Finite Group Databasebeta

← #69137

Subgroups — #69137

Order 384.

266 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 55 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#5592C211yesmaximal, characteristic
2#4745C211yesmaximal, characteristic, complemented
3#4740C211yesmaximal, characteristic, complemented
4C3×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
5#4536C211yesmaximal, characteristic, complemented
6#4475C211yesmaximal, characteristic, complemented
7#4471C211yesmaximal, characteristic, complemented
8#2122—31nomaximal, Sylow
9C22×#67C411yescharacteristic, complemented
10C2×#237C2211yescharacteristic
11C3×#120C2211yescharacteristic
12#953C411yescharacteristic, complemented
13#940—21no—
14#939—21no—
15#937C2211yescharacteristic
16D3×M16—21no—
17#885—21no—
18#884—21no—
19#882C411yescharacteristic, complemented
20#877—21no—
21#877C411yescharacteristic, complemented
22C3×#100C2211yescharacteristic
23C3×#99C2211yescharacteristic
24#834C2211yescharacteristic
25#833C2211yescharacteristic
26#824—21no—
27#806—21no—
28#535—31no—
29#413—31no—
30#410—31no—
31AΣL(1,𝔽16)—31no—
32AΣL(1,𝔽16)—31no—
33AΣL(1,𝔽16)—31no—
34AΣL(1,𝔽16)D311yescharacteristic, complemented
35C24×C3C2×C411yescharacteristic
36C23×D3C2×C411yescharacteristic
37C2×C3×D4—21no—
38C2×C3×D4C2311yescharacteristic
39C2×#67—21no—
40C2×#67—21no—
41C2×#67—21no—
42C2×#67—21no—
43C2×#67—41no—
44C2×#67—41no—
45C2×#67C2×C411yescharacteristic
46C2×#67D411yescharacteristic
47C22×Dic3C2×C411yescharacteristic
48C22×Dic3D411yescharacteristic
49#237—41no—
50#237—41no—
51C2×D12—21no—
52C2×C4×D3D411yescharacteristic
53C2×Dic6D411yescharacteristic
54C3×M16—21no—
55C3×C22⋊C4—21no—
56C3×C22⋊C4—21no—
57C3×C22⋊C4C2×C411yescharacteristic
58#217—21no—
59#217—21no—
60#217C2×C411yescharacteristic
61(C2×D3)⋊C4—21no—
62(C2×D3)⋊C4—21no—
63Dic3⋊C4—21no—
64Dic3⋊C4—41no—
65C4×Dic3—21no—
66C4×Dic3—41no—
67#208—21no—
68ΓL(1,𝔽25)—41no—
69D3×C8—41no—
70C2×#41—31no—
71C22×D4—31no—
72C2×M16—61no—
73#127—61no—
74#124—61no—
75#121—61no—
76#121—61no—
77#120—31no—
78#120—31no—
79#120—31no—
80#120C2×D311yescharacteristic
81#117—31no—
82#117—61no—
83#101—61no—
84#100—31no—
85#100C2×D311yescharacteristic
86#99—31no—
87#99—61no—
88#99C2×D311yescharacteristic
89C23×C3—21no—
90C23×C3—41no—
91C23×C3—81no—
92C23×C3C22×C411yescharacteristic
93C23×C3C2×D411yescharacteristic
94C22×D3—21no—
95C22×D3—41no—
96C22×D3—81no—
97C22×D3C22⋊C411yescharacteristic
98C3×D4—41no—
99C3×D4—81no—
100C2×C3×C4—21no—
101C2×C3×C4—41no—
102C2×C3×C4C2×D411yescharacteristic
103#67—41no—
104#67—41no—
105#67—42no—
106#67—41no—
107#67—41no—
108#67—81no—
109#67—81no—
110C2×Dic3—21no—
111C2×Dic3—21no—
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—21no—
115C2×Dic3—41no—
116C2×Dic3C22⋊C411yescharacteristic
117C2×Dic3C22⋊C411yescharacteristic
118C2×Dic3C22⋊C411yescharacteristic
119D12—81no—
120C4×D3—21no—
