Finite Group Databasebeta

← #13179893

Subgroups — #13179893

Order 1728.

3668 conjugacy classes of subgroups; only the 619 normal subgroups are shown.

normal subgroups only

This group has 3668 conjugacy classes of subgroups — too many to draw. Only its 619 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#41×C3⋊(C3⋊D3)C212yesmaximal, complemented
2C33×#135C211yesmaximal, characteristic, complemented
3#11723991C212yesmaximal, complemented
4#11723959C212yesmaximal, complemented
5#11723956C212yesmaximal, complemented
6#11723955C212yesmaximal, complemented
7#11723952C212yesmaximal, complemented
8#11723945C211yesmaximal, characteristic, complemented
9C2×#88850C211yesmaximal, characteristic, complemented
10#10596239—313nomaximal
11C33×#41C2212yescomplemented
12#89080C2212yes—
13Q8×C3⋊(C3⋊D3)C2212yescomplemented
14#89078—22no—
15#89078C2212yes—
16D4×C3⋊(C3⋊D3)—22no—
17D4×C3⋊(C3⋊D3)C2212yescomplemented
18#89076—22no—
19#89076C2212yescomplemented
20C2×C4×C3⋊(C3⋊D3)—22no—
21C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
22C33×Q16C2211yescharacteristic, complemented
23C33×SD16C2212yescomplemented
24C33×D8C2211yescharacteristic, complemented
25C2×C33×C8C2211yescharacteristic, complemented
26#88863C2212yescomplemented
27#88862C2212yescomplemented
28#88861C2212yescomplemented
29#88860C2212yescomplemented
30C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
31#88853C2211yescharacteristic, complemented
32#88852C2211yescharacteristic, complemented
33#88851C2212yescomplemented
34#88850C2212yescomplemented
35#88850C2212yescomplemented
36#41×C3⋊D3—326no—
37C32×#135D3113yescomplemented
38#64159—326no—
39#64127—326no—
40#64124—326no—
41#64123—326no—
42#64120—326no—
43#64113—313no—
44C2×#3743—313no—
45C22×C3⋊(C3⋊D3)—22no—
46C22×C3⋊(C3⋊D3)D411yescharacteristic
47C33×Q8C2312yes—
48C33×D4—22no—
49C33×D4C2312yes—
50C2×C33×C4—22no—
51C2×C33×C4C2311yescharacteristic
52#6312—22no—
53#6312—22no—
54#6312—22no—
55C2×C3⋊(C3⋊Dic3)—22no—
56C2×C3⋊(C3⋊Dic3)D411yescharacteristic
57C4⋊(C3⋊(C3⋊D3))—22no—
58C4⋊(C3⋊(C3⋊D3))C2312yes—
59C4×C3⋊(C3⋊D3)—22no—
60C4×C3⋊(C3⋊D3)—22no—
61C4×C3⋊(C3⋊D3)—22no—
62C4×C3⋊(C3⋊D3)D412yes—
63C4×C3⋊(C3⋊D3)C2312yes—
64#6308—22no—
65#6308C2312yes—
66C33×C8C2311yescharacteristic
67C33×C8C2311yescharacteristic
68C3⋊(C3⋊(C3⋊C8))C2312yes—
69#5768—913no—
70C32×#41C2×D3126yescomplemented
71#3832—326no—
72Q8×C3⋊D3—326no—
73#3830—326no—
74#3830—626no—
75D4×C3⋊D3—326no—
76D4×C3⋊D3—626no—
77#3828—326no—
78#3828—626no—
79C2×C4×C3⋊D3—313no—
80C2×C4×C3⋊D3—626no—
81C32×Q16C2×D3113yescomplemented
82C32×SD16C2×D3126yescomplemented
83C32×D8C2×D3113yescomplemented
84C2×C32×C8C2×D3113yescomplemented
85#3756—326no—
86#3755—326no—
87#3754—326no—
88#3753—326no—
89C2×C3⋊(C3⋊C8)—313no—
90#3746—313no—
91#3745—313no—
92#3744—326no—
93#3743—326no—
94#3743—326no—
95C22×C33—22no—
96C22×C33C2×D411yescharacteristic
97C2×C3⋊(C3⋊D3)—22no—
98C2×C3⋊(C3⋊D3)—41no—
99C2×C3⋊(C3⋊D3)—44no—
