Finite Group Databasebeta

← #69135

Subgroups — #69135

Order 384.

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

normal subgroups only

This group has 280 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#5589C211yesmaximal, characteristic
2D3×#100C211yesmaximal, characteristic, complemented
3D3×#99C211yesmaximal, characteristic, complemented
4C3×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
5#4536C211yesmaximal, characteristic, complemented
6#4476C211yesmaximal, characteristic, complemented
7#4472C211yesmaximal, characteristic, complemented
8#2121—31nomaximal, Sylow
9C22×#67C411yescharacteristic, complemented
10C2×D3×D4C2211yescharacteristic
11C3×#120C2211yescharacteristic
12#953C411yescharacteristic, complemented
13#938—21no—
14#937—21no—
15#935C2211yescharacteristic
16D3×M16—21no—
17#886C411yescharacteristic, complemented
18#884—21no—
19#883—21no—
20D3×C22⋊C4—21no—
21D3×C22⋊C4C411yescharacteristic, complemented
22C3×#100C2211yescharacteristic
23C3×#99C2211yescharacteristic
24#834C2211yescharacteristic
25#833C2211yescharacteristic
26#823—21no—
27#806—21no—
28#534—31no—
29C2×#100—31no—
30C2×#99—31no—
31#352—31no—
32#352—31no—
33AΣL(1,𝔽16)—31no—
34AΣL(1,𝔽16)D311yescharacteristic, complemented
35C24×C3C2×C411yescharacteristic, complemented
36C23×D3C2×C411yescharacteristic
37C23×D3D411yescharacteristic
38C2×C3×D4—21no—
39C2×C3×D4C2311yescharacteristic
40C2×#67—21no—
41C2×#67—21no—
42C2×#67—21no—
43C2×#67—41no—
44C2×#67—41no—
45C2×#67C2×C411yescharacteristic
46C2×#67D411yescharacteristic
47C22×Dic3C2×C411yescharacteristic, complemented
48D3×D4—41no—
49D3×D4—41no—
50C2×D12D411yescharacteristic
51C2×C4×D3—21no—
52C2×C4×D3—41no—
53C2×C4×D3D411yescharacteristic
54C3×M16—21no—
55C3×C22⋊C4—21no—
56C3×C22⋊C4—21no—
57C3×C22⋊C4C2×C411yescharacteristic, complemented
58#217—21no—
59#217—21no—
60#217—21no—
61#217C2×C411yescharacteristic, complemented
62(C2×D3)⋊C4—21no—
63(C2×D3)⋊C4—21no—
64(C2×D3)⋊C4—41no—
65C4⋊Dic3—21no—
66Dic3⋊C4—21no—
67#208—21no—
68ΓL(1,𝔽25)—41no—
69D3×C8—41no—
70C22×D4—31no—
71C22×D4—31no—
72C2×M16—61no—
73#123—31no—
74#123—31no—
75#123—61no—
76#121—61no—
77#121—61no—
78#120—31no—
79#120—61no—
80#120C2×D311yescharacteristic
81C2×C22⋊C4—31no—
82C2×C22⋊C4—61no—
83#100—31no—
84#100—61no—
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—21no—
96C22×D3—21no—
97C22×D3—21no—
98C22×D3—81no—
99C22×D3—81no—
100C22×D3C22⋊C411yescharacteristic
101C22×D3C22⋊C411yescharacteristic
102C22×D3C22⋊C411yescharacteristic
103C3×D4—41no—
104C3×D4—81no—
105C2×C3×C4—21no—
106C2×C3×C4—41no—
107C2×C3×C4C2×D411yescharacteristic
108#67—41no—
109#67—41no—
110#67—42no—
111#67—41no—
112#67—41no—
113#67—81no—
114C2×Dic3—21no—
115C2×Dic3—21no—
116C2×Dic3—41no—
117C2×Dic3—41no—
118C2×Dic3C22⋊C411yescharacteristic
119D12—41no—
120C4×D3—21no—
121C4×D3—21no—
122C4×D3—81no—
123C4×D3—82no—
124C3×C8—41no—
125C3⋊C8—41no—
126C24—31no—
127C24—31no—
128C24C4×D311yescharacteristic, complemented
129C2×D4—21no—
130C2×D4—31no—
131C2×D4—31no—
132C2×D4—31no—
133C2×D4—61no—
134C2×D4—61no—
