Finite Group Databasebeta

← #68763

Subgroups — #68763

Order 384.

282 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 67 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#5519C211yesmaximal, characteristic, complemented
2#5131C211yesmaximal, characteristic
3#5001C211yesmaximal, characteristic
4#4939C211yesmaximal, characteristic
5C3×#339C211yesmaximal, characteristic, complemented
6#4554C211yesmaximal, characteristic, complemented
7#4523C211yesmaximal, characteristic, complemented
8#2027—31nomaximal, Sylow
9C22×D12C2211yescharacteristic
10C2×C3×M16C2211yescharacteristic
11C3×#117C2211yescharacteristic
12C2×C3×C42C2211yescharacteristic
13C2×(C2×D3)⋊C4—22no—
14#926C2211yescharacteristic
15C2×C4⋊Dic3C2211yescharacteristic
16C2×#208C2211yescharacteristic
17#890—21no—
18#888—21no—
19#878—22no—
20#870—21no—
21#869—21no—
22#821—21no—
23#821—21no—
24#809—21no—
25#809—21no—
26#544—31no—
27#418—31no—
28#418—31no—
29#390—31no—
30#343—31no—
31#339—31no—
32#339D311yescharacteristic, complemented
33C23×D3—21no—
34C22×C3×C4—21no—
35C22×C3×C4C2311yescharacteristic
36C22×Dic3D412yes—
37C2×D12—21no—
38C2×D12—21no—
39C2×D12—21no—
40C2×D12—21no—
41C2×D12—21no—
42C2×D12Q812yescomplemented
43C2×Dic6—22no—
44C3×M16—21no—
45C3×M16—21no—
46C2×C3×C8—21no—
47C3×C4⋊C4D411yescharacteristic
48C3×C4⋊C4D411yescharacteristic
49C3×C22⋊C4—21no—
50C3×C42—21no—
51C3×C42—21no—
52C3×C42—21no—
53#217—21no—
54(C2×D3)⋊C4—44no—
55C4⋊Dic3—22no—
56C4⋊Dic3—21no—
57C4⋊Dic3—21no—
58C4⋊Dic3D411yescharacteristic
59C4⋊Dic3D411yescharacteristic
60Dic3⋊C4—22no—
61Dic3⋊C4—22no—
62C4×Dic3—21no—
63#208—21no—
64#208—21no—
65C2×C3⋊C8—21no—
66C22×D4—31no—
67C2×M16—31no—
68C2×M16C2×D311yescharacteristic
69C4⋊Q8—61no—
70C4⋊Q8—61no—
71#125—61no—
72#125—61no—
73C22⋊Q8—62no—
74#117—31no—
75#117C2×D311yescharacteristic
76C2×C4⋊C4—31no—
77C2×C22⋊C4—62no—
78C2×C42C2×D311yescharacteristic
79#102—61no—
80#102—61no—
81#102—61no—
82#102—61no—
83C23×C3C2×D411yescharacteristic
84C22×D3—22no—
85C22×D3—42no—
86C22×D3—42no—
87C22×D3—41no—
88C22×D3—41no—
89C2×C3×C4—22no—
90C2×C3×C4—21no—
91C2×C3×C4—21no—
92C2×C3×C4—41no—
93C2×C3×C4—41no—
94C2×C3×C4C2×D411yescharacteristic
95C2×C3×C4C2×D411yescharacteristic
96C2×C3×C4C2×Q811yescharacteristic
97C2×C3×C4#4111yescharacteristic
98C2×C3×C4#4111yescharacteristic
99C2×C3×C4#4111yescharacteristic
100C2×Dic3—21no—
101C2×Dic3—21no—
102C2×Dic3—22no—
103C2×Dic3—42no—
104D12—22no—
105D12—22no—
106D12—41no—
107D12—41no—
108Dic6—82no—
109C3×C8—41no—
110C3⋊C8—41no—
111C24—61no—
112C2×Q8—62no—
113C2×D4—32no—
114C2×D4—61no—
115C2×D4—61no—
116C2×D4—61no—
117C2×D4—61no—
118C2×D4—61no—
119C22×C4—21no—
120C22×C4—32no—
121C22×C4C22×D311yescharacteristic
122M16—21no—
123M16—21no—
124M16—61no—
125M16—61no—
126C2×C8—21no—
127C2×C8—61no—
128C4⋊C4—31no—
129C4⋊C4—31no—
130C4⋊C4—62no—
131C4⋊C4—62no—
132C4⋊C4—62no—
133C4⋊C4—61no—
134C4⋊C4—61no—
135C4⋊C4D1211yescharacteristic
136C4⋊C4D1211yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—61no—
139C22⋊C4—124no—
140C42—21no—
141C42—21no—
142C42—21no—
143C42—61no—
144C22×C3—21no—
145C22×C3—21no—
146C22×C3#12011yescharacteristic
147C22×C3C22⋊Q811yescharacteristic
148C22×C3#12411yescharacteristic
149C2×D3—44no—
150C2×D3—42no—
151C2×D3—82no—
152C2×D3—81no—
153C2×D3—81no—
154C3×C4—42no—
155C3×C4—41no—
156C3×C4—41no—
157C3×C4C22⋊Q811yescharacteristic
158C3×C4C22⋊Q811yescharacteristic
159C3×C4#12411yescharacteristic
160C3×C4#12411yescharacteristic
161Dic3—41no—
162Dic3—42no—
163Dic3—41no—
164C23—62no—
165C23—122no—
166C23—122no—
167C23—121no—
168C23—121no—
169C23D3×D411yescharacteristic
170Q8—242no—
171D4—62no—
172D4—62no—
173D4—121no—
174D4—121no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—61no—
181C2×C4—61no—
182C2×C4—62no—
183C2×C4—122no—
184C2×C4C2×D1211yescharacteristic
185C2×C4#23511yescharacteristic
186C2×C4#23511yescharacteristic
187C2×C4D3×D411yescharacteristic
188C2×C4#23711yescharacteristic
189C2×C4D3×Q811yescharacteristic
190C8—41no—
191C8—121no—
192C2×C3—21no—
193C2×C3—21no—
194C2×C3#39311yescharacteristic
195C2×C3#45311yescharacteristic
196C2×C3#45411yescharacteristic
197D3—82no—
198C22—21no—
199C22—21no—
200C22—124no—
201C22—122no—
202C22—242no—
203C22—241no—
204C22—241no—
205C22#88211yescharacteristic
206C22#88511yescharacteristic
207C22#89611yescharacteristic
208C4—42no—
209C4—41no—
210C4—41no—
211C4—122no—
212C4—121no—
213C4—121no—
214C4#87511yescharacteristic
215C4#88511yescharacteristic
216C4#89611yescharacteristic
217C4#89711yescharacteristic
218C3#202711yescharacteristic, complemented, Sylow
219C2—21no—
220C2—21no—
221C2—242no—
222C2#467311yescharacteristic
223C2#482211yescharacteristic
224C2#482711yescharacteristic
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#339C2
socleC22×C3#120
prime radicalC22×D12C22
hypercenter#339D3
hypocenterC3#2027
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
1. #5519C2semidirect
5. C3×#339C2semidirect
6. #4554C2semidirect
7. #4523C2semidirect
32. #339D3semidirect
42. C2×D12Q8semidirect
218. C3#2027semidirect

Maximal subgroups

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

subgroupindex
#55192
#51312
#50012
#49392
C3×#3392
#45542
#45232
#20273