Finite Group Databasebeta

← #69136

Subgroups — #69136

Order 384.

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

normal subgroups only

This group has 264 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#5586C211yesmaximal, characteristic
2#4745C211yesmaximal, characteristic, complemented
3#4740C211yesmaximal, characteristic, complemented
4C3×AΣL(1,𝔽16)C211yesmaximal, characteristic, complemented
5#4536C211yesmaximal, characteristic, complemented
6#4476C211yesmaximal, characteristic, complemented
7#4472C211yesmaximal, characteristic, complemented
8#2120—31nomaximal, Sylow
9C23×Dic3C411yescharacteristic, complemented
10C2×#237C2211yescharacteristic
11C3×#120C2211yescharacteristic
12#953C411yescharacteristic, complemented
13#939—21no—
14#935C2211yescharacteristic
15D4×Dic3—21no—
16D3×M16—21no—
17#886C411yescharacteristic, complemented
18#67⋊C4—21no—
19#878—21no—
20#877—21no—
21#877C411yescharacteristic, complemented
22C3×#100C2211yescharacteristic
23C3×#99C2211yescharacteristic
24#834C2211yescharacteristic
25#833C2211yescharacteristic
26#824—21no—
27#806—21no—
28#525—31no—
29#413—31no—
30#410—31no—
31#352—31no—
32#352—31no—
33AΣL(1,𝔽16)—31no—
34AΣL(1,𝔽16)D311yescharacteristic, complemented
35C24×C3C2×C411yescharacteristic
36C2×C3×D4—21no—
37C2×C3×D4C2311yescharacteristic
38C2×#67—21no—
39C2×#67C2×C411yescharacteristic
40C2×#67D411yescharacteristic
41C22×Dic3—21no—
42C22×Dic3—21no—
43C22×Dic3—41no—
44C22×Dic3—41no—
45C22×Dic3C2×C411yescharacteristic
46C22×Dic3C2×C411yescharacteristic
47C22×Dic3D411yescharacteristic
48#237—41no—
49#237—41no—
50C2×C4×D3D411yescharacteristic
51C2×Dic6D411yescharacteristic
52C3×M16—21no—
53C3×C22⋊C4—21no—
54C3×C22⋊C4—21no—
55C3×C22⋊C4C2×C411yescharacteristic
56#217—21no—
57#217—21no—
58#217—21no—
59#217C2×C411yescharacteristic
60(C2×D3)⋊C4—21no—
61C4⋊Dic3—21no—
62Dic3⋊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—
71C23×C4—31no—
72C2×M16—61no—
73#123—31no—
74#123—31no—
75C22⋊Q8—61no—
76#121—61no—
77#120—31no—
78#120C2×D311yescharacteristic
79C4×D4—61no—
80C4×D4—61no—
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—42no—
92C23×C3C22×C411yescharacteristic
93C23×C3C2×D411yescharacteristic
94C22×D3C22⋊C411yescharacteristic
95C3×D4—41no—
96C3×D4—42no—
97C2×C3×C4—21no—
98C2×C3×C4—41no—
99C2×C3×C4C2×D411yescharacteristic
100#67—41no—
101#67—41no—
102#67—42no—
103C2×Dic3—21no—
104C2×Dic3—21no—
105C2×Dic3—21no—
106C2×Dic3—21no—
107C2×Dic3—21no—
108C2×Dic3—21no—
109C2×Dic3—42no—
110C2×Dic3—41no—
111C2×Dic3—41no—
112C2×Dic3—41no—
113C2×Dic3—41no—
114C2×Dic3C22⋊C411yescharacteristic
115C2×Dic3C22⋊C411yescharacteristic
116C2×Dic3C22⋊C411yescharacteristic
117C4×D3—21no—
118C4×D3—21no—
119Dic6—41no—
120C3×C8—41no—
121C3⋊C8—41no—
122C24C4×D311yescharacteristic
123#41—121no—
124#41—121no—
125C2×Q8—31no—
126C2×D4—21no—
127C2×D4—31no—
128C2×D4—31no—
129C2×D4—61no—
130C2×D4C22×D311yescharacteristic
131C22×C4—31no—
132C22×C4—31no—
133C22×C4—31no—
134C22×C4—31no—
135C22×C4—61no—
136C22×C4—61no—
137C22×C4—121no—
138C22×C4—121no—
139M16—21no—
140M16—61no—
141M16—121no—
142C2×C8—121no—
143C4⋊C4—61no—
144C4⋊C4—61no—
145C4⋊C4—61no—
146C4⋊C4—121no—
147C22⋊C4—21no—
148C22⋊C4—21no—
149C22⋊C4—31no—
150C22⋊C4—61no—
151C22⋊C4—61no—
152C22⋊C4—61no—
153C22⋊C4—61no—
154C22⋊C4C4×D311yescharacteristic
155C42—61no—
156C42—121no—
157C22×C3—22no—
158C22×C3—21no—
159C22×C3—21no—
160C22×C3—41no—
161C22×C3—41no—
162C22×C3—41no—
163C22×C3—42no—
164C22×C3—42no—
165C22×C3—41no—
166C22×C3C2×C22⋊C411yescharacteristic
167C2×D3—21no—
168C2×D3—41no—
169C3×C4—21no—
170C3×C4—41no—
171C3×C4—81no—
172Dic3—21no—
173Dic3—21no—
174Dic3—41no—
175Dic3—41no—
176Dic3—81no—
177Dic3#9911yescharacteristic
178Dic3#9911yescharacteristic
179C23—21no—
180C23—31no—
181C23—41no—
182C23—42no—
183C23C2×C4×D311yescharacteristic
184C23D3×D411yescharacteristic
185Q8—121no—
186D4—41no—
187D4—42no—
188D4—121no—
189D4—121no—
190D4—122no—
191C2×C4—21no—
192C2×C4—31no—
193C2×C4—31no—
194C2×C4—31no—
195C2×C4—41no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—121no—
205C2×C4—121no—
206C2×C4—122no—
207C2×C4—121no—
208C2×C4—121no—
209C2×C4D3×D411yescharacteristic
210C8—41no—
211C8—121no—
212C2×C3—21no—
213C2×C3—41no—
214C2×C3—42no—
215C2×C3—41no—
216C2×C3C2×#9911yescharacteristic
217D3—41no—
218C22—22no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—42no—
225C22—42no—
226C22—41no—
227C22—61no—
228C22—121no—
229C22D3×C22⋊C411yescharacteristic
230C4—21no—
231C4—31no—
232C4—31no—
233C4—41no—
234C4—61no—
235C4—61no—
236C4—81no—
237C4—121no—
238C4—121no—
239C4—241no—
240C3#212011yescharacteristic, complemented, Sylow
241C2—21no—
242C2—41no—
243C2—42no—
244C2—41no—
245C2—121no—
246C2D3×#9911yescharacteristic
subgroupG/H
centerC2D3×#99
commutatorC23×C3C22×C4
FrattiniC2×D4C22×D3
FittingC3×AΣL(1,𝔽16)C2
socleC2×C3C2×#99
prime radical#5586C2
cosocleC2×C3×D4C23
hypercenterAΣL(1,𝔽16)D3
hypocenterC3#2120
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. #4476C2semidirect
7. #4472C2semidirect
9. C23×Dic3C4semidirect
12. #953C4semidirect
17. #886C4semidirect
21. #877C4semidirect
34. AΣL(1,𝔽16)D3semidirect
240. C3#2120semidirect

Maximal subgroups

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

subgroupindex
#55862
#47452
#47402
C3×AΣL(1,𝔽16)2
#45362
#44762
#44722
#21203