Finite Group Databasebeta

← #68762

Subgroups — #68762

Order 384.

394 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 75 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×#909C211yesmaximal, characteristic, complemented
2#5525C211yesmaximal, characteristic
3C2×#875C211yesmaximal, characteristic, complemented
4C2×#823C211yesmaximal, characteristic, complemented
5#5004C211yesmaximal, characteristic, complemented
6C2×#805C211yesmaximal, characteristic, complemented
7C3×#339C211yesmaximal, characteristic, complemented
8#2004—31nomaximal, Sylow
9C2×#235C2211yescharacteristic
10C22×D12C2211yescharacteristic
11C22×Dic6C2211yescharacteristic
12C2×C3×M16C2211yescharacteristic
13C3×#117C2211yescharacteristic
14C2×C3×C42C2211yescharacteristic
15C2×(C2×D3)⋊C4—22no—
16C2×#208C2211yescharacteristic
17#909—22no—
18#909—22no—
19C2×Dic12—21no—
20C2×#204—21no—
21#895—22no—
22#882—22no—
23#875—21no—
24#875—22no—
25#875—22no—
26#874—21no—
27#823—22no—
28#810—22no—
29#805—22no—
30C2×#137—31no—
31#535—31no—
32C2×#124—31no—
33C2×#104—31no—
34#419—31no—
35C2×#100—31no—
36#339D311yescharacteristic, complemented
37C23×D3D412yes—
38C22×C3×C4—21no—
39C22×C3×C4C2311yescharacteristic
40C2×#67—21no—
41C22×Dic3—21no—
42#235—21no—
43#235—21no—
44#235—21no—
45#235—21no—
46C2×D12—21no—
47C2×D12—21no—
48C2×D12—21no—
49C2×D12—21no—
50C2×D12—21no—
51C2×D12—21no—
52C2×D12—24no—
53C2×D12D411yescharacteristic
54C2×C4×D3—21no—
55C2×Dic6—21no—
56C2×Dic6—21no—
57C2×Dic6—21no—
58C2×Dic6—21no—
59C2×Dic6—21no—
60C2×Dic6D411yescharacteristic
61C2×Dic6D412yescomplemented
62C3×M16—21no—
63C3×M16D412yescomplemented
64C2×C3×C8—21no—
65C3×C4⋊C4D412yes—
66C3×C22⋊C4—21no—
67C3×C42—21no—
68C3×C42—21no—
69C3×C42D412yescomplemented
70(C2×D3)⋊C4—28no—
71(C2×D3)⋊C4—24no—
72#208—21no—
73#208—21no—
74C2×C3⋊C8—21no—
75Dic12—24no—
76#204—24no—
77C2×#41—31no—
78C22×Q8—31no—
79C22×D4—31no—
80#137—62no—
81#137—62no—
82C2×Q16—61no—
83C2×SD16—61no—
84C2×M16—31no—
85C2×M16C2×D311yescharacteristic
86#127—61no—
87#124—62no—
88#124—61no—
89#124—62no—
90#121—62no—
91#120—62no—
92#117C2×D311yescharacteristic
93C2×C22⋊C4—62no—
94C2×C42C2×D311yescharacteristic
95#104—62no—
96#103—62no—
97#100—62no—
98C23×C3C2×D411yescharacteristic
99C22×D3—22no—
100C22×D3—22no—
101C22×D3—21no—
102C22×D3—22no—
103C22×D3—44no—
104C2×C3×C4—22no—
105C2×C3×C4—22no—
106C2×C3×C4—42no—
107C2×C3×C4C2×D411yescharacteristic
108C2×C3×C4C2×D411yescharacteristic
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2×D411yescharacteristic
111C2×C3×C4C2×D411yescharacteristic
112C2×C3×C4#4111yescharacteristic
113#67—41no—
114#67—41no—
115C2×Dic3—22no—
116C2×Dic3—21no—
117C2×Dic3—41no—
118C2×Dic3—41no—
119D12—21no—
120D12—21no—
121D12—41no—
122D12—41no—
123D12—41no—
124D12—41no—
125D12—84no—
126C4×D3—41no—
127C4×D3—41no—
128Dic6—22no—
129Dic6—21no—
130Dic6—22no—
131Dic6—21no—
132Dic6—41no—
133Dic6—41no—
134C3×C8—22no—
135C3⋊C8—41no—
136C24—32no—
