Finite Group Databasebeta

← #69130

Subgroups — #69130

Order 384.

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

normal subgroups only

This group has 354 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#5590C211yesmaximal, characteristic, complemented
2#5032C211yesmaximal, characteristic, complemented
3#4747C211yesmaximal, characteristic, complemented
4#4741C211yesmaximal, characteristic, complemented
5C3×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
6#4475C211yesmaximal, characteristic, complemented
7#4474C211yesmaximal, characteristic, complemented
8#2196—31nomaximal, Sylow
9C22×#67—21no—
10#1004C2211yescharacteristic, complemented
11C2×D3×D4—21no—
12C3×#120C2211yescharacteristic
13#953—21no—
14#940C2211yescharacteristic
15#939—21no—
16#937C2211yescharacteristic
17#908—21no—
18#884—21no—
19#882—21no—
20#882—21no—
21#67⋊C4—21no—
22C3×#100C2211yescharacteristic, complemented
23C3×#99C2211yescharacteristic, complemented
24#834C2211yescharacteristic, complemented
25#825—21no—
26#806—21no—
27#806—21no—
28D42—31no—
29B(4,𝔽2)—31no—
30B(4,𝔽2)—31no—
31#453—31no—
32#353—31no—
33AΣL(1,𝔽16)—31no—
34AΣL(1,𝔽16)D311yescharacteristic, complemented
35C24×C3D411yescharacteristic, complemented
36C23×D3—21no—
37C23×D3D411yescharacteristic, complemented
38C2×C3×D4—21no—
39C2×C3×D4C2311yescharacteristic
40C2×#67—21no—
41C2×#67—21no—
42C2×#67—21no—
43C2×#67—21no—
44C2×#67—21no—
45C2×#67—21no—
46C2×#67—21no—
47C2×#67—44no—
48C2×#67—41no—
49C22×Dic3—21no—
50#237—41no—
51D3×D4—41no—
52D3×D4—44no—
53#235—21no—
54C2×D12—21no—
55C2×D12—41no—
56C2×D12D411yescharacteristic, complemented
57C2×C4×D3—21no—
58C3×M16—21no—
59C3×C22⋊C4—21no—
60C3×C22⋊C4—21no—
61C3×C22⋊C4D411yescharacteristic, complemented
62#217—21no—
63#217—41no—
64#217D411yescharacteristic, complemented
65(C2×D3)⋊C4—21no—
66(C2×D3)⋊C4—21no—
67(C2×D3)⋊C4—41no—
68Dic3⋊C4—21no—
69C4×Dic3D411yescharacteristic, complemented
70D24—41no—
71#204—41no—
72H(5,𝔽2)—31no—
73C22×D4—61no—
74C22×D4—61no—
75AGL(1,ℤ/8)—61no—
76#127—31no—
77#121—61no—
78#121—61no—
79#120—31no—
80#120—61no—
81#120—61no—
82#120—61no—
83#120C2×D311yescharacteristic
84C4×D4—61no—
85#104—61no—
86#100C2×D311yescharacteristic, complemented
87#99—31no—
88#99—61no—
89#99—61no—
90#99C2×D311yescharacteristic, complemented
91C23×C3—21no—
92C23×C3—41no—
93C23×C3—42no—
94C23×C3C2×D411yescharacteristic
95C23×C3C2×D411yescharacteristic
96C22×D3—21no—
97C22×D3—21no—
98C22×D3—21no—
99C22×D3—21no—
100C22×D3—42no—
101C22×D3—42no—
102C22×D3—41no—
103C22×D3—42no—
104C22×D3—81no—
105C3×D4—41no—
106C3×D4—42no—
107C2×C3×C4—21no—
108C2×C3×C4—41no—
109C2×C3×C4C2×D411yescharacteristic
110#67—24no—
111#67—42no—
112#67—42no—
113#67—42no—
114#67—41no—
115#67—41no—
116#67—42no—
117#67—41no—
118#67—82no—
119C2×Dic3—21no—
120C2×Dic3—21no—
121C2×Dic3—21no—
122C2×Dic3—21no—
123C2×Dic3—42no—
124C2×Dic3—41no—
125D12—21no—
126D12—41no—
127D12—42no—
128D12—82no—
129C4×D3—42no—
130C4×D3—41no—
131Dic6—21no—
132C3×C8—41no—
133C24—31no—
134C24—61no—
135C24D1211yescharacteristic, complemented
136#41—61no—
137#41—121no—
138C2×D4—21no—
139C2×D4—31no—
