Finite Group Databasebeta

← #68761

Subgroups — #68761

Order 384.

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

normal subgroups only

This group has 478 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×#908C211yesmaximal, characteristic, complemented
2#5525C211yesmaximal, characteristic, complemented
3C2×#874C211yesmaximal, characteristic, complemented
4C2×#823C211yesmaximal, characteristic, complemented
5#5001C211yesmaximal, characteristic, complemented
6C2×#805C211yesmaximal, characteristic, complemented
7C3×#339C211yesmaximal, characteristic, complemented
8#2002—31nomaximal, Sylow
9C2×#235C2211yescharacteristic
10C22×D12—22no—
11C22×D12C2211yescharacteristic
12C22×D12C2211yescharacteristic
13C2×C3×M16C2211yescharacteristic
14C3×#117C2211yescharacteristic
15C2×C3×C42C2211yescharacteristic
16C2×#208C2211yescharacteristic
17#908—22no—
18#908—22no—
19C2×D24—21no—
20C2×#204—21no—
21#895—22no—
22#882—22no—
23#875—21no—
24#874—22no—
25#874—21no—
26#874—22no—
27#823—22no—
28#809—22no—
29#805—22no—
30C2×AGL(1,ℤ/8)—31no—
31#535—31no—
32C2×#127—31no—
33C2×#104—31no—
34#418—31no—
35C2×#100—31no—
36#339D311yescharacteristic, complemented
37C23×D3—21no—
38C23×D3D412yes—
39C22×C3×C4—21no—
40C22×C3×C4C2311yescharacteristic
41C2×#67—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—28no—
51C2×D12—21no—
52C2×D12—21no—
53C2×D12—21no—
54C2×D12—21no—
55C2×D12—21no—
56C2×D12—21no—
57C2×D12—21no—
58C2×D12—24no—
59C2×D12—48no—
60C2×D12D411yescharacteristic, complemented
61C2×D12D412yescomplemented
62C2×C4×D3—21no—
63C2×Dic6D411yescharacteristic, complemented
64C3×M16—21no—
65C3×M16D412yescomplemented
66C2×C3×C8—21no—
67C3×C4⋊C4D412yescomplemented
68C3×C22⋊C4—21no—
69C3×C42—21no—
70C3×C42—21no—
71C3×C42D412yescomplemented
72(C2×D3)⋊C4—24no—
73#208—21no—
74#208—21no—
75C2×C3⋊C8—21no—
76D24—24no—
77#204—24no—
78C2×#41—31no—
79C22×D4—31no—
80C22×D4—31no—
81C22×D4—62no—
82AGL(1,ℤ/8)—62no—
83AGL(1,ℤ/8)—62no—
84C2×SD16—61no—
85C2×D8—61no—
86C2×M16—31no—
87C2×M16C2×D311yescharacteristic
88#127—62no—
89#127—61no—
90#127—62no—
91#124—61no—
92#121—62no—
93#120—62no—
94#117C2×D311yescharacteristic
95C2×C42C2×D311yescharacteristic
96#104—62no—
97#102—62no—
98#100—62no—
99C23×C3C2×D411yescharacteristic
100C22×D3—22no—
101C22×D3—21no—
102C22×D3—22no—
103C22×D3—22no—
104C22×D3—22no—
105C22×D3—44no—
106C22×D3—44no—
107C22×D3—41no—
108C22×D3—41no—
109C2×C3×C4—24no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4C2×D411yescharacteristic
113C2×C3×C4C2×D411yescharacteristic
114C2×C3×C4C2×D411yescharacteristic
115C2×C3×C4C2×D411yescharacteristic
116C2×C3×C4C2×D411yescharacteristic
117C2×C3×C4#4111yescharacteristic
118#67—41no—
119#67—41no—
120C2×Dic3—21no—
121D12—22no—
122D12—21no—
123D12—22no—
124D12—21no—
125D12—416no—
126D12—41no—
127D12—41no—
128D12—41no—
129D12—41no—
130D12—41no—
131D12—41no—
132D12—84no—
133C4×D3—41no—
134C4×D3—41no—
135Dic6—21no—
136Dic6—21no—
137C3×C8—22no—
138C3⋊C8—41no—
139C24—32no—
140C24—61no—
141#41—61no—
142#41—61no—
143#41—61no—
144#41—61no—
145C2×Q8—31no—
146C2×D4—31no—
147C2×D4—32no—
148C2×D4—68no—
