Finite Group Databasebeta

← #70979

Subgroups — #70979

Order 384.

266 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 51 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#5623C211yesmaximal, characteristic, complemented
2C3×#450C211yesmaximal, characteristic
3#5222C211yesmaximal, characteristic
4#5218C211yesmaximal, characteristic
5#4538C211yesmaximal, characteristic, complemented
6#4469C211yesmaximal, characteristic, complemented
7#4467C211yesmaximal, characteristic, complemented
8#1599—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C2×C3×SD16—21no—
11C3×C22⋊Q8C2211yescharacteristic
12C2×#217—21no—
13C2×#217—21no—
14#944—21no—
15C2×#214—21no—
16#924C2211yescharacteristic
17C2×Dic3⋊C4C2211yescharacteristic
18#889—21no—
19#889—21no—
20#888—21no—
21#878—21no—
22#877—21no—
23C3×#102—21no—
24C3×C22⋊C8C2211yescharacteristic
25#832—21no—
26C3⋊#95C2211yescharacteristic
27#830C2211yescharacteristic
28#544—31no—
29#450—31no—
30#450D311yescharacteristic
31#394—31no—
32#328—31no—
33#328—31no—
34#324—31no—
35C24×C3—21no—
36C2×C3×Q8D411yescharacteristic
37C2×C3×D4—21no—
38C2×C3×D4—21no—
39C2×C3×D4—42no—
40C22×C3×C4C2311yescharacteristic
41C22×Dic3—21no—
42C22×Dic3D411yescharacteristic
43C22×Dic3D411yescharacteristic
44C2×Dic6—21no—
45C2×Dic6D411yescharacteristic
46C3×SD16—42no—
47C2×C3×C8—21no—
48C3×C4⋊C4—21no—
49C3×C4⋊C4D411yescharacteristic
50C3×C22⋊C4—21no—
51#217—21no—
52#217—42no—
53#217—42no—
54#214—42no—
55C4⋊Dic3—21no—
56C4⋊Dic3D411yescharacteristic
57Dic3⋊C4—21no—
58Dic3⋊C4—21no—
59Dic3⋊C4—21no—
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63Dic3⋊C4—21no—
64C4×Dic3—21no—
65C4×Dic3—21no—
66C2×C3⋊C8—21no—
67C22×D4C2×D311yescharacteristic
68C2×SD16—21no—
69C2×SD16—61no—
70C4⋊Q8—61no—
71C4⋊Q8—61no—
72#125—61no—
73#125—61no—
74C22⋊Q8—31no—
75C22⋊Q8—61no—
76C22⋊Q8C2×D311yescharacteristic
77#117—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—41no—
88C23×C3—41no—
89C23×C3—41no—
90C23×C3—41no—
91C23×C3—41no—
92C23×C3—41no—
93C23×C3C2×D411yescharacteristic
94C3×Q8—41no—
95C3×D4—22no—
96C3×D4—22no—
97C3×D4—41no—
98C3×D4—41no—
99C2×C3×C4—21no—
100C2×C3×C4—21no—
101C2×C3×C4—41no—
102C2×C3×C4C2×D411yescharacteristic
103C2×C3×C4C2×D411yescharacteristic
104C2×Dic3—21no—
105C2×Dic3—21no—
106C2×Dic3—21no—
107C2×Dic3—21no—
108C2×Dic3—21no—
109C2×Dic3—41no—
110C2×Dic3—41no—
111C2×Dic3—41no—
112C2×Dic3—41no—
113C2×Dic3SD1612yescomplemented
114Dic6—41no—
115Dic6—81no—
116C3×C8—41no—
117C3⋊C8—41no—
118C24—21no—
119C2×Q8—31no—
120C2×Q8—61no—
121C2×Q8#6711yescharacteristic
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—42no—
125C22×C4—31no—
126C22×C4—31no—
127C22×C4—61no—
128C22×C4C22×D311yescharacteristic
129SD16—42no—
130SD16—122no—
131C2×C8—21no—
132C2×C8—61no—
133C4⋊C4—21no—
134C4⋊C4—31no—
135C4⋊C4—61no—
136C4⋊C4—61no—
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—122no—
147C22⋊C4—122no—
148C42—61no—
149C42—61no—
150C22×C3—21no—
151C22×C3—21no—
152C22×C3—42no—
153C22×C3—42no—
154C22×C3—42no—
155C22×C3—81no—
156C22×C3—81no—
157C22×C3—81no—
158C22×C3—81no—
159C22×C3#12011yescharacteristic
160C22×C3C2×SD1611yescharacteristic
161C22×C3#13711yescharacteristic
162C3×C4—21no—
163C3×C4—21no—
164C3×C4—41no—
165C3×C4—41no—
166Dic3—22no—
167Dic3—41no—
168Dic3—41no—
169Dic3—41no—
170Dic3—81no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23D3×D411yescharacteristic
179Q8—41no—
180Q8—121no—
181Q8—241no—
182D4—22no—
183D4—22no—
184D4—41no—
185D4—41no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—32no—
189C2×C4—41no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—61no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—121no—
196C2×C4—121no—
197C2×C4—121no—
198C2×C4—121no—
199C2×C4D3×D411yescharacteristic
200C2×C4C2×#6711yescharacteristic
201C8—41no—
202C8—121no—
203C2×C3—21no—
204C2×C3—21no—
205C2×C3—82no—
206C2×C3#44811yescharacteristic
207C2×C3#45411yescharacteristic
208C2×C3B(4,𝔽2)11yescharacteristic
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22D3×SD1611yescharacteristic
219C22#91511yescharacteristic
220C22#93711yescharacteristic
221C4—21no—
222C4—21no—
223C4—41no—
224C4—41no—
225C4—62no—
226C4—121no—
227C4—121no—
228C4—121no—
229C4—241no—
230C3#159911yescharacteristic, complemented, Sylow
231C2—21no—
232C2—21no—
233C2—82no—
234C2#503211yescharacteristic
235C2#517011yescharacteristic
236C2#520111yescharacteristic
subgroupG/H
centerC22#937
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#450C2
socleC22×C3#120
prime radicalC22×C3×D4C22
hypercenter#450D3
hypocenterC3#1599
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #5623C2semidirect
5. #4538C2semidirect
6. #4469C2semidirect
7. #4467C2semidirect
113. C2×Dic3SD16semidirect
230. C3#1599semidirect

Maximal subgroups

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

subgroupindex
#56232
C3×#4502
#52222
#52182
#45382
#44692
#44672
#15993