Finite Group Databasebeta

← #79213

Subgroups — #79213

Order 384.

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

normal subgroups only

This group has 504 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
1C2×#908C211yesmaximal, characteristic, complemented
2C2×#907C211yesmaximal, characteristic, complemented
3#5525C211yesmaximal, characteristic, complemented
4#5510C211yesmaximal, characteristic, complemented
5C3×#428C211yesmaximal, characteristic, complemented
6#5002C211yesmaximal, characteristic, complemented
7C2×#809C211yesmaximal, characteristic, complemented
8#4886C212yesmaximal, complemented
9#4883C212yesmaximal, complemented
10#4865C212yesmaximal, complemented
11#4864C212yesmaximal, complemented
12#3062—31nomaximal, Sylow
13C2×#235C2211yescharacteristic
14C2×#235C2211yescharacteristic
15C22×D12C2211yescharacteristic
16C2×C3×M16C2211yescharacteristic
17C3×#117C2211yescharacteristic
18C2×C3×C4⋊C4C2211yescharacteristic
19#930—22no—
20C22×C3⋊C8C2211yescharacteristic
21#908—24no—
22#907—24no—
23C2×D24C2212yes—
24C2×#204C2212yes—
25C2×ΓL(1,𝔽25)C2212yescomplemented
26C2×D3×C8C2212yescomplemented
27#897—21no—
28#897C2212yescomplemented
29#895—21no—
30#895C2212yes—
31#895C2212yes—
32#895C2212yescomplemented
33#882—22no—
34#875—21no—
35#874—21no—
36C3×#107C2212yescomplemented
37C3×#106C2212yescomplemented
38#810C2211yescharacteristic, complemented
39#810C2211yescharacteristic, complemented
40#809C2212yes—
41#809C2212yes—
42#809C2211yescharacteristic, complemented
43#809C2211yescharacteristic, complemented
44C2×AGL(1,ℤ/8)—31no—
45C2×#131—31no—
46#537—31no—
47#535—31no—
48#469—32no—
49#468—32no—
50#466—32no—
51#465—32no—
52#428D311yescharacteristic, complemented
53#416—31no—
54C2×#102—31no—
55C23×D3—21no—
56C22×C3×C4—21no—
57C22×C3×C4C2311yescharacteristic
58C2×#67—21no—
59C2×#67D412yes—
60#235—22no—
61#235—22no—
62#235—22no—
63#235—22no—
64C2×D12—22no—
65C2×D12—22no—
66C2×D12—21no—
67C2×D12—22no—
68C2×D12—22no—
69C2×D12—22no—
70C2×D12D411yescharacteristic
71C2×D12C2312yes—
72C2×D12C2311yescharacteristic
73C2×C4×D3—21no—
74C2×C4×D3C2312yes—
75C2×Dic6D411yescharacteristic
76C2×Dic6C2311yescharacteristic
77C3×M16D414yescomplemented
78C2×C3×C8C2312yes—
79C3×C4⋊C4—21no—
80C3×C4⋊C4C2312yes—
81C3×C4⋊C4C2311yescharacteristic
82C3×C4⋊C4C2311yescharacteristic
83C3×C22⋊C4—21no—
84C3×C42—21no—
85(C2×D3)⋊C4—22no—
86(C2×D3)⋊C4—22no—
87(C2×D3)⋊C4—22no—
88(C2×D3)⋊C4—22no—
89C4⋊Dic3—22no—
90C2×C3⋊C8—22no—
91C2×C3⋊C8C2311yescharacteristic
92C2×C3⋊C8C2311yescharacteristic
93D24—24no—
94#204—24no—
95ΓL(1,𝔽25)—24no—
96D3×C8—24no—
97C2×#41—31no—
98C2×#41—31no—
99C22×D4—31no—
100AGL(1,ℤ/8)—64no—
101C2×SD16—32no—
102C2×D8—32no—
103#131—64no—
104C2×M16—32no—
105C2×M16C2×D311yescharacteristic
106C22×C8—31no—
107C22×C8—32no—
108#127—61no—
109#124—61no—
110C22⋊Q8—32no—
111C22⋊Q8—61no—
112#121—32no—
113#121—32no—
114#121—32no—
115#121—61no—
116#121—62no—
117#120—62no—
118#117C2×D311yescharacteristic
119C2×C4⋊C4C2×D311yescharacteristic
120#107C2×D312yescomplemented
121#106C2×D312yescomplemented
122#103—31no—
123#103—31no—
124#102—32no—
125#102—32no—
126#102—31no—
127#102—31no—
128C23×C3C2×D411yescharacteristic
129C22×D3—21no—
130C22×D3—22no—
131C22×D3—42no—
132C22×D3—81no—
133C22×D3—81no—
134C22×D3C2×D412yes—
135C2×C3×C4—22no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4—81no—
139C2×C3×C4C2×D412yes—
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4#4112yes—
142C2×C3×C4C2411yescharacteristic
143#67—24no—
144#67—42no—
145C2×Dic3—21no—
146C2×Dic3C2×D412yes—
147D12—22no—
148D12—22no—
149D12—21no—
