Finite Group Databasebeta

← #70164

Subgroups — #70164

Order 384.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×#890C211yesmaximal, characteristic, complemented
2C2×C3×#102C211yesmaximal, characteristic
3#5221C211yesmaximal, characteristic
4C2×#832C211yesmaximal, characteristic
5#4551C211yesmaximal, characteristic, complemented
6#4550C211yesmaximal, characteristic, complemented
7#4529C211yesmaximal, characteristic, complemented
8#2025—31nomaximal, Sylow
9C22×C3×D4C2211yescharacteristic
10C22×C3×C8C2211yescharacteristic
11C2×C3×C4⋊C4C2211yescharacteristic
12C2×#217—21no—
13C2×#217—21no—
14C2×C4⋊Dic3—21no—
15C2×C4⋊Dic3C2211yescharacteristic
16C2×C4⋊Dic3C2211yescharacteristic
17C2×Dic3⋊C4—21no—
18C2×C4×Dic3C2211yescharacteristic
19C22×C3⋊C8C2211yescharacteristic
20#890—22no—
21#890—22no—
22C3×#102—22no—
23#832—22no—
24C2×#125—31no—
25C2×#102—31no—
26C2×#102D311yescharacteristic
27#390—31no—
28#340—31no—
29#340—31no—
30#336—31no—
31C24×C3—21no—
32C2×C3×D4—21no—
33C2×C3×D4—21no—
34C2×C3×D4—21no—
35C2×C3×D4—21no—
36C2×C3×D4—21no—
37C2×C3×D4Q812yes—
38C22×C3×C4—21no—
39C22×C3×C4C2311yescharacteristic
40C22×Dic3—21no—
41C22×Dic3—21no—
42C22×Dic3D411yescharacteristic
43C22×Dic3D411yescharacteristic
44C2×C3×C8—21no—
45C2×C3×C8—21no—
46C2×C3×C8—21no—
47C3×C4⋊C4—21no—
48C3×C4⋊C4D412yes—
49#217—42no—
50#217—42no—
51C4⋊Dic3—24no—
52C4⋊Dic3—21no—
53C4⋊Dic3—21no—
54C4⋊Dic3—21no—
55C4⋊Dic3D412yes—
56Dic3⋊C4—24no—
57C4×Dic3—21no—
58C4×Dic3—21no—
59C2×C3⋊C8—21no—
60C2×C3⋊C8—21no—
61C2×C3⋊C8—21no—
62C22×D4C2×D311yescharacteristic
63C22×C8—31no—
64C22×C8C2×D311yescharacteristic
65#125—62no—
66#125—62no—
67C2×C4⋊C4—31no—
68C2×C4⋊C4—31no—
69C2×C4⋊C4—61no—
70C2×C4⋊C4—61no—
71C2×C4⋊C4C2×D311yescharacteristic
72C2×C22⋊C4—61no—
73C2×C22⋊C4—61no—
74C2×C42—31no—
75#102—22no—
76#102—62no—
77C23×C3—22no—
78C23×C3—41no—
79C23×C3—41no—
80C23×C3—41no—
81C23×C3—41no—
82C23×C3—41no—
83C23×C3—41no—
84C23×C3C2×D411yescharacteristic
85C3×D4—22no—
86C3×D4—22no—
87C3×D4—41no—
88C3×D4—41no—
89C2×C3×C4—22no—
90C2×C3×C4—41no—
91C2×C3×C4—41no—
92C2×C3×C4C2×D411yescharacteristic
93C2×C3×C4C2×D411yescharacteristic
94C2×C3×C4C2×Q811yescharacteristic
95C2×C3×C4#4111yescharacteristic
96C2×C3×C4#4111yescharacteristic
97C2×C3×C4#4111yescharacteristic
98C2×Dic3—21no—
99C2×Dic3—21no—
100C2×Dic3—22no—
101C2×Dic3—21no—
102C2×Dic3—21no—
103C2×Dic3—21no—
104C2×Dic3—21no—
105C2×Dic3—41no—
106C2×Dic3—41no—
107C2×Dic3—41no—
108C2×Dic3—41no—
109C2×Dic3—41no—
110C3×C8—41no—
111C3⋊C8—41no—
112C24—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4Dic612yes—
119C22×C4—21no—
120C22×C4—31no—
121C22×C4—31no—
122C22×C4—61no—
123C22×C4—61no—
124C22×C4C22×D311yescharacteristic
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—61no—
129C2×C8—61no—
130C2×C8—61no—
131C4⋊C4—21no—
132C4⋊C4—32no—
