Finite Group Databasebeta

← #70970

Subgroups — #70970

Order 384.

272 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 49 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#5602C211yesmaximal, characteristic, complemented
2C3×#447C211yesmaximal, characteristic
3#5222C211yesmaximal, characteristic, complemented
4#5218C211yesmaximal, characteristic, complemented
5#4537C211yesmaximal, characteristic, complemented
6#4469C211yesmaximal, characteristic, complemented
7#4467C211yesmaximal, characteristic, complemented
8#1597—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C2×C3×D8—21no—
11C3×#121C2211yescharacteristic
12C2×#217—21no—
13C2×#217—21no—
14#935—21no—
15#935—21no—
16D4×Dic3—21no—
17C2×#214—21no—
18#924C2211yescharacteristic
19C2×Dic3⋊C4C2211yescharacteristic
20#889—21no—
21#879—21no—
22#878—21no—
23C3×#102—21no—
24C3×C22⋊C8C2211yescharacteristic
25#832—21no—
26C3⋊#95C2211yescharacteristic
27#830C2211yescharacteristic
28#539—31no—
29#450—31no—
30#447D311yescharacteristic
31#394—31no—
32#328—31no—
33#327—31no—
34#324—31no—
35C24×C3—21no—
36C2×C3×D4—21no—
37C2×C3×D4—21no—
38C2×C3×D4—21no—
39C2×C3×D4—42no—
40C2×C3×D4D411yescharacteristic
41C22×C3×C4C2311yescharacteristic
42C22×Dic3—21no—
43C22×Dic3—21no—
44C22×Dic3D411yescharacteristic
45C22×Dic3D411yescharacteristic
46C2×Dic6D411yescharacteristic
47C3×D8—42no—
48C2×C3×C8—21no—
49C3×C4⋊C4D411yescharacteristic
50C3×C22⋊C4—21no—
51#217—21no—
52#217—21no—
53#217—21no—
54#217—21no—
55#217—42no—
56#217—42no—
57#214—42no—
58C4⋊Dic3—21no—
59C4⋊Dic3D411yescharacteristic
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63Dic3⋊C4—21no—
64Dic3⋊C4—21no—
65C4×Dic3—21no—
66C2×C3⋊C8—21no—
67C22×D4C2×D311yescharacteristic
68C2×SD16—61no—
69C2×D8—21no—
70#126—61no—
71#125—61no—
72#123—61no—
73#123—61no—
74C22⋊Q8—31no—
75C22⋊Q8—61no—
76#121C2×D311yescharacteristic
77C4×D4—61no—
78C2×C4⋊C4—31no—
79C2×C22⋊C4—61no—
80C2×C22⋊C4—61no—
81#102—21no—
82#102—61no—
83C22⋊C8—31no—
84C22⋊C8C2×D311yescharacteristic
85#95—31no—
86C23×C3—22no—
87C23×C3—21no—
88C23×C3—41no—
89C23×C3—41no—
90C23×C3—41no—
91C23×C3—41no—
92C23×C3—41no—
93C23×C3—41no—
94C23×C3C2×D411yescharacteristic
95C3×D4—22no—
96C3×D4—22no—
97C3×D4—41no—
98C3×D4—41no—
99C3×D4—41no—
100C3×D4—41no—
101C3×D4—41no—
102C2×C3×C4—21no—
103C2×C3×C4—41no—
104C2×C3×C4C2×D411yescharacteristic
105C2×C3×C4C2×D411yescharacteristic
106C2×Dic3—21no—
107C2×Dic3—21no—
108C2×Dic3—21no—
109C2×Dic3—21no—
110C2×Dic3—21no—
111C2×Dic3—21no—
112C2×Dic3—41no—
113C2×Dic3—41no—
114C2×Dic3—41no—
115C2×Dic3—41no—
116C2×Dic3—81no—
117Dic6—41no—
118C3×C8—41no—
119C3⋊C8—41no—
120C24—21no—
121C2×Q8—31no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—42no—
126C2×D4#6711yescharacteristic
127C22×C4—31no—
128C22×C4—31no—
129C22×C4—61no—
130C22×C4—61no—
131C22×C4C22×D311yescharacteristic
132SD16—122no—
133D8—42no—
134C2×C8—21no—
135C2×C8—61no—
136C4⋊C4—31no—
137C4⋊C4—61no—
138C4⋊C4—61no—
139C4⋊C4—61no—
140C4⋊C4—61no—
141C4⋊C4—61no—
142C4⋊C4—61no—
143C4⋊C4#6711yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—61no—
146C22⋊C4—61no—
147C22⋊C4—61no—
148C22⋊C4—61no—
149C22⋊C4—122no—
150C22⋊C4—122no—
151C42—61no—
152C22×C3—21no—
153C22×C3—21no—
154C22×C3—42no—
155C22×C3—42no—
156C22×C3—41no—
157C22×C3—41no—
158C22×C3—41no—
159C22×C3—42no—
160C22×C3—81no—
161C22×C3—81no—
162C22×C3—81no—
163C22×C3—81no—
164C22×C3#12011yescharacteristic
165C22×C3#13511yescharacteristic
166C22×C3AGL(1,ℤ/8)11yescharacteristic
167C3×C4—21no—
168C3×C4—21no—
169C3×C4—41no—
170Dic3—41no—
171Dic3—41no—
172Dic3—41no—
173Dic3—41no—
174Dic3—81no—
175C23—22no—
176C23—21no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23D3×D411yescharacteristic
184Q8—121no—
185D4—22no—
186D4—22no—
187D4—41no—
188D4—41no—
189D4—41no—
190D4—41no—
191D4—41no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—61no—
199C2×C4—61no—
200C2×C4—121no—
201C2×C4—121no—
202C2×C4—121no—
203C2×C4—121no—
204C2×C4—241no—
205C2×C4D3×D411yescharacteristic
206C2×C4C2×#6711yescharacteristic
207C8—41no—
208C8—121no—
209C2×C3—21no—
210C2×C3—21no—
211C2×C3—81no—
212C2×C3—82no—
213C2×C3#44911yescharacteristic
214C2×C3#45411yescharacteristic
215C2×C3B(4,𝔽2)11yescharacteristic
216C22—21no—
217C22—21no—
218C22—42no—
219C22—42no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—42no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22—81no—
228C22#91111yescharacteristic
229C22#91211yescharacteristic
230C22#93711yescharacteristic
231C4—21no—
232C4—21no—
233C4—41no—
234C4—121no—
235C4—121no—
236C4—121no—
237C4—121no—
238C4—241no—
239C3#159711yescharacteristic, complemented, Sylow
240C2—21no—
241C2—21no—
242C2—82no—
243C2—81no—
244C2#503211yescharacteristic
245C2#515811yescharacteristic
246C2#520111yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
Fitting, prime radicalC3×#447C2
socleC22×C3#120
hypercenter#447D3
hypocenterC3#1597
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #5602C2semidirect
3. #5222C2semidirect
4. #5218C2semidirect
5. #4537C2semidirect
6. #4469C2semidirect
7. #4467C2semidirect
239. C3#1597semidirect

Maximal subgroups

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

subgroupindex
#56022
C3×#4472
#52222
#52182
#45372
#44692
#44672
#15973