Finite Group Databasebeta

← #71888

Subgroups — #71888

Order 384.

272 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C3×#539C211yesmaximal, characteristic, complemented
2#5046C211yesmaximal, characteristic, complemented
3#5036C211yesmaximal, characteristic
4#4991C211yesmaximal, characteristic, complemented
5#4538C211yesmaximal, characteristic, complemented
6#4537C211yesmaximal, characteristic, complemented
7#4528C211yesmaximal, characteristic, complemented
8#1597—31nomaximal, Sylow
9C22×D12C2211yescharacteristic
10C3×#126—21no—
11C3×#125—21no—
12C3×#123—21no—
13C3×#123—21no—
14C3×C22⋊Q8—21no—
15C3×C22⋊Q8C2211yescharacteristic
16C3×#121C2211yescharacteristic
17C3×C4×D4—21no—
18C2×C3×C4⋊C4C2211yescharacteristic
19C2×#215—21no—
20C2×#213—21no—
21C2×(C2×D3)⋊C4—21no—
22C2×(C2×D3)⋊C4—21no—
23C3⋊#95C2211yescharacteristic
24#830C2211yescharacteristic
25#830C2211yescharacteristic
26#809—21no—
27#809—21no—
28#539D311yescharacteristic, complemented
29#450—31no—
30#447—31no—
31#394—31no—
32#328—31no—
33#327—31no—
34#324—31no—
35C23×D3—21no—
36C2×C3×Q8D411yescharacteristic
37C2×C3×D4—21no—
38C2×C3×D4D411yescharacteristic
39C22×C3×C4—21no—
40C22×C3×C4D411yescharacteristic
41C22×C3×C4D411yescharacteristic
42C22×C3×C4C2311yescharacteristic
43C22×Dic3—21no—
44C2×D12—21no—
45C2×D12—21no—
46C2×D12—42no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4—21no—
49C3×C4⋊C4—21no—
50C3×C4⋊C4—21no—
51C3×C4⋊C4—21no—
52C3×C4⋊C4—21no—
53C3×C4⋊C4D411yescharacteristic
54C3×C4⋊C4D411yescharacteristic
55C3×C22⋊C4—21no—
56C3×C22⋊C4—21no—
57C3×C22⋊C4—21no—
58C3×C22⋊C4—21no—
59C3×C22⋊C4—21no—
60C3×C42—21no—
61#215—42no—
62#213—42no—
63(C2×D3)⋊C4—42no—
64(C2×D3)⋊C4—42no—
65C2×C3⋊C8—21no—
66C2×C3⋊C8—21no—
67C22×D4—31no—
68C2×SD16—61no—
69C2×D8—61no—
70#126—21no—
71#125—21no—
72#123—21no—
73#123—21no—
74C22⋊Q8—21no—
75C22⋊Q8C2×D311yescharacteristic
76#121C2×D311yescharacteristic
77C4×D4—21no—
78C2×C4⋊C4C2×D311yescharacteristic
79C2×C22⋊C4—61no—
80C2×C22⋊C4—61no—
81#102—61no—
82#102—61no—
83C22⋊C8—31no—
84C22⋊C8—31no—
85#95—31no—
86C23×C3—21no—
87C23×C3C2×D411yescharacteristic
88C22×D3—22no—
89C22×D3—41no—
90C22×D3—41no—
91C22×D3—41no—
92C22×D3—41no—
93C22×D3—41no—
94C22×D3—41no—
95C3×Q8—41no—
96C3×D4—41no—
97C3×D4—41no—
98C3×D4—41no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—21no—
102C2×C3×C4—21no—
103C2×C3×C4—21no—
104C2×C3×C4—21no—
105C2×C3×C4—21no—
106C2×C3×C4—41no—
107C2×C3×C4—41no—
108C2×C3×C4—81no—
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2×D411yescharacteristic
111C2×Dic3—41no—
112C2×Dic3—41no—
113C2×Dic3—41no—
114D12—22no—
115D12—22no—
116D12—41no—
117D12—41no—
118C3⋊C8—41no—
119C3⋊C8—41no—
120C24—61no—
121C2×Q8#6711yescharacteristic
122C2×D4—21no—
123C2×D4—61no—
124C2×D4—61no—
125C2×D4—122no—
126C2×D4#6711yescharacteristic
127C22×C4—21no—
128C22×C4—61no—
129C22×C4#6711yescharacteristic
130C22×C4#6711yescharacteristic
131C22×C4C22×D311yescharacteristic
132SD16—122no—
133D8—122no—
134C2×C8—61no—
135C2×C8—61no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4#6711yescharacteristic
143C4⋊C4#6711yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—21no—
147C22⋊C4—21no—
148C22⋊C4—21no—
149C22⋊C4—122no—
150C22⋊C4—122no—
151C42—21no—
152C22×C3—21no—
153C22×C3—21no—
154C22×C3—41no—
155C22×C3—41no—
156C22×C3—41no—
157C22×C3#12011yescharacteristic
158C22×C3#13511yescharacteristic
159C22×C3AGL(1,ℤ/8)11yescharacteristic
160C2×D3—42no—
161C2×D3—42no—
162C2×D3—42no—
163C2×D3—81no—
164C2×D3—81no—
165C2×D3—81no—
166C2×D3—81no—
167C3×C4—21no—
168C3×C4—21no—
169C3×C4—41no—
170C3×C4—41no—
171C3×C4—41no—
172C3×C4—41no—
173C3×C4—41no—
174Dic3—81no—
175C23—21no—
176C23—62no—
177C23—121no—
178C23—121no—
179C23—121no—
180C23—121no—
181C23—121no—
182C23—121no—
183C23C2×#6711yescharacteristic
184Q8—41no—
185D4—41no—
186D4—41no—
187D4—41no—
188D4—62no—
189D4—62no—
190D4—121no—
191D4—121no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—121no—
203C2×C4—121no—
204C2×C4—121no—
205C2×C4C2×#6711yescharacteristic
206C2×C4C2×#6711yescharacteristic
207C8—121no—
208C8—121no—
209C2×C3—21no—
210C2×C3—21no—
211C2×C3—81no—
212C2×C3#44911yescharacteristic
213C2×C3#45411yescharacteristic
214C2×C3B(4,𝔽2)11yescharacteristic
215D3—82no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—122no—
222C22—122no—
223C22—122no—
224C22—241no—
225C22—241no—
226C22—241no—
227C22—241no—
228C22#94911yescharacteristic
229C22#95011yescharacteristic
230C22#95311yescharacteristic
231C4—21no—
232C4—21no—
233C4—41no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—241no—
239C3#159711yescharacteristic, complemented, Sylow
240C2—21no—
241C2—21no—
242C2—81no—
243C2—242no—
244C2#523811yescharacteristic
245C2#524411yescharacteristic
246C2#524511yescharacteristic
subgroupG/H
centerC22#953
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#539C2
socleC22×C3#120
prime radical#5036C2
hypercenter#539D3
hypocenterC3#1597
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. C3×#539C2semidirect
2. #5046C2semidirect
4. #4991C2semidirect
5. #4538C2semidirect
6. #4537C2semidirect
7. #4528C2semidirect
28. #539D3semidirect
239. C3#1597semidirect

Maximal subgroups

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

subgroupindex
C3×#5392
#50462
#50362
#49912
#45382
#45372
#45282
#15973