121C4×D3—21no—
122Dic6—41no—
123C3×C8—41no—
124C3⋊C8—41no—
125C24—31no—
126C24C4×D311yescharacteristic
127#41—121no—
128#41—121no—
129C2×Q8—31no—
130C2×D4—21no—
131C2×D4—31no—
132C2×D4—31no—
133C2×D4—61no—
134C2×D4—61no—
135C2×D4—61no—
136C2×D4—61no—
137C2×D4—61no—
138C2×D4—121no—
139C2×D4—121no—
140C2×D4C22×D311yescharacteristic
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—31no—
144M16—21no—
145M16—61no—
146M16—121no—
147C2×C8—121no—
148C4⋊C4—61no—
149C4⋊C4—121no—
150C22⋊C4—21no—
151C22⋊C4—21no—
152C22⋊C4—31no—
153C22⋊C4—61no—
154C22⋊C4—61no—
155C22⋊C4—61no—
156C22⋊C4—61no—
157C22⋊C4C4×D311yescharacteristic
158C42—61no—
159C42—121no—
160C22×C3—21no—
161C22×C3—21no—
162C22×C3—41no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3—41no—
166C22×C3—41no—
167C22×C3—82no—
168C22×C3C2×C22⋊C411yescharacteristic
169C2×D3—21no—
170C2×D3—41no—
171C2×D3—41no—
172C2×D3—41no—
173C2×D3—41no—
174C2×D3—82no—
175C3×C4—21no—
176C3×C4—41no—
177C3×C4—81no—
178Dic3—21no—
179Dic3—21no—
180Dic3—41no—
181Dic3—81no—
182Dic3#9911yescharacteristic
183Dic3#9911yescharacteristic
184C23—21no—
185C23—31no—
186C23—41no—
187C23—61no—
188C23—81no—
189C23—121no—
190C23—241no—
191C23C2×C4×D311yescharacteristic
192C23D3×D411yescharacteristic
193Q8—121no—
194D4—41no—
195D4—81no—
196D4—121no—
197D4—121no—
198D4—121no—
199D4—121no—
200D4—122no—
201D4—241no—
202D4—241no—
203D4—241no—
204C2×C4—21no—
205C2×C4—31no—
206C2×C4—31no—
207C2×C4—31no—
208C2×C4—41no—
209C2×C4—61no—
210C2×C4—61no—
211C2×C4—61no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—121no—
217C2×C4D3×D411yescharacteristic
218C8—41no—
219C8—121no—
220C2×C3—21no—
221C2×C3—41no—
222C2×C3—41no—
223C2×C3—81no—
224C2×C3C2×#9911yescharacteristic
225D3—41no—
226D3—81no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—61no—
235C22—82no—
236C22—121no—
237C22—121no—
238C22—121no—
239C22—121no—
240C22—242no—
241C22D3×C22⋊C411yescharacteristic
242C4—21no—
243C4—31no—
244C4—31no—
245C4—41no—
246C4—61no—
247C4—61no—
248C4—81no—
249C4—121no—
250C4—241no—
251C3#212211yescharacteristic, complemented, Sylow
252C2—21no—
253C2—41no—
254C2—41no—
255C2—81no—
256C2—121no—
257C2—241no—
258C2D3×#9911yescharacteristic
subgroupG/H
centerC2D3×#99
commutatorC23×C3C22×C4
FrattiniC2×D4C22×D3
FittingC3×AΣL(1,𝔽16)C2
socleC2×C3C2×#99
prime radical#5592C2
cosocleC2×C3×D4C23
hypercenterAΣL(1,𝔽16)D3
hypocenterC3#2122
LCSG ▷ C23×C3 ▷ 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
2. #4745C2semidirect
3. #4740C2semidirect
4. C3×AΣL(1,𝔽16)C2semidirect
5. #4536C2semidirect
6. #4475C2semidirect
7. #4471C2semidirect
9. C22×#67C4semidirect
12. #953C4semidirect
19. #882C4semidirect
21. #877C4semidirect
34. AΣL(1,𝔽16)D3semidirect
251. C3#2122semidirect

Maximal subgroups

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

subgroupindex
#55922
#47452
#47402
C3×AΣL(1,𝔽16)2
#45362
#44752
#44712
#21223