Finite Group Databasebeta

← #83812

Subgroups — #83812

Order 384.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#568C211yesmaximal, characteristic, complemented
2C2×#950C211yesmaximal, characteristic
3#5745C211yesmaximal, characteristic, complemented
4#5236C211yesmaximal, characteristic, complemented
5#5145C214yesmaximal, complemented
6#5144C214yesmaximal, complemented
7#5139C211yesmaximal, characteristic, complemented
8C2×#822C211yesmaximal, characteristic, complemented
9#5122C211yesmaximal, characteristic, complemented
10#2957—31nomaximal, Sylow
11C2×C3×#41C2211yescharacteristic
12C2×#235C2211yescharacteristic
13C3×#131—22no—
14C3×#131C2214yescomplemented
15C2×C3×M16—22no—
16C2×C3×M16C2211yescharacteristic
17C2×C3×M16C2211yescharacteristic
18#950C412yescomplemented
19#950C412yescomplemented
20#950C2212yes—
21#950C2212yes—
22C2×#216C411yescharacteristic, complemented
23C2×#215C411yescharacteristic, complemented
24C2×#214C411yescharacteristic, complemented
25C2×#213C411yescharacteristic, complemented
26#926C2211yescharacteristic
27C2×#208C2211yescharacteristic
28C22×C3⋊C8C2211yescharacteristic
29#907—22no—
30D3×M16—22no—
31#901C2214yescomplemented
32#837C2214yescomplemented
33#825C2214yescomplemented
34#822C2212yescomplemented
35#822C2212yescomplemented
36#815C2212yescomplemented
37C8×Dic3C2212yescomplemented
38C2×#135—31no—
39#568—31no—
40#568D311yescharacteristic, complemented
41#444—34no—
42#443—34no—
43C2×#108—31no—
44#421—31no—
45#421—31no—
46#405—31no—
47C3×#41—21no—
48C3×#41—21no—
49C3×#41C2×C412yes—
50C3×#41C2312yes—
51C2×C3×Q8C2×C411yescharacteristic
52C2×C3×D4—21no—
53C2×C3×D4C2×C411yescharacteristic
54C22×C3×C4—21no—
55C22×C3×C4C2311yescharacteristic
56C2×#67—21no—
57#235—21no—
58#235—21no—
59#235C2×C412yes—
60#235C2312yes—
61C2×D12C2×C411yescharacteristic
62C2×C4×D3—21no—
63C2×Dic6C2×C411yescharacteristic
64C3×M16—21no—
65C3×M16—24no—
66C3×M16D412yes—
67C3×M16D412yes—
68C3×M16C2312yes—
69C2×C3×C8—21no—
70C2×C3×C8—24no—
71C2×C3×C8C2312yes—
72#217—21no—
73#216C2×C412yes—
74#216C2×C412yes—
75#215C2×C412yes—
76#215C2×C412yes—
77#214C2×C412yes—
78#214C2×C412yes—
79#213C2×C412yes—
80#213C2×C412yes—
81C4⋊Dic3—21no—
82C4×Dic3C2312yes—
83#208—21no—
84#208C2312yes—
85C2×C3⋊C8—21no—
86C2×C3⋊C8C2×C411yescharacteristic
87C2×C3⋊C8C2×C411yescharacteristic
88C2×C3⋊C8C2×C411yescharacteristic
89C2×C3⋊C8C2×C411yescharacteristic
90C2×C3⋊C8C2311yescharacteristic
91C2×C3⋊C8C2311yescharacteristic
92ΓL(1,𝔽25)—24no—
93D3×C8—24no—
94C2×#41—31no—
95C2×#41C2×D311yescharacteristic
96#135—32no—
97#135—32no—
98#135—32no—
99#135—32no—
100C2×Q16—31no—
101C2×SD16—31no—
102C2×SD16—31no—
103C2×D8—31no—
104#131—22no—
105#131—34no—
106#131—62no—
107#131C2×D314yescomplemented
108C2×M16—22no—
109C2×M16—31no—
110C2×M16—62no—
111C2×M16C2×D311yescharacteristic
112C2×M16C2×D311yescharacteristic
113C22×C8—31no—
114#117—31no—
115#108—32no—
116#108—32no—
117#104—34no—
118#104—34no—
119#97—32no—
120C4×C8—32no—
121C23×C3—21no—
122C23×C3#4111yescharacteristic
123C22×D3—21no—
124C3×Q8—21no—
125C3×Q8C22×C412yes—
126C3×D4—22no—
127C3×D4—22no—
128C3×D4—21no—
129C3×D4C22×C412yes—
130C2×C3×C4—21no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4C22×C411yescharacteristic
134C2×C3×C4C22×C411yescharacteristic
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4#4111yescharacteristic
138C2×C3×C4C2411yescharacteristic
139#67—22no—
140#67—22no—
141C2×Dic3—21no—
142C2×Dic3—22no—
143D12—21no—
144D12C22×C412yes—
145C4×D3—22no—
146C4×D3—22no—
147Dic6—21no—
148Dic6C22×C412yes—
149C3×C8—22no—
150C3×C8C2×D414yes—