150D12—22no—
151D12—21no—
152D12—42no—
153D12—82no—
154D12—82no—
155D12—82no—
156C4×D3—22no—
157C4×D3—22no—
158C4×D3—42no—
159Dic6—22no—
160Dic6—21no—
161Dic6—21no—
162C3×C8C2×D414yes—
163C3⋊C8—21no—
164C3⋊C8—21no—
165C24—61no—
166#41—62no—
167#41—62no—
168#41—62no—
169#41—62no—
170C2×Q8—31no—
171C2×Q8—31no—
172C2×D4—32no—
173C2×D4—31no—
174C2×D4—31no—
175C2×D4—32no—
176C2×D4—62no—
177C2×D4—62no—
178C2×D4—62no—
179C2×D4—61no—
180C2×D4—62no—
181C2×D4—61no—
182C2×D4—62no—
183C22×C4—21no—
184C22×C4—32no—
185C22×C4—61no—
186C22×C4C22×D311yescharacteristic
187SD16—64no—
188D8—64no—
189M16—64no—
190M16D1214yescomplemented
191C2×C8—31no—
192C2×C8—31no—
193C2×C8—62no—
194C2×C8—64no—
195C2×C8C22×D312yes—
196C4⋊C4—21no—
197C4⋊C4—62no—
198C4⋊C4C22×D312yes—
199C4⋊C4C22×D311yescharacteristic
200C4⋊C4C22×D311yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—62no—
203C22⋊C4—62no—
204C22⋊C4—62no—
205C22⋊C4—62no—
206C42—21no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3#12112yes—
210C22×C3C22×D411yescharacteristic
211C2×D3—22no—
212C2×D3—22no—
213C2×D3—22no—
214C2×D3—41no—
215C2×D3—42no—
216C2×D3—41no—
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—41no—
220C2×D3—84no—
221C3×C4—41no—
222C3×C4—42no—
223C3×C4—41no—
224C3×C4#12112yes—
225C3×C4C22×D411yescharacteristic
226C3×C4C2×#4111yescharacteristic
227Dic3—22no—
228Dic3—41no—
229C23—32no—
230C23—61no—
231C23—62no—
232C23—122no—
233C23—241no—
234C23—241no—
235C23D3×D411yescharacteristic
236Q8—62no—
237Q8—61no—
238Q8—61no—
239D4—64no—
240D4—62no—
241D4—61no—
242D4—62no—
243D4—62no—
244D4—61no—
245D4—122no—
246D4—122no—
247D4—242no—
248D4—242no—
249D4—242no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—32no—
254C2×C4—62no—
255C2×C4—62no—
256C2×C4—61no—
257C2×C4—81no—
258C2×C4—122no—
259C2×C4C2×D1212yes—
260C2×C4D3×D411yescharacteristic
261C2×C4#23912yes—
262C2×C4C23×D311yescharacteristic
263C8—61no—
264C8—61no—
265C8C2×D1214yes—
266C2×C3—22no—
267C2×C3C2×#12111yescharacteristic
268C2×C3#57611yescharacteristic
269C2×C3#57711yescharacteristic
270D3—42no—
271D3—81no—
272D3—82no—
273C22—21no—
274C22—21no—
275C22—62no—
276C22—62no—
277C22—62no—
278C22—121no—
279C22—122no—
280C22—121no—
281C22—122no—
282C22—122no—
283C22—121no—
284C22—244no—
285C22#89512yes—
286C22C2×D3×D411yescharacteristic
287C4—42no—
288C4—41no—
289C4—41no—
290C4—62no—
291C4—121no—
292C4#89512yes—
293C4C22×D1211yescharacteristic
294C4C2×#23911yescharacteristic
295C3#306211yescharacteristic, complemented, Sylow
296C2—22no—
297C2—122no—
298C2—242no—
299C2—241no—
300C2C2×#89511yescharacteristic
301C2#577411yescharacteristic
302C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#428C2
socleC22×C3C22×D4
hypercenter#428D3
hypocenterC3#3062
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#908C2semidirect
2. C2×#907C2semidirect
3. #5525C2semidirect
4. #5510C2semidirect
5. C3×#428C2semidirect
6. #5002C2semidirect
7. C2×#809C2semidirect
8. #4886C2semidirect
9. #4883C2semidirect
10. #4865C2semidirect
11. #4864C2semidirect
25. C2×ΓL(1,𝔽25)C22semidirect
26. C2×D3×C8C22semidirect
28. #897C22semidirect
32. #895C22semidirect
36. C3×#107C22semidirect
37. C3×#106C22semidirect
38. #810C22semidirect
39. #810C22semidirect
42. #809C22semidirect
43. #809C22semidirect
52. #428D3semidirect
77. C3×M16D4semidirect
120. #107C2×D3semidirect
121. #106C2×D3semidirect
190. M16D12semidirect
295. C3#3062semidirect

Maximal subgroups

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

subgroupindex
C2×#9082
C2×#9072
#55252
#55102
C3×#4282
#50022
C2×#8092
#48862
#48832
#48652
#48642
#30623