133C4⋊C4—64no—
134C4⋊C4—64no—
135C4⋊C4—61no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4#6712yes—
139C22⋊C4—122no—
140C22⋊C4—122no—
141C42—61no—
142C42—61no—
143C22×C3—42no—
144C22×C3—42no—
145C22×C3—42no—
146C22×C3—81no—
147C22×C3—81no—
148C22×C3—81no—
149C22×C3—81no—
150C22×C3#12011yescharacteristic
151C22×C3C22⋊Q811yescharacteristic
152C22×C3#12411yescharacteristic
153C22×C3#13511yescharacteristic
154C22×C3#13511yescharacteristic
155C22×C3AGL(1,ℤ/8)11yescharacteristic
156C22×C3AGL(1,ℤ/8)11yescharacteristic
157C3×C4—42no—
158C3×C4C22⋊Q811yescharacteristic
159C3×C4C22⋊Q811yescharacteristic
160C3×C4#12411yescharacteristic
161C3×C4#12411yescharacteristic
162Dic3—41no—
163Dic3—41no—
164Dic3—42no—
165Dic3—81no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23D3×D411yescharacteristic
174D4—22no—
175D4—22no—
176D4—41no—
177D4—41no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—61no—
182C2×C4—61no—
183C2×C4—61no—
184C2×C4—61no—
185C2×C4—61no—
186C2×C4—61no—
187C2×C4—62no—
188C2×C4—121no—
189C2×C4—121no—
190C2×C4—121no—
191C2×C4—121no—
192C2×C4—121no—
193C2×C4C2×Dic611yescharacteristic
194C2×C4#23511yescharacteristic
195C2×C4D3×D411yescharacteristic
196C2×C4#23711yescharacteristic
197C2×C4#23711yescharacteristic
198C2×C4C2×#6711yescharacteristic
199C8—41no—
200C8—121no—
201C2×C3—82no—
202C2×C3#39311yescharacteristic
203C2×C3#44911yescharacteristic
204C2×C3#44911yescharacteristic
205C2×C3#47811yescharacteristic
206C2×C3#47811yescharacteristic
207C2×C3#48811yescharacteristic
208C2×C3#49011yescharacteristic
209C22—42no—
210C22—42no—
211C22—42no—
212C22—81no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22#87811yescharacteristic
217C22#88511yescharacteristic
218C22#91111yescharacteristic
219C22#91211yescharacteristic
220C22#91411yescharacteristic
221C22#91611yescharacteristic
222C22#93711yescharacteristic
223C4—42no—
224C4—121no—
225C4—121no—
226C4—122no—
227C4—241no—
228C4#87811yescharacteristic
229C4#88511yescharacteristic
230C4#92411yescharacteristic
231C4#93611yescharacteristic
232C3#202511yescharacteristic, complemented, Sylow
233C2—82no—
234C2#476311yescharacteristic
235C2#476411yescharacteristic
236C2#476611yescharacteristic
237C2#476811yescharacteristic
238C2#494711yescharacteristic
239C2#515811yescharacteristic
240C2#517211yescharacteristic
subgroupG/H
centerC23D3×D4
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC2×C3×#102C2
socleC23×C3C2×D4
prime radicalC22×C3×D4C22
hypercenterC2×#102D3
hypocenterC3#2025
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#890C2semidirect
5. #4551C2semidirect
6. #4550C2semidirect
7. #4529C2semidirect
232. C3#2025semidirect

Maximal subgroups

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

subgroupindex
C2×#8902
C2×C3×#1022
#52212
C2×#8322
#45512
#45502
#45292
#20253