Finite Group Databasebeta

← #68325

Subgroups — #68325

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
2#5046C211yesmaximal, characteristic
3#4731C211yesmaximal, characteristic
4#4672C211yesmaximal, characteristic
5C3×#328C211yesmaximal, characteristic, complemented
6#4469C211yesmaximal, characteristic, complemented
7#4452C211yesmaximal, characteristic, complemented
8#1599—31nomaximal, Sylow
9C22×D12C2211yescharacteristic
10C3×C22⋊Q8C2211yescharacteristic
11#944—21no—
12C2×#215—21no—
13C2×(C2×D3)⋊C4—21no—
14C2×(C2×D3)⋊C4—21no—
15#924C2211yescharacteristic
16C2×Dic3⋊C4C2211yescharacteristic
17C2×#204—21no—
18#889—21no—
19#889—21no—
20#888—21no—
21#878—21no—
22#877—21no—
23C3×C22⋊C8C2211yescharacteristic
24C3×#95C2211yescharacteristic
25#830C2211yescharacteristic
26#821—21no—
27#809—21no—
28#544—31no—
29#450—31no—
30#450—31no—
31#394—31no—
32#328—31no—
33#328D311yescharacteristic, complemented
34#324—31no—
35C23×D3—21no—
36C2×C3×Q8D411yescharacteristic
37C22×C3×C4—21no—
38C22×C3×C4C2311yescharacteristic
39C22×Dic3D411yescharacteristic
40C22×Dic3D411yescharacteristic
41C2×D12—21no—
42C2×D12—21no—
43C2×D12—42no—
44C2×Dic6—21no—
45C2×Dic6D411yescharacteristic
46C2×C3×C8—21no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4D411yescharacteristic
49C3×C22⋊C4—21no—
50#217—21no—
51#215—42no—
52(C2×D3)⋊C4—42no—
53(C2×D3)⋊C4—42no—
54C4⋊Dic3—21no—
55C4⋊Dic3D411yescharacteristic
56Dic3⋊C4—21no—
57Dic3⋊C4—21no—
58Dic3⋊C4—21no—
59Dic3⋊C4—21no—
60Dic3⋊C4—21no—
61Dic3⋊C4—21no—
62Dic3⋊C4—21no—
63C4×Dic3—21no—
64C4×Dic3—21no—
65C2×C3⋊C8—21no—
66#204—42no—
67C22×D4—31no—
68C2×SD16—61no—
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—61no—
82#102—61no—
83C22⋊C8—31no—
84C22⋊C8C2×D311yescharacteristic
85#95C2×D311yescharacteristic
86C23×C3C2×D411yescharacteristic
87C22×D3—22no—
88C22×D3—41no—
89C22×D3—41no—
90C22×D3—41no—
91C22×D3—41no—
92C22×D3—41no—
93C22×D3—41no—
94C3×Q8—41no—
95C2×C3×C4—21no—
96C2×C3×C4—21no—
97C2×C3×C4—41no—
98C2×C3×C4—41no—
99C2×C3×C4—41no—
100C2×C3×C4—41no—
101C2×C3×C4C2×D411yescharacteristic
102C2×C3×C4C2×D411yescharacteristic
103C2×Dic3—21no—
104C2×Dic3—21no—
105C2×Dic3—21no—
106C2×Dic3—21no—
107C2×Dic3—21no—
108C2×Dic3—41no—
109C2×Dic3SD1612yescomplemented
110D12—22no—
111D12—22no—
112D12—41no—
113D12—41no—
114Dic6—41no—
115Dic6—81no—
116C3×C8—41no—
117C3⋊C8—41no—
118C24—61no—
119C2×Q8—31no—
120C2×Q8—61no—
121C2×Q8D1211yescharacteristic
122C2×D4—61no—
123C2×D4—61no—
124C2×D4—122no—
125C22×C4—21no—
126C22×C4—31no—
127C22×C4—31no—
128C22×C4C22×D311yescharacteristic
129SD16—122no—
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⋊C4D1211yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—61no—
146C22⋊C4—122no—
147C22⋊C4—122no—
148C42—61no—
149C42—61no—
150C22×C3—21no—
151C22×C3—21no—
152C22×C3#12011yescharacteristic
153C22×C3C2×SD1611yescharacteristic
154C22×C3#13711yescharacteristic
155C2×D3—42no—
156C2×D3—42no—
157C2×D3—42no—
158C2×D3—81no—
159C2×D3—81no—
160C2×D3—81no—
161C2×D3—81no—
162C3×C4—21no—
163C3×C4—21no—
164C3×C4—41no—
165C3×C4—41no—
166C3×C4—81no—
167Dic3—22no—
168Dic3—41no—
169Dic3—41no—
170Dic3—41no—
171C23—62no—
172C23—121no—
173C23—121no—
174C23—121no—
175C23—121no—
176C23—121no—
177C23—121no—
178C23D3×D411yescharacteristic
179Q8—41no—
180Q8—121no—
181Q8—241no—
182D4—62no—
183D4—62no—
184D4—121no—
185D4—121no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—32no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—61no—
194C2×C4—61no—
195C2×C4—61no—
196C2×C4—61no—
197C2×C4—61no—
198C2×C4—121no—
199C2×C4C2×D1211yescharacteristic
200C2×C4D3×D411yescharacteristic
201C8—41no—
202C8—121no—
203C2×C3—21no—
204C2×C3—21no—
205C2×C3#44811yescharacteristic
206C2×C3#45411yescharacteristic
207C2×C3B(4,𝔽2)11yescharacteristic
208D3—82no—
209C22—21no—
210C22—21no—
211C22—122no—
212C22—122no—
213C22—122no—
214C22—241no—
215C22—241no—
216C22—241no—
217C22—241no—
218C22#88211yescharacteristic
219C22D3×SD1611yescharacteristic
220C22#91811yescharacteristic
221C4—21no—
222C4—21no—
223C4—41no—
224C4—41no—
225C4—62no—
226C4—81no—
227C4—121no—
228C4—121no—
229C4—121no—
230C3#159911yescharacteristic, complemented, Sylow
231C2—21no—
232C2—21no—
233C2—242no—
234C2#474111yescharacteristic
235C2#480611yescharacteristic
236C2#482711yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#328C2
socleC22×C3#120
prime radicalC22×D12C22
hypercenter#328D3
hypocenterC3#1599
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #5623C2semidirect
5. C3×#328C2semidirect
6. #4469C2semidirect
7. #4452C2semidirect
33. #328D3semidirect
109. 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
#50462
#47312
#46722
C3×#3282
#44692
#44522
#15993