Finite Group Databasebeta

← #70998

Subgroups — #70998

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
2C3×B(4,𝔽2)C211yesmaximal, characteristic
3#5243C211yesmaximal, characteristic, complemented
4#5242C211yesmaximal, characteristic, complemented
5#4536C211yesmaximal, characteristic, complemented
6#4473C211yesmaximal, characteristic, complemented
7#4472C211yesmaximal, characteristic, complemented
8#2197—31nomaximal, Sylow
9C3×H(5,𝔽2)C2211yescharacteristic
10C3×#120—21no—
11C3×#120C2211yescharacteristic
12C2×#217—21no—
13#951—21no—
14#936C2211yescharacteristic
15#935—21no—
16#935—21no—
17#935C2211yescharacteristic
18D4×Dic3—21no—
19#879—21no—
20#878—21no—
21#877—21no—
22C3×#99—21no—
23C3×#99C2211yescharacteristic
24#837—21no—
25#834—21no—
26#834C2211yescharacteristic
27#833C2211yescharacteristic
28#551—31no—
29#458—31no—
30B(4,𝔽2)D311yescharacteristic
31#455—31no—
32#354—31no—
33#352—31no—
34AΣL(1,𝔽16)—31no—
35C24×C3D411yescharacteristic
36C3×#41—21no—
37C3×#41—41no—
38C2×C3×D4—21no—
39C2×C3×D4—21no—
40C2×C3×D4—21no—
41C2×C3×D4—41no—
42C2×C3×D4—41no—
43C2×C3×D4C2311yescharacteristic
44C22×Dic3—21no—
45C22×Dic3—21no—
46C22×Dic3D411yescharacteristic, complemented
47C2×Dic6D411yescharacteristic, complemented
48C3×C22⋊C4—21no—
49C3×C22⋊C4—21no—
50C3×C22⋊C4—41no—
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#216—41no—
61#214—41no—
62C4⋊Dic3—21no—
63Dic3⋊C4—21no—
64Dic3⋊C4—21no—
65Dic3⋊C4—21no—
66Dic3⋊C4—41no—
67C4×Dic3—21no—
68C4×Dic3D411yescharacteristic, complemented
69#208—21no—
70H(5,𝔽2)C2×D311yescharacteristic
71#137—61no—
72#126—61no—
73#124—31no—
74#123—31no—
75#123—61no—
76#123—61no—
77C22⋊Q8—61no—
78#120—21no—
79#120C2×D311yescharacteristic
80C4×D4—61no—
81#117—61no—
82C2×C22⋊C4—61no—
83#104—61no—
84#100—31no—
85#99—21no—
86#99—31no—
87#99—61no—
88#99C2×D311yescharacteristic
89C23×C3—21no—
90C23×C3—21no—
91C23×C3—21no—
92C23×C3—41no—
93C23×C3—42no—
94C23×C3C2×D411yescharacteristic
95C23×C3C2×D411yescharacteristic
96C3×Q8—21no—
97C3×D4—21no—
98C3×D4—41no—
99C3×D4—42no—
100C3×D4—41no—
101C3×D4—41no—
102C3×D4—41no—
103C3×D4—82no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—21no—
107C2×C3×C4—41no—
108C2×C3×C4—41no—
109C2×C3×C4C2×D411yescharacteristic
110C2×Dic3—21no—
111C2×Dic3—21no—
112C2×Dic3—21no—
113C2×Dic3—21no—
114C2×Dic3—42no—
115C2×Dic3—41no—
116C2×Dic3—41no—
117C2×Dic3—81no—
118Dic6—41no—
119C3⋊C8—41no—
120C24#6711yescharacteristic
121#41—21no—
122#41—41no—
123C2×Q8—31no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4C22×D311yescharacteristic
130C22×C4—31no—
131C22×C4—61no—
132C22×C4—61no—
133Q16—121no—
134SD16—121no—
135M16—61no—
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—41no—
145C22⋊C4—61no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—61no—
150C22⋊C4—61no—
151C22⋊C4—121no—
152C22⋊C4#6711yescharacteristic
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—41no—
164C22×C3—81no—
165C22×C3—81no—
166C22×C3—81no—
167C22×C3#12011yescharacteristic
168C3×C4—21no—
169C3×C4—41no—
170C3×C4—41no—
171C3×C4—81no—
172Dic3—41no—
173Dic3—41no—
174Dic3—41no—
175Dic3—41no—
176Dic3—81no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—41no—
181C23—42no—
182C23D3×D411yescharacteristic
183C23C2×#6711yescharacteristic
184Q8—21no—
185Q8—121no—
186D4—21no—
187D4—41no—
188D4—41no—
189D4—41no—
190D4—42no—
191D4—41no—
192D4—82no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—122no—
203C2×C4—121no—
204C2×C4—121no—
205C2×C4—241no—
206C2×C4D3×D411yescharacteristic
207C8—121no—
208C2×C3—21no—
209C2×C3—41no—
210C2×C3—41no—
211C2×C3—42no—
212C2×C3—81no—
213C2×C3B(4,𝔽2)11yescharacteristic
214C22—22no—
215C22—21no—
216C22—21no—
217C22—41no—
218C22—41no—
219C22—41no—
220C22—42no—
221C22—41no—
222C22—41no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22#93711yescharacteristic
227C4—21no—
228C4—41no—
229C4—41no—
230C4—81no—
231C4—121no—
232C4—121no—
233C4—121no—
234C4—121no—
235C4—241no—
236C3#219711yescharacteristic, complemented, Sylow
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—81no—
242C2#503211yescharacteristic
subgroupG/H
centerC2#5032
commutator, cosocleC2×C3×D4C23
FrattiniC2×D4C22×D3
Fitting, prime radicalC3×B(4,𝔽2)C2
socleC2×C3B(4,𝔽2)
hypercenterB(4,𝔽2)D3
hypocenterC3#2197
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ B(4,𝔽2)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #5587C2semidirect
3. #5243C2semidirect
4. #5242C2semidirect
5. #4536C2semidirect
6. #4473C2semidirect
7. #4472C2semidirect
46. C22×Dic3D4semidirect
47. C2×Dic6D4semidirect
59. #217D4semidirect
68. 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
C3×B(4,𝔽2)2
#52432
#52422
#45362
#44732
#44722
#21973