137#41—61no—
138#41—61no—
139#41—61no—
140#41—61no—
141C2×Q8—31no—
142C2×Q8—32no—
143C2×Q8—61no—
144C2×Q8—61no—
145C2×Q8—61no—
146C2×Q8—61no—
147C2×Q8—61no—
148C2×D4—31no—
149C2×D4—64no—
150C2×D4—61no—
151C2×D4—61no—
152C2×D4—61no—
153C2×D4—61no—
154C2×D4—61no—
155C2×D4—61no—
156C2×D4—61no—
157C22×C4—21no—
158C22×C4—61no—
159C22×C4—61no—
160C22×C4C22×D311yescharacteristic
161Q16—64no—
162SD16—64no—
163M16—21no—
164M16—61no—
165M16—61no—
166M16D1212yescomplemented
167C2×C8—21no—
168C2×C8—61no—
169C4⋊C4D1212yes—
170C22⋊C4—21no—
171C22⋊C4—68no—
172C22⋊C4—64no—
173C42—21no—
174C42—21no—
175C42D1212yescomplemented
176C22×C3—22no—
177C22×C3#12011yescharacteristic
178C22×C3#12011yescharacteristic
179C22×C3#12111yescharacteristic
180C2×D3—44no—
181C2×D3—41no—
182C2×D3—42no—
183C2×D3—42no—
184C2×D3—44no—
185C2×D3—82no—
186C3×C4—42no—
187C3×C4—42no—
188C3×C4#12011yescharacteristic
189C3×C4#12111yescharacteristic
190C3×C4#12111yescharacteristic
191C3×C4#12711yescharacteristic
192Dic3—42no—
193Dic3—41no—
194C23—62no—
195C23—62no—
196C23—61no—
197C23—62no—
198C23—124no—
199C23D3×D411yescharacteristic
200Q8—62no—
201Q8—61no—
202Q8—62no—
203Q8—61no—
204Q8—121no—
205Q8—121no—
206D4—61no—
207D4—61no—
208D4—121no—
209D4—121no—
210D4—121no—
211D4—121no—
212D4—121no—
213D4—121no—
214D4—244no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—62no—
219C2×C4—61no—
220C2×C4—121no—
221C2×C4—121no—
222C2×C4—121no—
223C2×C4—121no—
224C2×C4C2×D1211yescharacteristic
225C2×C4C2×D1211yescharacteristic
226C2×C4C2×D1211yescharacteristic
227C2×C4D3×D411yescharacteristic
228C2×C4D3×D411yescharacteristic
229C2×C4#23911yescharacteristic
230C8—22no—
231C8—121no—
232C2×C3—21no—
233C2×C3—21no—
234C2×C3#39211yescharacteristic
235C2×C3#45512yes—
236D3—81no—
237D3—82no—
238C22—22no—
239C22—124no—
240C22—121no—
241C22—122no—
242C22—122no—
243C22—124no—
244C22—242no—
245C22#88211yescharacteristic
246C22#88211yescharacteristic
247C22#89511yescharacteristic
248C4—42no—
249C4—42no—
250C4—121no—
251C4—122no—
252C4#87411yescharacteristic
253C4#88211yescharacteristic
254C4#89511yescharacteristic
255C4#89511yescharacteristic
256C3#200411yescharacteristic, complemented, Sylow
257C2—21no—
258C2—21no—
259C2—242no—
260C2—241no—
261C2#467111yescharacteristic
262C2#482512yes—
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#339C2
socleC22×C3#120
prime radical#5525C2
hypercenter#339D3
hypocenterC3#2004
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. C2×#909C2semidirect
3. C2×#875C2semidirect
4. C2×#823C2semidirect
5. #5004C2semidirect
6. C2×#805C2semidirect
7. C3×#339C2semidirect
36. #339D3semidirect
61. C2×Dic6D4semidirect
63. C3×M16D4semidirect
69. C3×C42D4semidirect
166. M16D12semidirect
175. C42D12semidirect
256. C3#2004semidirect

Maximal subgroups

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

subgroupindex
C2×#9092
#55252
C2×#8752
C2×#8232
#50042
C2×#8052
C3×#3392
#20043