Finite Group Databasebeta

← #69132

Subgroups — #69132

Order 384.

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

normal subgroups only

This group has 258 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#5587C211yesmaximal, characteristic, complemented
2#5032C211yesmaximal, characteristic
3#4746C211yesmaximal, characteristic, complemented
4#4742C211yesmaximal, characteristic, complemented
5C3×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
6#4476C211yesmaximal, characteristic, complemented
7#4473C211yesmaximal, characteristic, complemented
8#2197—31nomaximal, Sylow
9#1004C2211yescharacteristic
10C3×#120C2211yescharacteristic
11#953—21no—
12C2×#217—21no—
13#936C2211yescharacteristic
14#935—21no—
15#935C2211yescharacteristic
16D4×Dic3—21no—
17#909—21no—
18#886—21no—
19#879—21no—
20#878—21no—
21#877—21no—
22C3×#100C2211yescharacteristic
23C3×#99C2211yescharacteristic
24#834C2211yescharacteristic
25#825—21no—
26#806—21no—
27#806—21no—
28#551—31no—
29#458—31no—
30B(4,𝔽2)—31no—
31#455—31no—
32#354—31no—
33#352—31no—
34AΣL(1,𝔽16)D311yescharacteristic, complemented
35C24×C3D411yescharacteristic
36C2×C3×D4—21no—
37C2×C3×D4C2311yescharacteristic
38C2×#67—21no—
39C2×#67—21no—
40C2×#67—41no—
41C22×Dic3—21no—
42C22×Dic3—21no—
43C22×Dic3D411yescharacteristic, complemented
44#237—41no—
45D3×D4—41no—
46#235—21no—
47C2×Dic6D411yescharacteristic, complemented
48C3×M16—21no—
49C3×C22⋊C4—21no—
50C3×C22⋊C4—21no—
51C3×C22⋊C4D411yescharacteristic
52#217—21no—
53#217—21no—
54#217—21no—
55#217—21no—
56#217—21no—
57#217—21no—
58#217—41no—
59#217D411yescharacteristic, complemented
60(C2×D3)⋊C4—41no—
61C4⋊Dic3—21no—
62C4⋊Dic3—41no—
63Dic3⋊C4—21no—
64Dic3⋊C4—21no—
65Dic3⋊C4—21no—
66C4×Dic3—21no—
67C4×Dic3D411yescharacteristic, complemented
68Dic12—41no—
69#204—41no—
70H(5,𝔽2)—31no—
71#137—61no—
72#126—61no—
73#124—31no—
74#123—31no—
75#123—61no—
76#123—61no—
77C22⋊Q8—61no—
78#120—61no—
79#120C2×D311yescharacteristic
80C4×D4—61no—
81#117—61no—
82C2×C22⋊C4—61no—
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—42no—
92C23×C3C2×D411yescharacteristic
93C23×C3C2×D411yescharacteristic
94C22×D3—21no—
95C22×D3—21no—
96C3×D4—41no—
97C3×D4—42no—
98C2×C3×C4—21no—
99C2×C3×C4—41no—
100C2×C3×C4C2×D411yescharacteristic
101#67—41no—
102#67—41no—
103#67—41no—
104#67—82no—
105C2×Dic3—21no—
106C2×Dic3—21no—
107C2×Dic3—21no—
108C2×Dic3—21no—
109C2×Dic3—21no—
110C2×Dic3—21no—
111C2×Dic3—42no—
112C2×Dic3—41no—
113C2×Dic3—41no—
114C2×Dic3—81no—
115D12—21no—
116C4×D3—41no—
117Dic6—21no—
118Dic6—41no—
119C3×C8—41no—
120C24D1211yescharacteristic
121#41—61no—
122#41—121no—
123C2×Q8—31no—
124C2×D4—21no—
125C2×D4—61no—
126C2×D4—61no—
127C2×D4—121no—
128C2×D4—121no—
129C2×D4C22×D311yescharacteristic
130C22×C4—31no—
131C22×C4—61no—
132C22×C4—61no—
133Q16—121no—
134SD16—121no—
135M16—21no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4—61no—
139C4⋊C4—61no—
140C4⋊C4—121no—
141C22⋊C4—21no—
142C22⋊C4—21no—
143C22⋊C4—31no—
144C22⋊C4—61no—
145C22⋊C4—61no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—121no—
151C22⋊C4—121no—
152C22⋊C4D1211yescharacteristic
153C42—31no—
154C42—61no—
155C22×C3—22no—
156C22×C3—21no—
157C22×C3—21no—
158C22×C3—41no—
159C22×C3—41no—
160C22×C3—41no—
161C22×C3—42no—
162C22×C3—41no—
163C22×C3—81no—
164C22×C3#12011yescharacteristic
165C2×D3—41no—
166C2×D3—81no—
167C2×D3—81no—
168C3×C4—21no—
169C3×C4—41no—
170C3×C4—81no—
171Dic3—41no—
172Dic3—41no—
173Dic3—41no—
174Dic3—41no—
175Dic3—41no—
176Dic3—81no—
177C23—21no—
178C23—41no—
179C23—42no—
180C23—61no—
181C23—61no—
182C23C2×D1211yescharacteristic
183C23D3×D411yescharacteristic
184Q8—61no—
185Q8—121no—
186D4—41no—
187D4—42no—
188D4—61no—
189D4—121no—
190D4—121no—
191D4—121no—
192D4—242no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—121no—
202C2×C4—122no—
203C2×C4—121no—
204C2×C4—121no—
205C2×C4—241no—
206C2×C4D3×D411yescharacteristic
207C8—41no—
208C2×C3—21no—
209C2×C3—41no—
210C2×C3—41no—
211C2×C3—42no—
212C2×C3B(4,𝔽2)11yescharacteristic
213D3—81no—
214C22—22no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—42no—
221C22—41no—
222C22—81no—
223C22—121no—
224C22—241no—
225C22—241no—
226C22#88211yescharacteristic
227C4—21no—
228C4—41no—
229C4—81no—
230C4—121no—
231C4—121no—
232C4—121no—
233C4—121no—
234C4—121no—
235C4—241no—
236C3#219711yescharacteristic, complemented, Sylow
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—241no—
242C2#474111yescharacteristic
subgroupG/H
centerC2#4741
commutator, cosocleC2×C3×D4C23
FrattiniC2×D4C22×D3
FittingC3×AΣL(1,𝔽16)C2
socleC2×C3B(4,𝔽2)
prime radical#5032C2
hypercenterAΣL(1,𝔽16)D3
hypocenterC3#2197
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. #5587C2semidirect
3. #4746C2semidirect
4. #4742C2semidirect
5. C3×AΣL(1,𝔽16)C2semidirect
6. #4476C2semidirect
7. #4473C2semidirect
34. AΣL(1,𝔽16)D3semidirect
43. C22×Dic3D4semidirect
47. C2×Dic6D4semidirect
59. #217D4semidirect
67. C4×Dic3D4semidirect
236. C3#2197semidirect

Maximal subgroups

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

subgroupindex
#55872
#50322
#47462
#47422
C3×AΣL(1,𝔽16)2
#44762
#44732
#21973