#909—21no—
18#885—21no—
19#885—21no—
20#885—21no—
21#882—21no—
22C3×#100C2211yescharacteristic
23C3×#99C2211yescharacteristic
24#834C2211yescharacteristic
25#825—21no—
26#806—21no—
27#806—21no—
28H(3,𝔽4)—31no—
29B(4,𝔽2)—31no—
30B(4,𝔽2)—31no—
31#455—31no—
32#354—31no—
33AΣL(1,𝔽16)—31no—
34AΣL(1,𝔽16)D311yescharacteristic, complemented
35C24×C3D411yescharacteristic, complemented
36C23×D3D411yescharacteristic, complemented
37C2×C3×D4—21no—
38C2×C3×D4C2311yescharacteristic
39C2×#67—21no—
40C2×#67—21no—
41C2×#67—21no—
42C2×#67—21no—
43C2×#67—21no—
44C2×#67—41no—
45#237—41no—
46D3×D4—41no—
47#235—21no—
48C2×D12—41no—
49C2×Dic6—21no—
50C2×Dic6D411yescharacteristic, complemented
51C3×M16—21no—
52C3×C22⋊C4—21no—
53C3×C22⋊C4—21no—
54C3×C22⋊C4D411yescharacteristic, complemented
55#217—21no—
56#217—21no—
57#217—21no—
58#217—21no—
59#217—21no—
60#217—41no—
61#217D411yescharacteristic, complemented
62(C2×D3)⋊C4—21no—
63(C2×D3)⋊C4—21no—
64(C2×D3)⋊C4—21no—
65(C2×D3)⋊C4—41no—
66C4×Dic3—21no—
67C4×Dic3D411yescharacteristic, complemented
68Dic12—41no—
69#204—41no—
70H(5,𝔽2)—31no—
71#137—61no—
72#124—31no—
73#124—61no—
74#124—61no—
75#124—61no—
76#124—61no—
77#120—31no—
78#120—61no—
79#120—61no—
80#120—61no—
81#120—61no—
82#120C2×D311yescharacteristic
83#104—61no—
84#100C2×D311yescharacteristic
85#99—31no—
86#99—61no—
87#99—61no—
88#99C2×D311yescharacteristic
89C23×C3—21no—
90C23×C3—41no—
91C23×C3—81no—
92C23×C3C2×D411yescharacteristic
93C23×C3C2×D411yescharacteristic
94C22×D3—21no—
95C22×D3—21no—
96C22×D3—21no—
97C22×D3—41no—
98C22×D3—81no—
99C3×D4—41no—
100C3×D4—81no—
101C2×C3×C4—21no—
102C2×C3×C4—41no—
103C2×C3×C4C2×D411yescharacteristic
104#67—41no—
105#67—41no—
106#67—41no—
107#67—81no—
108#67—81no—
109#67—81no—
110#67—82no—
111C2×Dic3—21no—
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—21no—
115C2×Dic3—21no—
116C2×Dic3—41no—
117D12—21no—
118D12—82no—
119C4×D3—41no—
120Dic6—21no—
121Dic6—41no—
122Dic6—81no—
123C3×C8—41no—
124C24—31no—
125C24D1211yescharacteristic, complemented
126#41—61no—
127#41—121no—
128C2×Q8—31no—
129C2×Q8—61no—
130C2×D4—21no—
131C2×D4—61no—
132C2×D4—61no—
133C2×D4—61no—
134C2×D4—61no—
135C2×D4—61no—
136C2×D4—121no—
137C2×D4—121no—
138C2×D4—121no—
139C2×D4C22×D311yescharacteristic
140Q16—121no—
141SD16—121no—
142M16—21no—
143C22⋊C4—21no—
144C22⋊C4—21no—
145C22⋊C4—31no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—61no—
151C22⋊C4—61no—
152C22⋊C4—61no—
153C22⋊C4—61no—
154C22⋊C4—121no—
155C22⋊C4—121no—
156C22⋊C4D1211yescharacteristic, complemented
157C42—31no—
158C42—61no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3—41no—
162C22×C3—41no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3—42no—
166C22×C3—41no—
167C22×C3—81no—
168C22×C3#12011yescharacteristic
169C2×D3—41no—
170C2×D3—41no—
171C2×D3—42no—
172C2×D3—41no—
173C2×D3—41no—
174C2×D3—81no—
175C2×D3—81no—
176C2×D3—81no—
177C3×C4—21no—
178C3×C4—41no—
179C3×C4—81no—
180Dic3—41no—
181Dic3—41no—
182Dic3—41no—
183Dic3—41no—
184Dic3—81no—
185C23—21no—
186C23—41no—
187C23—61no—
188C23—61no—
189C23—61no—
190C23—81no—
191C23—121no—
192C23—241no—
193C23C2×D1211yescharacteristic
194C23D3×D411yescharacteristic
195Q8—61no—
196Q8—121no—
197Q8—241no—
198D4—41no—
199D4—61no—
200D4—81no—
201D4—121no—
202D4—121no—
203D4—121no—
204D4—242no—
205D4—241no—
206D4—241no—
207D4—241no—
208D4—242no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—121no—
217C2×C4—121no—
218C2×C4D3×D411yescharacteristic
219C8—41no—
220C2×C3—21no—
221C2×C3—41no—
222C2×C3—41no—
223C2×C3—81no—
224C2×C3B(4,𝔽2)11yescharacteristic
225D3—81no—
226D3—81no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—42no—
234C22—41no—
235C22—81no—
236C22—121no—
237C22—121no—
238C22—122no—
239C22—121no—
240C22—121no—
241C22—241no—
242C22—241no—
243C22—241no—
244C22#88211yescharacteristic
245C4—21no—
246C4—41no—
247C4—81no—
248C4—121no—
249C4—121no—
250C4—121no—
251C4—121no—
252C4—241no—
253C3#219911yescharacteristic, complemented, Sylow
254C2—21no—
255C2—41no—
256C2—41no—
257C2—81no—
258C2—241no—
259C2—241no—
260C2#474111yescharacteristic
subgroupG/H
centerC2#4741
commutator, cosocleC2×C3×D4C23
FrattiniC2×D4C22×D3
FittingC3×AΣL(1,𝔽16)C2
socleC2×C3B(4,𝔽2)
hypercenterAΣL(1,𝔽16)D3
hypocenterC3#2199
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
1. #5594C2semidirect
2. #5032C2semidirect
3. #4746C2semidirect
4. #4741C2semidirect
5. C3×AΣL(1,𝔽16)C2semidirect
6. #4475C2semidirect
7. #4473C2semidirect
34. AΣL(1,𝔽16)D3semidirect
35. C24×C3D4semidirect
36. C23×D3D4semidirect
50. C2×Dic6D4semidirect
54. C3×C22⋊C4D4semidirect
61. #217D4semidirect
67. C4×Dic3D4semidirect
125. C24D12semidirect
156. C22⋊C4D12semidirect
253. C3#2199semidirect

Maximal subgroups

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

subgroupindex
#55942
#50322
#47462
#47412
C3×AΣL(1,𝔽16)2
#44752
#44732
#21993