Finite Group Databasebeta

← #10951981

Subgroups — #10951981

Order 768.

668 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 135 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#82019C212yesmaximal, complemented
2#81791C212yesmaximal, complemented
3C3×#2246C211yesmaximal, characteristic, complemented
4#71472C212yesmaximal, complemented
5#71471C214yesmaximal, complemented
6#71295C212yesmaximal, complemented
7#69067C211yesmaximal, characteristic, complemented
8#69056C211yesmaximal, characteristic, complemented
9#33858—31nomaximal, Sylow
10#5757—22no—
11#5757C2214yescomplemented
12C2×C3×D16C2212yescomplemented
13C2×C3×D16C2212yescomplemented
14C3×#493C2212yescomplemented
15#5160—22no—
16#5160C2214yescomplemented
17#5147C2214yescomplemented
18#5077—24no—
19#4912—22no—
20#4911—24no—
21D3×D16—22no—
22#4904C2211yescharacteristic
23#4901C2212yescomplemented
24#4700C2211yescharacteristic
25#4700C2212yescomplemented
26C3×C4×C16C2211yescharacteristic, complemented
27#4563C2214yescomplemented
28#4510C2212yescomplemented
29#4508C2212yescomplemented
30#4506C2211yescharacteristic, complemented
31#4463C2211yescharacteristic, complemented
32#3415—32no—
33#3292—32no—
34#2246D311yescharacteristic, complemented
35#2221—32no—
36#2213—32no—
37#2192—34no—
38#2178—31no—
39#2170—31no—
40#1004—24no—
41C2×C3×D8—22no—
42C2×C3×D8C2314yes—
43C3×#127—22no—
44#932—24no—
45#932—24no—
46#912—24no—
47#912—42no—
48#911—44no—
49#911—44no—
50D3×D8—24no—
51D3×D8—42no—
52#904D411yescharacteristic
53#904C2312yes—
54#901—22no—
55#901C2312yes—
56C3×D16—22no—
57C3×D16—22no—
58C3×D16—42no—
59C2×C3×C16C2311yescharacteristic
60C2×C3×C16C2311yescharacteristic
61C3×C4×C8C2311yescharacteristic
62#833—24no—
63#827—24no—
64#826—24no—
65#819D411yescharacteristic
66#819C2312yes—
67#812C2312yes—
68#805—22no—
69#805D412yes—
70Dic24D411yescharacteristic, complemented
71#800D412yescomplemented
72D48D411yescharacteristic, complemented
73#798—22no—
74D3×C16—22no—
75#576—34no—
76#576—62no—
77#509—34no—
78#509—64no—
79#508—31no—
80#508—62no—
81C2×D16—62no—
82C2×D16C2×D312yescomplemented
83C2×D16C2×D312yescomplemented
84#504—32no—
85#493C2×D312yescomplemented
86#472—34no—
87#472—62no—
88#453—64no—
89#443—31no—
90#443—32no—
91#368—31no—
92#364—31no—
93#361—34no—
94#359—32no—
95#350—32no—
96C4×C16C2×D311yescharacteristic, complemented
97C2×C3×D4—24no—
98C2×C3×D4—82no—
99C2×#67—84no—
100#237—44no—
101#237—44no—
102D3×D4—44no—
103D3×D4—44no—
104#235—22no—
105#235C2×D412yes—
106C3×D8—24no—
107C3×D8—24no—
108C3×D8—42no—
109C3×D8—42no—
110C2×C3×C8C2×D411yescharacteristic
111C2×C3×C8C2411yescharacteristic
112C3×C42C2×D411yescharacteristic
113#214—44no—
114#214—44no—
115#213—44no—
116#213—44no—
117#208—22no—
118#208C2×D412yes—
119Dic12D811yescharacteristic
120Dic12C2×D412yes—
121D24D811yescharacteristic
122D24C2×D412yes—
123#204—22no—
124#204D812yes—
125ΓL(1,𝔽25)—22no—
126ΓL(1,𝔽25)—42no—
127D3×C8—22no—
128D3×C8—42no—
129C3×C16—21no—
130C3×C16C2×D411yescharacteristic
131C3×C16C2×D411yescharacteristic
132C3⋊C16—22no—
133H(5,𝔽2)—64no—
134AGL(1,ℤ/8)—64no—
135AGL(1,ℤ/8)—64no—
136AGL(1,ℤ/8)—124no—
137AGL(1,ℤ/8)—124no—
138#135—31no—
139#135—32no—
140#135—64no—
141#135—122no—
142C2×D8—22no—
143C2×D8—64no—
144C2×D8—122no—
145C2×D8C22×D314yes—
146#131—32no—
147#131—62no—
148#127—22no—
149Q32—31no—
150SD32—32no—
151SD32—64no—
152D16—22no—
153D16—22no—
154D16—31no—
155D16—42no—
156D16—64no—
157M32—32no—
158M32—62no—
159C2×C16—62no—