100C2×C3⋊(C3⋊D3)C2×D412yes—
101C33×C4—22no—
102C33×C4C2×D411yescharacteristic
103C33×C4C2411yescharacteristic
104C3⋊(C3⋊Dic3)—22no—
105C3⋊(C3⋊Dic3)C2×D412yes—
106D3×#41—926no—
107C3×#135C3⋊D3113yescomplemented
108#950—926no—
109#918—926no—
110#915—926no—
111#914—926no—
112#911—926no—
113#904—913no—
114C2×ΓL(1,𝔽25)—913no—
115C22×C3⋊D3—313no—
116C22×C3⋊D3—626no—
117C32×Q8C22×D3126yes—
118C32×D4—226no—
119C32×D4C22×D3126yes—
120C2×C32×C4—226no—
121C2×C32×C4C22×D3113yes—
122#638—626no—
123#638—626no—
124#638—626no—
125C2×C3⋊Dic3—313no—
126C2×C3⋊Dic3—626no—
127C4⋊(C3⋊D3)—326no—
128C4⋊(C3⋊D3)—626no—
129C4×C3⋊D3—326no—
130C4×C3⋊D3—326no—
131C4×C3⋊D3—626no—
132C4×C3⋊D3—626no—
133C4×C3⋊D3—626no—
134#634—326no—
135#634—626no—
136C32×C8C22×D3113yes—
137C32×C8C22×D3113yes—
138C3⋊(C3⋊C8)—326no—
139#575—271noSylow
140C2×C33—21no—
141C2×C33—42no—
142C2×C33C22×D411yescharacteristic
143C3⋊(C3⋊D3)—22no—
144C3⋊(C3⋊D3)—42no—
145C3×#41C2×C3⋊D3126yescomplemented
146#239—926no—
147D3×Q8—926no—
148#237—926no—
149#237—1826no—
150D3×D4—926no—
151D3×D4—1826no—
152#235—926no—
153#235—1826no—
154C2×C4×D3—913no—
155C2×C4×D3—1826no—
156C3×Q16C2×C3⋊D3113yescomplemented
157C3×SD16C2×C3⋊D3126yescomplemented
158C3×D8C2×C3⋊D3113yescomplemented
159C2×C3×C8C2×C3⋊D3113yescomplemented
160#216—926no—
161#215—926no—
162#214—926no—
163#213—926no—
164C2×C3⋊C8—913no—
165Dic12—913no—
166D24—913no—
167#204—926no—
168ΓL(1,𝔽25)—926no—
169ΓL(1,𝔽25)—926no—
170C22×C32—226no—
171C22×C32D3×D4113yes—
172C2×C3⋊D3—326no—
173C2×C3⋊D3—626no—
174C2×C3⋊D3—1213no—
175C2×C3⋊D3—1252no—
176C32×C4—226no—
177C32×C4D3×D4113yes—
178C32×C4C23×D3113yes—
179C3⋊Dic3—326no—
180C3⋊Dic3—626no—
181C2×#41—272no—
182#137—272no—
183#137—272no—
184AGL(1,ℤ/8)—272no—
185AGL(1,ℤ/8)—272no—
186#135—272no—
187#135—271no—
188#135C3⋊(C3⋊D3)11yescharacteristic, complemented
189C2×M16—271no—
190C33#57511yescharacteristic, complemented, Sylow
191C22×D3—913no—
192C22×D3—1826no—
193C3×Q8C22×C3⋊D3126yes—
194C3×D4—226no—
195C3×D4C22×C3⋊D3126yes—
196C2×C3×C4—226no—
197C2×C3×C4C22×C3⋊D3113yes—
198#67—1826no—
199#67—1826no—
200#67—1826no—
201C2×Dic3—913no—
202C2×Dic3—1826no—
203D12—926no—
204D12—1826no—
205C4×D3—926no—
206C4×D3—926no—
207C4×D3—1826no—
208C4×D3—1826no—
209C4×D3—1826no—
210Dic6—926no—
211Dic6—1826no—
212C3×C8C22×C3⋊D3113yes—
213C3×C8C22×C3⋊D3113yes—
214C3⋊C8—926no—
215C2×C32—213no—
216C2×C32—426no—
217C2×C32C2×D3×D4113yes—
218C3⋊D3—626no—
219C3⋊D3—1226no—
220#41—272no—
221#41—272no—
222#41—272no—
223#41—542no—
224#41—542no—
225#41C2×C3⋊(C3⋊D3)12yescomplemented
226C2×Q8—272no—
227C2×D4—272no—
228C2×D4—542no—
229C22×C4—271no—
230C22×C4—542no—
231Q16—272no—
232Q16—271no—
233Q16C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
234SD16—272no—
235SD16—272no—
236SD16—272no—
237SD16C2×C3⋊(C3⋊D3)12yescomplemented
238D8—272no—
239D8—271no—
240D8C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
241M16—272no—
242M16—272no—
243C2×C8—271no—
