Finite Group Databasebeta

← #69028

Subgroups — #69028

Order 384.

380 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 71 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#5748C211yesmaximal, characteristic, complemented
2C2×#908C211yesmaximal, characteristic, complemented
3#5525C211yesmaximal, characteristic, complemented
4#5139C211yesmaximal, characteristic, complemented
5#5131C211yesmaximal, characteristic, complemented
6#5006C211yesmaximal, characteristic, complemented
7C3×#344C211yesmaximal, characteristic, complemented
8#2008—31nomaximal, Sylow
9C2×#235C2211yescharacteristic
10C2×#235C2211yescharacteristic
11C22×D12C2211yescharacteristic
12C2×C3×M16C2211yescharacteristic
13C2×C3×M16C2211yescharacteristic
14C3×#117C2211yescharacteristic
15#926C2211yescharacteristic
16#909—21no—
17#909—21no—
18#908—22no—
19#908—21no—
20#908—21no—
21#908—22no—
22#904—22no—
23C2×D24—21no—
24C2×#204—21no—
25#895—22no—
26#882—22no—
27#875—21no—
28#874—21no—
29#825—22no—
30#821—22no—
31#806—22no—
32#575—31no—
33C2×AGL(1,ℤ/8)—31no—
34#535—31no—
35#421—31no—
36#418—31no—
37#410—31no—
38#344D311yescharacteristic, complemented
39C23×D3—21no—
40C22×C3×C4C2311yescharacteristic
41C2×#67—21no—
42C2×#67—21no—
43#235—21no—
44#235—21no—
45#235—21no—
46#235—21no—
47#235—21no—
48#235—21no—
49#235—21no—
50#235—21no—
51C2×D12—21no—
52C2×D12—21no—
53C2×D12—21no—
54C2×D12—21no—
55C2×D12—21no—
56C2×D12—24no—
57C2×D12D411yescharacteristic
58C2×D12D412yes—
59C2×D12D411yescharacteristic, complemented
60C2×C4×D3—21no—
61C2×C4×D3—21no—
62C2×Dic6D411yescharacteristic
63C2×Dic6D411yescharacteristic, complemented
64C3×M16—21no—
65C3×M16—21no—
66C3×M16—21no—
67C3×M16D412yescomplemented
68C2×C3×C8—21no—
69C2×C3×C8D412yescomplemented
70C3×C4⋊C4—21no—
71C3×C22⋊C4D412yes—
72C3×C42—21no—
73#217—21no—
74(C2×D3)⋊C4—24no—
75C4⋊Dic3—21no—
76C4×Dic3—21no—
77Dic12—22no—
78D24—24no—
79D24—22no—
80#204—24no—
81#204—22no—
82#204—22no—
83C2×#41—31no—
84C2×#41—31no—
85C22×D4—31no—
86#137—61no—
87#137—61no—
88AGL(1,ℤ/8)—62no—
89AGL(1,ℤ/8)—61no—
90AGL(1,ℤ/8)—61no—
91AGL(1,ℤ/8)—62no—
92#135—62no—
93C2×SD16—61no—
94C2×D8—61no—
95C2×M16C2×D311yescharacteristic
96C2×M16C2×D311yescharacteristic
97#127—61no—
98#124—61no—
99#121—62no—
100#120—62no—
101#117—31no—
102#117C2×D311yescharacteristic
103#104—62no—
104#102—62no—
105#99—62no—
106C23×C3C2×D411yescharacteristic
107C22×D3—22no—
108C22×D3—21no—
109C22×D3—21no—
110C22×D3—41no—
111C22×D3—41no—
112C22×D3—82no—
113C2×C3×C4—22no—
114C2×C3×C4C2×D411yescharacteristic
115C2×C3×C4C2×D411yescharacteristic
116C2×C3×C4C2×D411yescharacteristic
117C2×C3×C4C2×D411yescharacteristic
118C2×C3×C4C2×D411yescharacteristic
119C2×C3×C4#4111yescharacteristic
120#67—41no—
121#67—41no—
122#67—41no—
123#67—41no—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—41no—
127D12—22no—
128D12—21no—
129D12—21no—
130D12—21no—
131D12—22no—
132D12—21no—
133D12—41no—
134D12—41no—
135D12—84no—
136C4×D3—41no—
137C4×D3—41no—
138C4×D3—41no—
139C4×D3—41no—
140Dic6—21no—
141Dic6—21no—
142Dic6—21no—
143Dic6—21no—
144C3×C8—22no—
145C3×C8—22no—
146C24—61no—
147#41—61no—
