Finite Group Databasebeta

← #83412

Subgroups — #83412

Order 384.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C3×#536C211yesmaximal, characteristic
2C2×#951C211yesmaximal, characteristic
3C22×#216C211yesmaximal, characteristic, complemented
4#5516C211yesmaximal, characteristic, complemented
5#5233C211yesmaximal, characteristic
6#5225C211yesmaximal, characteristic, complemented
7#5090C214yesmaximal, complemented
8#5054C214yesmaximal, complemented
9#4999C211yesmaximal, characteristic, complemented
10#3044—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C22×C3×Q8C2211yescharacteristic
13C22×Dic6C2211yescharacteristic
14C3×C4⋊Q8C2212yes—
15C3×#124C2212yes—
16C3×#123—22no—
17C3×C22⋊Q8—22no—
18C3×#117C2211yescharacteristic
19#951—24no—
20C2×#216—24no—
21C2×#216C2214yes—
22C2×#216C2212yes—
23C2×#214C2212yes—
24#926C2211yescharacteristic
25C2×#208C2211yescharacteristic
26C22×C3⋊C8C2211yescharacteristic
27#887—22no—
28#877—22no—
29C4×Dic6C2214yes—
30#835C2214yes—
31#835C2212yes—
32#832C2212yes—
33C4⋊(C3⋊C8)C2214yes—
34C2×#137—31no—
35C22×Q16—31no—
36#536D311yescharacteristic
37#519—31no—
38#464—34no—
39#462—34no—
40#424—31no—
41#419—31no—
42#416—31no—
43C3×#41—22no—
44C3×#41—22no—
45C2×C3×Q8—21no—
46C2×C3×Q8—22no—
47C2×C3×Q8—22no—
48C2×C3×Q8C2312yes—
49C2×C3×Q8C2311yescharacteristic
50C2×C3×D4—21no—
51C2×C3×D4C2311yescharacteristic
52C22×C3×C4—21no—
53C22×C3×C4—21no—
54C22×C3×C4C2311yescharacteristic
55C22×Dic3—21no—
56C2×Dic6—21no—
57C2×Dic6—22no—
58C2×Dic6D414yes—
59C2×Dic6C2312yes—
60C3×C4⋊C4—24no—
61C3×C4⋊C4D412yes—
62C3×C22⋊C4—24no—
63C3×C22⋊C4D412yes—
64C3×C42C2312yes—
65#217—21no—
66#216—28no—
67#216—24no—
68#214—24no—
69C4⋊Dic3C2312yes—
70Dic3⋊C4—24no—
71C4×Dic3—24no—
72C4×Dic3—21no—
73#208—22no—
74C2×C3⋊C8—22no—
75C2×C3⋊C8C2312yes—
76C2×C3⋊C8C2312yes—
77C2×#41C2×D311yescharacteristic
78C22×Q8—31no—
79C22×Q8C2×D311yescharacteristic
80#137—64no—
81C2×Q16—34no—
82C2×Q16—32no—
83C2×Q16—64no—
84C2×SD16—32no—
85C2×M16—31no—
86C22×C8—31no—
87C4⋊Q8C2×D312yes—
88#124C2×D312yes—
89#123—22no—
90C22⋊Q8—22no—
91C4×Q8—34no—
92C4×Q8—62no—
93#117—31no—
94#117—62no—
95#117C2×D311yescharacteristic
96C4⋊C8—34no—
97#103—32no—
98#103—34no—
99#102—32no—
100C23×C3—21no—
101C23×C3C2×D411yescharacteristic
102C3×Q8—22no—
103C3×Q8—21no—
104C3×Q8—22no—
105C3×Q8—21no—
106C3×Q8—42no—
107C3×D4—21no—
108C3×D4—21no—
109C3×D4—42no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—42no—
113C2×C3×C4—42no—
114C2×C3×C4C2×D412yes—
115C2×C3×C4C2×D414yes—
116C2×C3×C4C2×D411yescharacteristic
117C2×C3×C4#4112yes—
118C2×C3×C4C2411yescharacteristic
119C2×Dic3—24no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122Dic6—24no—
123Dic6—22no—
124Dic6C2×D414yes—
125C3⋊C8—22no—
126C3⋊C8—22no—
127#41—22no—
128#41—22no—
129C2×Q8—21no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—34no—
133C2×Q8—32no—
134C2×Q8—62no—
135C2×Q8—61no—
136C2×Q8C22×D312yes—
137C2×Q8C22×D311yescharacteristic
138C2×D4—21no—
139C2×D4C22×D311yescharacteristic
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—61no—
143C22×C4C22×D311yescharacteristic
144Q16—68no—
145Q16—64no—
146SD16—64no—
147M16—62no—
148C2×C8—32no—
149C2×C8—32no—
150C2×C8—62no—
151C4⋊C4—24no—
152C4⋊C4—32no—
153C4⋊C4—64no—
154C4⋊C4#6712yes—
155C22⋊C4—24no—
156C22⋊C4—61no—
157C22⋊C4#6712yes—
158C42—61no—
159C42—64no—
160C42C22×D312yes—
161C22×C3—22no—
162C22×C3—41no—
163C22×C3—42no—
164C22×C3#12112yes—
165C22×C3C22×D411yescharacteristic
166C3×C4—24no—
167C3×C4—41no—
168C3×C4—42no—
169C3×C4#12112yes—
170C3×C4C22×D411yescharacteristic
171C3×C4C2×#4111yescharacteristic
172Dic3—24no—
173Dic3—42no—
174C23—21no—
175C23C2×#6711yescharacteristic
176Q8—22no—
177Q8—21no—
178Q8—22no—
179Q8—21no—
180Q8—34no—
181Q8—42no—
182Q8—64no—
183Q8—62no—
184D4—21no—
185D4—21no—
186D4—42no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—64no—
192C2×C4—62no—
193C2×C4—62no—
194C2×C4D3×D412yes—
195C2×C4#23712yes—
196C2×C4C2×#6714yes—
197C2×C4C2×#6711yescharacteristic
198C2×C4C23×D311yescharacteristic
199C8—62no—
200C8—62no—
201C2×C3—22no—
202C2×C3—81no—
203C2×C3C2×#12111yescharacteristic
204C2×C3#57812yes—
205C22—22no—
206C22—41no—
207C22—42no—
208C22#93812yes—
209C22C22×#6711yescharacteristic
210C4—24no—
211C4—41no—
212C4—42no—
213C4—64no—
214C4—122no—
215C4#93812yes—
216C4C2×D3×D411yescharacteristic
217C4C2×#23711yescharacteristic
218C3#304411yescharacteristic, complemented, Sylow
219C2—22no—
220C2—81no—
221C2C2×#93811yescharacteristic
222C2#583812yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#536C2
socleC22×C3C22×D4
prime radicalC2×C3×#41C22
hypercenter#536D3
hypocenterC3#3044
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
3. C22×#216C2semidirect
4. #5516C2semidirect
6. #5225C2semidirect
7. #5090C2semidirect
8. #5054C2semidirect
9. #4999C2semidirect
218. C3#3044semidirect

Maximal subgroups

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

subgroupindex
C3×#5362
C2×#9512
C22×#2162
#55162
#52332
#52252
#50902
#50542
#49992
#30443