149C2×D4—64no—
150C2×D4—61no—
151C2×D4—61no—
152C2×D4—61no—
153C2×D4—61no—
154C2×D4—61no—
155C2×D4—61no—
156C2×D4—61no—
157C2×D4—61no—
158C2×D4—61no—
159C2×D4—61no—
160C2×D4—61no—
161C2×D4—61no—
162C2×D4—128no—
163C22×C4—21no—
164C22×C4—61no—
165C22×C4C22×D311yescharacteristic
166SD16—64no—
167D8—64no—
168M16—21no—
169M16—61no—
170M16—61no—
171M16D1212yescomplemented
172C2×C8—21no—
173C2×C8—61no—
174C4⋊C4D1212yescomplemented
175C22⋊C4—21no—
176C22⋊C4—64no—
177C42—21no—
178C42—21no—
179C42D1212yescomplemented
180C22×C3—22no—
181C22×C3#12011yescharacteristic
182C22×C3#12011yescharacteristic
183C22×C3#12111yescharacteristic
184C2×D3—44no—
185C2×D3—44no—
186C2×D3—41no—
187C2×D3—42no—
188C2×D3—42no—
189C2×D3—42no—
190C2×D3—44no—
191C2×D3—82no—
192C2×D3—82no—
193C2×D3—82no—
194C3×C4—24no—
195C3×C4—42no—
196C3×C4#12011yescharacteristic
197C3×C4#12111yescharacteristic
198C3×C4#12111yescharacteristic
199C3×C4#12711yescharacteristic
200Dic3—41no—
201C23—62no—
202C23—62no—
203C23—62no—
204C23—61no—
205C23—62no—
206C23—124no—
207C23—124no—
208C23—121no—
209C23—121no—
210C23D3×D411yescharacteristic
211Q8—61no—
212Q8—61no—
213D4—61no—
214D4—62no—
215D4—62no—
216D4—61no—
217D4—1216no—
218D4—121no—
219D4—121no—
220D4—121no—
221D4—121no—
222D4—121no—
223D4—121no—
224D4—121no—
225D4—121no—
226D4—244no—
227C2×C4—24no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—61no—
231C2×C4—121no—
232C2×C4—121no—
233C2×C4C2×D1211yescharacteristic
234C2×C4C2×D1211yescharacteristic
235C2×C4C2×D1211yescharacteristic
236C2×C4D3×D411yescharacteristic
237C2×C4D3×D411yescharacteristic
238C2×C4#23911yescharacteristic
239C8—22no—
240C8—121no—
241C2×C3—21no—
242C2×C3—21no—
243C2×C3#39211yescharacteristic
244C2×C3#45312yes—
245D3—81no—
246D3—82no—
247D3—82no—
248C22—22no—
249C22—124no—
250C22—124no—
251C22—121no—
252C22—122no—
253C22—122no—
254C22—122no—
255C22—124no—
256C22—242no—
257C22—242no—
258C22—242no—
259C22#88211yescharacteristic
260C22#88211yescharacteristic
261C22#89511yescharacteristic
262C4—24no—
263C4—42no—
264C4—121no—
265C4#87411yescharacteristic
266C4#88211yescharacteristic
267C4#89511yescharacteristic
268C4#89511yescharacteristic
269C3#200211yescharacteristic, complemented, Sylow
270C2—21no—
271C2—21no—
272C2—242no—
273C2—242no—
274C2—241no—
275C2#467111yescharacteristic
276C2#482212yes—
subgroupG/H
centerC22#882
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#339C2
socleC22×C3#120
hypercenter#339D3
hypocenterC3#2002
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×#908C2semidirect
2. #5525C2semidirect
3. C2×#874C2semidirect
4. C2×#823C2semidirect
5. #5001C2semidirect
6. C2×#805C2semidirect
7. C3×#339C2semidirect
36. #339D3semidirect
60. C2×D12D4semidirect
61. C2×D12D4semidirect
63. C2×Dic6D4semidirect
65. C3×M16D4semidirect
67. C3×C4⋊C4D4semidirect
71. C3×C42D4semidirect
171. M16D12semidirect
174. C4⋊C4D12semidirect
179. C42D12semidirect
269. C3#2002semidirect

Maximal subgroups

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

subgroupindex
C2×#9082
#55252
C2×#8742
C2×#8232
#50012
C2×#8052
C3×#3392
#20023