148#41—61no—
149#41—61no—
150#41—61no—
151#41—61no—
152#41—61no—
153#41—61no—
154#41—61no—
155C2×Q8—31no—
156C2×Q8—31no—
157C2×D4—32no—
158C2×D4—31no—
159C2×D4—31no—
160C2×D4—64no—
161C2×D4—61no—
162C2×D4—61no—
163C2×D4—61no—
164C2×D4—61no—
165C2×D4—61no—
166C2×D4—61no—
167C2×D4—61no—
168C22×C4—61no—
169C22×C4—61no—
170C22×C4C22×D311yescharacteristic
171Q16—62no—
172SD16—64no—
173SD16—62no—
174SD16—62no—
175D8—64no—
176D8—62no—
177M16—21no—
178M16—21no—
179M16—21no—
180M16D1212yescomplemented
181C2×C8—21no—
182C2×C8D1212yescomplemented
183C4⋊C4—21no—
184C4⋊C4—61no—
185C22⋊C4—64no—
186C22⋊C4—61no—
187C22⋊C4D1212yes—
188C42—21no—
189C42—61no—
190C22×C3—41no—
191C22×C3#12011yescharacteristic
192C22×C3#12111yescharacteristic
193C22×C3#12711yescharacteristic
194C2×D3—42no—
195C2×D3—44no—
196C2×D3—41no—
197C2×D3—41no—
198C2×D3—42no—
199C2×D3—82no—
200C2×D3—82no—
201C2×D3—81no—
202C3×C4—42no—
203C3×C4#12011yescharacteristic
204C3×C4#12011yescharacteristic
205C3×C4#12111yescharacteristic
206C3×C4#12111yescharacteristic
207Dic3—41no—
208Dic3—41no—
209Dic3—81no—
210C23—62no—
211C23—61no—
212C23—61no—
213C23—121no—
214C23—121no—
215C23—242no—
216C23C2×D1211yescharacteristic
217Q8—61no—
218Q8—61no—
219Q8—61no—
220Q8—61no—
221D4—61no—
222D4—62no—
223D4—61no—
224D4—61no—
225D4—62no—
226D4—61no—
227D4—121no—
228D4—121no—
229D4—121no—
230D4—121no—
231D4—121no—
232D4—121no—
233D4—244no—
234C2×C4—22no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—121no—
238C2×C4—121no—
239C2×C4—121no—
240C2×C4—121no—
241C2×C4—121no—
242C2×C4C2×D1211yescharacteristic
243C2×C4C2×D1211yescharacteristic
244C2×C4D3×D411yescharacteristic
245C2×C4D3×D411yescharacteristic
246C2×C4D3×D411yescharacteristic
247C2×C4#23911yescharacteristic
248C8—22no—
249C8—22no—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3#39211yescharacteristic
254D3—81no—
255D3—82no—
256D3—81no—
257C22—41no—
258C22—122no—
259C22—124no—
260C22—121no—
261C22—121no—
262C22—122no—
263C22—241no—
264C22—242no—
265C22—242no—
266C22#87411yescharacteristic
267C22#88211yescharacteristic
268C22#89511yescharacteristic
269C4—42no—
270C4—121no—
271C4—121no—
272C4—241no—
273C4#88211yescharacteristic
274C4#88211yescharacteristic
275C4#89511yescharacteristic
276C4#89511yescharacteristic
277C3#200811yescharacteristic, complemented, Sylow
278C2—21no—
279C2—21no—
280C2—21no—
281C2—242no—
282C2—241no—
283C2—241no—
284C2#467111yescharacteristic
subgroupG/H
centerC2#4671
commutator, cosocleC22×C3×C4C23
FrattiniC22×C4C22×D3
FittingC3×#344C2
socleC2×C3#392
hypercenter#344D3
hypocenterC3#2008
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
1. #5748C2semidirect
2. C2×#908C2semidirect
3. #5525C2semidirect
4. #5139C2semidirect
5. #5131C2semidirect
6. #5006C2semidirect
7. C3×#344C2semidirect
38. #344D3semidirect
59. C2×D12D4semidirect
63. C2×Dic6D4semidirect
67. C3×M16D4semidirect
69. C2×C3×C8D4semidirect
180. M16D12semidirect
182. C2×C8D12semidirect
277. C3#2008semidirect

Maximal subgroups

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

subgroupindex
#57482
C2×#9082
#55252
#51392
#51312
#50062
C3×#3442
#20083