244C2×C8C2×C3⋊(C3⋊D3)11yescharacteristic, complemented
245C22×C3—226no—
246C22×C3D4×C3⋊D3113yes—
247C2×D3—926no—
248C2×D3—1826no—
249C2×D3—3613no—
250C2×D3—3652no—
251C3×C4—226no—
252C3×C4D4×C3⋊D3113yes—
253C3×C4C23×C3⋊D3113yes—
254Dic3—926no—
255Dic3—1826no—
256C32#5768113yescomplemented
257C23—271no—
258C23—542no—
259Q8—272no—
260Q8—542no—
261Q8C22×C3⋊(C3⋊D3)12yes—
262D4—22no—
263D4—272no—
264D4—542no—
265D4—542no—
266D4—542no—
267D4—542no—
268D4C22×C3⋊(C3⋊D3)12yes—
269C2×C4—22no—
270C2×C4—271no—
271C2×C4—272no—
272C2×C4—272no—
273C2×C4—542no—
274C2×C4—542no—
275C2×C4—542no—
276C2×C4—542no—
277C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
278C8—272no—
279C8C22×C3⋊(C3⋊D3)11yescharacteristic
280C8C22×C3⋊(C3⋊D3)11yescharacteristic
281C2×C3—213no—
282C2×C3—426no—
283C2×C3C2×D4×C3⋊D3113yes—
284D3—1826no—
285D3—3626no—
286C22—22no—
287C22—272no—
288C22—542no—
289C22—1081no—
290C22—1084no—
291C22D4×C3⋊(C3⋊D3)11yescharacteristic
292C4—22no—
293C4—272no—
294C4—542no—
295C4D4×C3⋊(C3⋊D3)11yescharacteristic
296C4C23×C3⋊(C3⋊D3)11yescharacteristic
297C3#10596239113yescomplemented
298C2—21no—
299C2—42no—
300C2—542no—
301C2—1082no—
302C2C2×D4×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC4D4×C3⋊(C3⋊D3)
commutator, cosocleC33×C4C24
FrattiniC4C23×C3⋊(C3⋊D3)
FittingC33×#135C2
socleC2×C33C22×D4
hypercenter#135C3⋊(C3⋊D3)
hypocenterC33#575
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #41×C3⋊(C3⋊D3)C2semidirect
2. C33×#135C2semidirect
3. #11723991C2semidirect
4. #11723959C2semidirect
5. #11723956C2semidirect
6. #11723955C2semidirect
7. #11723952C2semidirect
8. #11723945C2semidirect
9. C2×#88850C2semidirect
11. C33×#41C22semidirect
13. Q8×C3⋊(C3⋊D3)C22semidirect
17. D4×C3⋊(C3⋊D3)C22semidirect
19. #89076C22semidirect
22. C33×Q16C22semidirect
23. C33×SD16C22semidirect
24. C33×D8C22semidirect
25. C2×C33×C8C22semidirect
26. #88863C22semidirect
27. #88862C22semidirect
28. #88861C22semidirect
29. #88860C22semidirect
30. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
31. #88853C22semidirect
32. #88852C22semidirect
33. #88851C22semidirect
34. #88850C22semidirect
35. #88850C22semidirect
37. C32×#135D3semidirect
70. C32×#41C2×D3semidirect
81. C32×Q16C2×D3semidirect
82. C32×SD16C2×D3semidirect
83. C32×D8C2×D3semidirect
84. C2×C32×C8C2×D3semidirect
107. C3×#135C3⋊D3semidirect
145. C3×#41C2×C3⋊D3semidirect
156. C3×Q16C2×C3⋊D3semidirect
157. C3×SD16C2×C3⋊D3semidirect
158. C3×D8C2×C3⋊D3semidirect
159. C2×C3×C8C2×C3⋊D3semidirect
188. #135C3⋊(C3⋊D3)semidirect
190. C33#575semidirect
225. #41C2×C3⋊(C3⋊D3)semidirect
233. Q16C2×C3⋊(C3⋊D3)semidirect
237. SD16C2×C3⋊(C3⋊D3)semidirect
240. D8C2×C3⋊(C3⋊D3)semidirect
244. C2×C8C2×C3⋊(C3⋊D3)semidirect
256. C32#5768semidirect
297. C3#10596239semidirect

Maximal subgroups

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

subgroupindex
#41×C3⋊(C3⋊D3)2
C33×#1352
#117239912
#117239592
#117239562
#117239552
#117239522
#117239452
C2×#888502
#105962393