Finite Group Databasebeta

← #80716

Subgroups — #80716

Order 384.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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×D3×D8C211yesmaximal, characteristic, complemented
2C2×#901C211yesmaximal, characteristic, complemented
3#5607C212yesmaximal, complemented
4C3×#466C211yesmaximal, characteristic, complemented
5#5157C211yesmaximal, characteristic, complemented
6#5115C211yesmaximal, characteristic, complemented
7#5039C212yesmaximal, complemented
8#4883C211yesmaximal, characteristic, complemented
9#4880C211yesmaximal, characteristic, complemented
10#4782C212yesmaximal, complemented
11#4770C212yesmaximal, complemented
12#3058—31nomaximal, Sylow
13C2×D3×D4C2212yes—
14C2×#235C2211yescharacteristic
15C2×C3×D8C2211yescharacteristic
16C22×C3×C8C2211yescharacteristic, complemented
17C3×#121C2212yescomplemented
18#940—22no—
19#938C2212yescomplemented
20#937—22no—
21C2×#213C2212yes—
22#930C2212yescomplemented
23C2×#208C2211yescharacteristic, complemented
24D3×D8—24no—
25C2×D24C2211yescharacteristic
26#901—24no—
27C2×ΓL(1,𝔽25)C2211yescharacteristic
28C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
29C2×D3×C8C2211yescharacteristic
30C2×D3×C8C2211yescharacteristic, complemented
31#895C2212yescomplemented
32#892C2212yes—
33#885—22no—
34#882—22no—
35C3×#107C2211yescharacteristic, complemented
36C3×#102C2212yescomplemented
37#832C2212yescomplemented
38#821C2212yescomplemented
39#818C2211yescharacteristic, complemented
40#809C2212yescomplemented
41#808C2212yescomplemented
42C22×D8—31no—
43C2×#131—31no—
44#535—32no—
45#469—32no—
46#469—32no—
47#466—31no—
48#466—31no—
49#466—31no—
50#466D311yescharacteristic, complemented
51#427—31no—
52#418—32no—
53C23×D3—22no—
54C2×C3×D4—22no—
55C2×C3×D4C2312yes—
56C22×C3×C4C2311yescharacteristic
57C2×#67—22no—
58C2×#67—22no—
59C2×#67—22no—
60C2×#67D411yescharacteristic
61C2×#67D411yescharacteristic
62D3×D4—24no—
63D3×D4—24no—
64#235—22no—
65#235—22no—
66C2×D12—22no—
67C2×D12D411yescharacteristic
68C2×D12C2312yes—
69C2×C4×D3C2311yescharacteristic
70C2×C4×D3C2311yescharacteristic
71C2×Dic6D411yescharacteristic
72C3×D8—22no—
73C2×C3×C8—22no—
74C2×C3×C8C2311yescharacteristic
75C2×C3×C8C2311yescharacteristic
76C3×C4⋊C4C2312yes—
77C3×C22⋊C4—22no—
78#217—22no—
79#213—24no—
80(C2×D3)⋊C4—22no—
81(C2×D3)⋊C4—22no—
82(C2×D3)⋊C4—22no—
83C4⋊Dic3C2312yes—
84C4×Dic3—22no—
85#208—22no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88D24—22no—
89ΓL(1,𝔽25)—22no—
90ΓL(1,𝔽25)—22no—
91D3×C8—22no—
92D3×C8D414yescomplemented
93C2×#41—31no—
94C22×D4—32no—
95C2×D8—32no—
96C2×D8—31no—
97C2×D8—64no—
98C2×D8C2×D311yescharacteristic
99#131—64no—
100C2×M16—31no—
101C2×M16—31no—
102C2×M16—31no—
103C22×C8—31no—
104C22×C8—31no—
105C22×C8C2×D311yescharacteristic, complemented
106#127—62no—
107#124—62no—
108#121—32no—
109#121—32no—
110#121—32no—
111#121C2×D312yescomplemented
112#120—62no—
113#120—62no—
114#117—32no—
115#107—31no—
116#107C2×D311yescharacteristic, complemented
117#106—32no—
118#102—32no—
119#102—32no—
120#102—32no—
121#102C2×D312yescomplemented
122C23×C3—22no—
123C23×C3C2×D411yescharacteristic
124C22×D3—22no—
125C22×D3—44no—
126C22×D3—82no—
127C22×D3—82no—
128C22×D3C2×D411yescharacteristic
129C22×D3C2×D411yescharacteristic
130C3×D4—22no—
131C3×D4—22no—
132C3×D4—82no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4C2411yescharacteristic
138#67—22no—
139#67—22no—
140#67—44no—
141#67—82no—
142#67—82no—
143C2×Dic3—22no—
144C2×Dic3C2×D411yescharacteristic
145C2×Dic3C2×D411yescharacteristic
146D12—22no—
147D12—22no—
148D12—22no—
149D12—82no—
150C4×D3—21no—