135C2×D4—61no—
136C2×D4—121no—
137C2×D4—121no—
138C2×D4—121no—
139C2×D4—121no—
140C2×D4C22×D311yescharacteristic
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—61no—
144C22×C4—121no—
145M16—21no—
146M16—61no—
147M16—121no—
148C2×C8—121no—
149C4⋊C4—61no—
150C4⋊C4—61no—
151C22⋊C4—21no—
152C22⋊C4—21no—
153C22⋊C4—31no—
154C22⋊C4—61no—
155C22⋊C4—61no—
156C22⋊C4—61no—
157C22⋊C4—61no—
158C22⋊C4—61no—
159C22⋊C4—121no—
160C22⋊C4C4×D311yescharacteristic, complemented
161C22×C3—21no—
162C22×C3—21no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3—41no—
166C22×C3—41no—
167C22×C3—41no—
168C22×C3—82no—
169C22×C3C2×C22⋊C411yescharacteristic
170C2×D3—21no—
171C2×D3—21no—
172C2×D3—41no—
173C2×D3—42no—
174C2×D3—41no—
175C2×D3—41no—
176C2×D3—81no—
177C2×D3—81no—
178C2×D3—81no—
179C2×D3#9911yescharacteristic
180C2×D3#9911yescharacteristic
181C3×C4—21no—
182C3×C4—41no—
183C3×C4—81no—
184Dic3—21no—
185Dic3—41no—
186Dic3—41no—
187Dic3—81no—
188C23—21no—
189C23—31no—
190C23—31no—
191C23—31no—
192C23—41no—
193C23—61no—
194C23—61no—
195C23—61no—
196C23—61no—
197C23—81no—
198C23—241no—
199C23—241no—
200C23C2×C4×D311yescharacteristic
201C23D3×D411yescharacteristic
202D4—41no—
203D4—81no—
204D4—121no—
205D4—121no—
206D4—121no—
207D4—121no—
208D4—121no—
209D4—122no—
210D4—241no—
211C2×C4—21no—
212C2×C4—31no—
213C2×C4—41no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—121no—
219C2×C4—121no—
220C2×C4—241no—
221C2×C4—242no—
222C2×C4D3×D411yescharacteristic
223C8—41no—
224C8—121no—
225C2×C3—21no—
226C2×C3—41no—
227C2×C3—41no—
228C2×C3—81no—
229C2×C3C2×#9911yescharacteristic
230D3—21no—
231D3—21no—
232D3—41no—
233D3—41no—
234C22—21no—
235C22—21no—
236C22—31no—
237C22—31no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—41no—
243C22—61no—
244C22—61no—
245C22—82no—
246C22—121no—
247C22—121no—
248C22—121no—
249C22—122no—
250C22—241no—
251C22—241no—
252C22—241no—
253C22D3×C22⋊C411yescharacteristic
254C4—21no—
255C4—41no—
256C4—61no—
257C4—81no—
258C4—121no—
259C4—121no—
260C4—241no—
261C3#212111yescharacteristic, complemented, Sylow
262C2—21no—
263C2—41no—
264C2—41no—
265C2—61no—
266C2—61no—
267C2—81no—
268C2—121no—
269C2—121no—
270C2D3×#9911yescharacteristic
subgroupG/H
centerC2D3×#99
commutatorC23×C3C22×C4
FrattiniC2×D4C22×D3
FittingC3×AΣL(1,𝔽16)C2
socleC2×C3C2×#99
prime radical#5589C2
cosocleC2×C3×D4C23
hypercenterAΣL(1,𝔽16)D3
hypocenterC3#2121
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. D3×#100C2semidirect
3. D3×#99C2semidirect
4. C3×AΣL(1,𝔽16)C2semidirect
5. #4536C2semidirect
6. #4476C2semidirect
7. #4472C2semidirect
9. C22×#67C4semidirect
12. #953C4semidirect
17. #886C4semidirect
21. D3×C22⋊C4C4semidirect
34. AΣL(1,𝔽16)D3semidirect
35. C24×C3C2×C4semidirect
47. C22×Dic3C2×C4semidirect
57. C3×C22⋊C4C2×C4semidirect
61. #217C2×C4semidirect
128. C24C4×D3semidirect
160. C22⋊C4C4×D3semidirect
261. C3#2121semidirect

Maximal subgroups

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

subgroupindex
#55892
D3×#1002
D3×#992
C3×AΣL(1,𝔽16)2
#45362
#44762
#44722
#21213