151C3⋊C8—22no—
152C3⋊C8C22×C411yescharacteristic
153C3⋊C8C22×C411yescharacteristic
154C3⋊C8C22×C411yescharacteristic
155C3⋊C8C22×C411yescharacteristic
156#41—21no—
157#41—21no—
158#41—32no—
159#41—32no—
160#41—61no—
161#41—61no—
162#41C4×D312yes—
163#41C22×D312yes—
164C2×Q8—31no—
165C2×Q8C4×D311yescharacteristic
166C2×D4—21no—
167C2×D4—31no—
168C2×D4—61no—
169C2×D4C4×D311yescharacteristic
170C22×C4—21no—
171C22×C4—61no—
172C22×C4C22×D311yescharacteristic
173Q16—32no—
174Q16—32no—
175SD16—32no—
176SD16—32no—
177SD16—32no—
178SD16—32no—
179D8—32no—
180D8—32no—
181M16—21no—
182M16—24no—
183M16—32no—
184M16—61no—
185M16—64no—
186M16#6712yes—
187M16#6712yes—
188M16C22×D312yes—
189C2×C8—21no—
190C2×C8—24no—
191C2×C8—31no—
192C2×C8—31no—
193C2×C8—31no—
194C2×C8—31no—
195C2×C8—31no—
196C2×C8—31no—
197C2×C8—61no—
198C2×C8—64no—
199C2×C8C22×D312yes—
200C4⋊C4—61no—
201C22⋊C4—61no—
202C42—32no—
203C22×C3—22no—
204C22×C3—41no—
205C22×C3—81no—
206C22×C3C4×D411yescharacteristic
207C22×C3C4×D411yescharacteristic
208C22×C3C2×#4111yescharacteristic
209C2×D3—22no—
210C2×D3—81no—
211C3×C4—22no—
212C3×C4C4×D411yescharacteristic
213C3×C4C4×D411yescharacteristic
214C3×C4C23×C411yescharacteristic
215C3×C4C22×D411yescharacteristic
216Dic3—22no—
217Dic3—42no—
218C23—21no—
219C23—61no—
220C23#23511yescharacteristic
221Q8—21no—
222Q8—32no—
223Q8—61no—
224Q8C2×C4×D312yes—
225D4—22no—
226D4—22no—
227D4—21no—
228D4—32no—
229D4—62no—
230D4—62no—
231D4—61no—
232D4C2×C4×D312yes—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—61no—
237C2×C4—62no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4C2×C4×D311yescharacteristic
241C2×C4C2×C4×D311yescharacteristic
242C2×C4#23511yescharacteristic
243C2×C4C2×#6711yescharacteristic
244C2×C4C2×#6711yescharacteristic
245C2×C4C23×D311yescharacteristic
246C8—22no—
247C8—31no—
248C8—31no—
249C8—31no—
250C8—31no—
251C8—62no—
252C8C2×#6714yes—
253C2×C3—21no—
254C2×C3—21no—
255C2×C3—21no—
256C2×C3—42no—
257C2×C3C2×C4×D411yescharacteristic
258D3—42no—
259C22—22no—
260C22—41no—
261C22—62no—
262C22—81no—
263C22—241no—
264C22C4×#6711yescharacteristic
265C22C4×#6711yescharacteristic
266C22C2×#23511yescharacteristic
267C4—22no—
268C4—62no—
269C4—122no—
270C4C4×#6711yescharacteristic
271C4C4×#6711yescharacteristic
272C4C22×C4×D311yescharacteristic
273C4C22×#6711yescharacteristic
274C3#295711yescharacteristic, complemented, Sylow
275C2—21no—
276C2—21no—
277C2—21no—
278C2—42no—
279C2—122no—
280C2C2×C4×#6711yescharacteristic
subgroupG/H
centerC4C22×#67
commutatorC3×C4C23×C4
FrattiniC2×C4C23×D3
FittingC3×#568C2
socleC2×C3C2×C4×D4
prime radicalC2×#950C2
cosocleC2×C3×C4C24
hypercenter#568D3
hypocenterC3#2957
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C2×M16 ◁ #568

Complements

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

NH ≅ G/Ntype
1. C3×#568C2semidirect
3. #5745C2semidirect
4. #5236C2semidirect
5. #5145C2semidirect
6. #5144C2semidirect
7. #5139C2semidirect
8. C2×#822C2semidirect
9. #5122C2semidirect
14. C3×#131C22semidirect
18. #950C4semidirect
19. #950C4semidirect
22. C2×#216C4semidirect
23. C2×#215C4semidirect
24. C2×#214C4semidirect
25. C2×#213C4semidirect
31. #901C22semidirect
32. #837C22semidirect
33. #825C22semidirect
34. #822C22semidirect
35. #822C22semidirect
36. #815C22semidirect
37. C8×Dic3C22semidirect
40. #568D3semidirect
107. #131C2×D3semidirect
274. C3#2957semidirect

Maximal subgroups

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

subgroupindex
C3×#5682
C2×#9502
#57452
#52362
#51452
#51442
#51392
C2×#8222
#51222
#29573