151C4×D3—21no—
152C4×D3C2×D412yes—
153C4×D3#4112yes—
154Dic6—22no—
155C3×C8—21no—
156C3×C8C2×D412yes—
157C3⋊C8—21no—
158C3⋊C8C2×D412yes—
159C24—62no—
160#41—62no—
161#41—62no—
162C2×Q8—31no—
163C2×D4—22no—
164C2×D4—31no—
165C2×D4—31no—
166C2×D4—31no—
167C2×D4—32no—
168C2×D4—62no—
169C2×D4—64no—
170C2×D4—64no—
171C2×D4—62no—
172C2×D4—62no—
173C2×D4—62no—
174C2×D4C22×D312yes—
175C22×C4—31no—
176C22×C4—31no—
177C22×C4C22×D311yescharacteristic
178D8—22no—
179D8—64no—
180D8—62no—
181M16—62no—
182M16—62no—
183M16—62no—
184C2×C8—22no—
185C2×C8—31no—
186C2×C8—34no—
187C2×C8—31no—
188C2×C8—62no—
189C2×C8C22×D311yescharacteristic
190C2×C8C22×D311yescharacteristic
191C4⋊C4—32no—
192C4⋊C4C22×D312yes—
193C22⋊C4—22no—
194C22⋊C4—62no—
195C22⋊C4—62no—
196C22⋊C4—62no—
197C22⋊C4—62no—
198C42—62no—
199C22×C3—21no—
200C22×C3—21no—
201C22×C3—21no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3—42no—
205C22×C3C22×D411yescharacteristic
206C2×D3—21no—
207C2×D3—21no—
208C2×D3—21no—
209C2×D3—21no—
210C2×D3—21no—
211C2×D3—42no—
212C2×D3—42no—
213C2×D3—42no—
214C2×D3—88no—
215C2×D3#12112yes—
216C3×C4—21no—
217C3×C4—42no—
218C3×C4C22×D411yescharacteristic
219C3×C4C2×#4111yescharacteristic
220Dic3—21no—
221Dic3—42no—
222Dic3#12112yes—
223C23—22no—
224C23—31no—
225C23—31no—
226C23—62no—
227C23—124no—
228C23—242no—
229C23—242no—
230C23D3×D411yescharacteristic
231Q8—62no—
232D4—22no—
233D4—22no—
234D4—62no—
235D4—62no—
236D4—62no—
237D4—62no—
238D4—62no—
239D4—82no—
240D4—124no—
241D4—242no—
242D4—242no—
243D4—242no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—31no—
248C2×C4—32no—
249C2×C4—32no—
250C2×C4—31no—
251C2×C4—61no—
252C2×C4—61no—
253C2×C4—62no—
254C2×C4D3×D411yescharacteristic
255C2×C4C23×D311yescharacteristic
256C8—21no—
257C8—32no—
258C8—61no—
259C8D3×D412yes—
260C2×C3—41no—
261C2×C3—82no—
262C2×C3C2×#12111yescharacteristic
263C2×C3#57611yescharacteristic
264C2×C3#57611yescharacteristic
265D3—22no—
266D3—41no—
267D3—82no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—32no—
272C22—42no—
273C22—42no—
274C22—42no—
275C22—61no—
276C22—61no—
277C22—61no—
278C22—61no—
279C22—61no—
280C22—122no—
281C22—122no—
282C22—122no—
283C22—248no—
284C22C2×D3×D411yescharacteristic
285C4—21no—
286C4—32no—
287C4—42no—
288C4—61no—
289C4—122no—
290C4C2×D3×D411yescharacteristic
291C4D3×#4111yescharacteristic
292C3#305811yescharacteristic, complemented, Sylow
293C2—41no—
294C2—62no—
295C2—82no—
296C2—121no—
297C2—242no—
298C2D3×#12111yescharacteristic
299C2#575711yescharacteristic
300C2#576911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#466C2
socleC22×C3C22×D4
hypercenter#466D3
hypocenterC3#3058
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×D3×D8C2semidirect
2. C2×#901C2semidirect
3. #5607C2semidirect
4. C3×#466C2semidirect
5. #5157C2semidirect
6. #5115C2semidirect
7. #5039C2semidirect
8. #4883C2semidirect
9. #4880C2semidirect
10. #4782C2semidirect
11. #4770C2semidirect
16. C22×C3×C8C22semidirect
17. C3×#121C22semidirect
19. #938C22semidirect
22. #930C22semidirect
23. C2×#208C22semidirect
28. C2×ΓL(1,𝔽25)C22semidirect
30. C2×D3×C8C22semidirect
31. #895C22semidirect
35. C3×#107C22semidirect
36. C3×#102C22semidirect
37. #832C22semidirect
38. #821C22semidirect
39. #818C22semidirect
40. #809C22semidirect
41. #808C22semidirect
50. #466D3semidirect
92. D3×C8D4semidirect
105. C22×C8C2×D3semidirect
111. #121C2×D3semidirect
116. #107C2×D3semidirect
121. #102C2×D3semidirect
292. C3#3058semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D82
C2×#9012
#56072
C3×#4662
#51572
#51152
#50392
#48832
#48802
#47822
#47702
#30583