160C2×C16C22×D311yescharacteristic
161C2×C16C22×D311yescharacteristic
162#108—31no—
163#108—32no—
164#104—32no—
165#104—62no—
166#100—64no—
167C4×C8C22×D311yescharacteristic
168C23×C3—44no—
169C22×D3—84no—
170C3×D4—44no—
171C3×D4—44no—
172C3×D4—164no—
173C2×C3×C4—21no—
174C2×C3×C4C22×D411yescharacteristic
175#67—22no—
176#67—42no—
177#67—84no—
178#67—84no—
179C2×Dic3—84no—
180D12—22no—
181D12C2×D812yes—
182C4×D3—22no—
183C4×D3—42no—
184Dic6—22no—
185Dic6C2×D812yes—
186C3×C8C2×D811yescharacteristic
187C3×C8C2×D811yescharacteristic
188C3×C8C22×D411yescharacteristic
189C3×C8H(5,𝔽2)11yescharacteristic
190C3⋊C8—22no—
191C3⋊C8—42no—
192#41—32no—
193#41—62no—
194#41—124no—
195#41—124no—
196C2×D4—24no—
197C2×D4—82no—
198C2×D4—124no—
199C2×D4—124no—
200C2×D4—244no—
201Q16—31no—
202Q16—32no—
203SD16—32no—
204SD16—62no—
205SD16—124no—
206SD16—124no—
207D8—24no—
208D8—24no—
209D8—31no—
210D8—32no—
211D8—42no—
212D8—42no—
213D8—124no—
214D8—124no—
215M16—32no—
216M16—62no—
217M16—62no—
218M16—122no—
219C2×C8—62no—
220C2×C8—122no—
221C2×C8D3×D411yescharacteristic
222C2×C8C23×D311yescharacteristic
223C42D3×D411yescharacteristic
224C16—21no—
225C16—62no—
226C16D3×D411yescharacteristic
227C16D3×D411yescharacteristic
228C22×C3—84no—
229C22×C3—164no—
230C22×C3#57611yescharacteristic
231C2×D3—22no—
232C2×D3—42no—
233C2×D3—324no—
234C3×C4—22no—
235C3×C4D4211yescharacteristic
236C3×C4C22×D811yescharacteristic
237Dic3—22no—
238Dic3—42no—
239C23—44no—
240C23—244no—
241Q8—32no—
242Q8—62no—
243D4—32no—
244D4—44no—
245D4—44no—
246D4—62no—
247D4—62no—
248D4—122no—
249D4—164no—
250D4—244no—
251D4—244no—
252C2×C4—21no—
253C2×C4—62no—
254C2×C4—122no—
255C2×C4—244no—
256C2×C4C2×D3×D411yescharacteristic
257C8—62no—
258C8—122no—
259C8D3×D811yescharacteristic
260C8D3×D811yescharacteristic
261C8C2×D3×D411yescharacteristic
262C8#100411yescharacteristic
263C2×C3—21no—
264C2×C3—164no—
265C2×C3D4×D811yescharacteristic
266D3—42no—
267D3—82no—
268C22—62no—
269C22—84no—
270C22—122no—
271C22—164no—
272C22—964no—
273C22#575711yescharacteristic
274C4—22no—
275C4—62no—
276C4—122no—
277C4#571711yescharacteristic
278C4C2×D3×D811yescharacteristic
279C3#3385811yescharacteristic, complemented, Sylow
280C2—21no—
281C2—122no—
282C2—164no—
283C2—242no—
284C2#8178911yescharacteristic
subgroupG/H
centerC2#81789
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2246C2
socleC2×C3D4×D8
hypercenter#2246D3
hypocenterC3#33858
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2246

Complements

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

NH ≅ G/Ntype
1. #82019C2semidirect
2. #81791C2semidirect
3. C3×#2246C2semidirect
4. #71472C2semidirect
5. #71471C2semidirect
6. #71295C2semidirect
7. #69067C2semidirect
8. #69056C2semidirect
11. #5757C22semidirect
12. C2×C3×D16C22semidirect
13. C2×C3×D16C22semidirect
14. C3×#493C22semidirect
16. #5160C22semidirect
17. #5147C22semidirect
23. #4901C22semidirect
25. #4700C22semidirect
26. C3×C4×C16C22semidirect
27. #4563C22semidirect
28. #4510C22semidirect
29. #4508C22semidirect
30. #4506C22semidirect
31. #4463C22semidirect
34. #2246D3semidirect
70. Dic24D4semidirect
71. #800D4semidirect
72. D48D4semidirect
82. C2×D16C2×D3semidirect
83. C2×D16C2×D3semidirect
85. #493C2×D3semidirect
96. C4×C16C2×D3semidirect
279. C3#33858semidirect

Maximal subgroups

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

subgroupindex
#820192
#817912
C3×#22462
#714722
#714712
#712952
#690672
#690562
#338583