140C2×D4—61no—
141C2×D4—61no—
142C2×D4—61no—
143C2×D4—61no—
144C2×D4—61no—
145C2×D4—61no—
146C2×D4—61no—
147C2×D4—61no—
148C2×D4—121no—
149C2×D4—121no—
150C2×D4—124no—
151C2×D4—124no—
152C2×D4—121no—
153C2×D4C22×D311yescharacteristic
154C22×C4—61no—
155C22×C4—61no—
156SD16—121no—
157D8—121no—
158M16—21no—
159C4⋊C4—61no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—31no—
163C22⋊C4—61no—
164C22⋊C4—61no—
165C22⋊C4—61no—
166C22⋊C4—121no—
167C22⋊C4—121no—
168C22⋊C4D1211yescharacteristic, complemented
169C42—31no—
170C22×C3—22no—
171C22×C3—21no—
172C22×C3—21no—
173C22×C3—41no—
174C22×C3—41no—
175C22×C3—41no—
176C22×C3—42no—
177C22×C3—41no—
178C22×C3—81no—
179C22×C3#12011yescharacteristic
180C2×D3—22no—
181C2×D3—41no—
182C2×D3—41no—
183C2×D3—42no—
184C2×D3—41no—
185C2×D3—41no—
186C2×D3—41no—
187C2×D3—41no—
188C2×D3—81no—
189C2×D3—84no—
190C2×D3—81no—
191C2×D3—81no—
192C3×C4—21no—
193C3×C4—41no—
194C3×C4—81no—
195Dic3—22no—
196Dic3—41no—
197Dic3—41no—
198Dic3—81no—
199C23—21no—
200C23—41no—
201C23—42no—
202C23—61no—
203C23—61no—
204C23—61no—
205C23—61no—
206C23—121no—
207C23—122no—
208C23—122no—
209C23—122no—
210C23—241no—
211C23C2×D1211yescharacteristic
212C23D3×D411yescharacteristic
213Q8—61no—
214D4—41no—
215D4—42no—
216D4—61no—
217D4—64no—
218D4—121no—
219D4—122no—
220D4—122no—
221D4—122no—
222D4—122no—
223D4—121no—
224D4—121no—
225D4—122no—
226D4—121no—
227D4—242no—
228D4—242no—
229C2×C4—21no—
230C2×C4—41no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—61no—
234C2×C4—61no—
235C2×C4—121no—
236C2×C4—122no—
237C2×C4—122no—
238C2×C4—121no—
239C2×C4D3×D411yescharacteristic
240C8—41no—
241C2×C3—21no—
242C2×C3—41no—
243C2×C3—41no—
244C2×C3—42no—
245C2×C3B(4,𝔽2)11yescharacteristic
246D3—42no—
247D3—81no—
248D3—81no—
249C22—22no—
250C22—21no—
251C22—21no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22—42no—
256C22—41no—
257C22—62no—
258C22—81no—
259C22—121no—
260C22—121no—
261C22—121no—
262C22—122no—
263C22—121no—
264C22—121no—
265C22—121no—
266C22—241no—
267C22—244no—
268C22—241no—
269C22—241no—
270C22#88211yescharacteristic
271C4—21no—
272C4—41no—
273C4—62no—
274C4—81no—
275C4—121no—
276C4—121no—
277C4—241no—
278C3#219611yescharacteristic, complemented, Sylow
279C2—21no—
280C2—41no—
281C2—41no—
282C2—42no—
283C2—122no—
284C2—241no—
285C2—241no—
286C2#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#2196
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. #5590C2semidirect
2. #5032C2semidirect
3. #4747C2semidirect
4. #4741C2semidirect
5. C3×AΣL(1,𝔽16)C2semidirect
6. #4475C2semidirect
7. #4474C2semidirect
10. #1004C22semidirect
22. C3×#100C22semidirect
23. C3×#99C22semidirect
24. #834C22semidirect
34. AΣL(1,𝔽16)D3semidirect
35. C24×C3D4semidirect
37. C23×D3D4semidirect
56. C2×D12D4semidirect
61. C3×C22⋊C4D4semidirect
64. #217D4semidirect
69. C4×Dic3D4semidirect
86. #100C2×D3semidirect
90. #99C2×D3semidirect
135. C24D12semidirect
168. C22⋊C4D12semidirect
278. C3#2196semidirect

Maximal subgroups

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

subgroupindex
#55902
#50322
#47472
#47412
C3×AΣL(1,𝔽16)2
#44752
#44742
#21963