Finite Group Databasebeta

← #13160003

Subgroups — #13160003

Order 1728.

872 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 162 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#11723942C211yesmaximal, characteristic, complemented
2C3×#64110C211yesmaximal, characteristic, complemented
3#11723369C211yesmaximal, characteristic
4C32×#815C211yesmaximal, characteristic, complemented
5#11722058C211yesmaximal, characteristic, complemented
6#11722048C211yesmaximal, characteristic, complemented
7#11722046C211yesmaximal, characteristic, complemented
8#10592416—31nomaximal
9#10590924—34nomaximal
10#89076C2211yescharacteristic
11C3×#3828C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13#88855C2211yescharacteristic
14#88850—21no—
15C8×C3⋊(C3⋊D3)—21no—
16C3×#3748C2211yescharacteristic
17C3×#3743—21no—
18C3×C8×C3⋊D3—21no—
19C32×C4×Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88800C411yescharacteristic, complemented
22#88797C411yescharacteristic, complemented
23#88796C411yescharacteristic, complemented
24#88791C411yescharacteristic, complemented
25#64113—31no—
26#64110—31no—
27#64110—34no—
28#64110—64no—
29#64110D311yescharacteristic, complemented
30C3×#901—34no—
31#63829—34no—
32C32×#97—31no—
33#63646—31no—
34#63632—31no—
35#63632—31no—
36C3×#815D314yescomplemented
37#63577—34no—
38#63570—34no—
39#63569—34no—
40C2×C33×C4C2311yescharacteristic
41#6312—21no—
42C4⋊(C3⋊(C3⋊D3))C2×C411yescharacteristic
43C4×C3⋊(C3⋊D3)—21no—
44#6308C2×C411yescharacteristic
45C3×#638—21no—
46C3×C4⋊(C3⋊D3)C2×C411yescharacteristic
47C3×C4×C3⋊D3—21no—
48C3×#634C2×C411yescharacteristic
49C2×C32×Dic3—21no—
50C33×C8D411yescharacteristic
51C33×C8D411yescharacteristic
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)—21no—
54C32×C3⋊C8C2×C411yescharacteristic
55C32×C3⋊C8C2×C411yescharacteristic
56#5145—91no—
57#4717—94no—
58#3828—31no—
59#3828—34no—
60#3828—64no—
61#3828C2×D311yescharacteristic
62C3×#235—34no—
63C2×C32×C8—24no—
64C2×C32×C8—31no—
65C2×C32×C8C2×D314yes—
66C2×C32×C8C2×D311yescharacteristic
67C32×C42—31no—
68#3748—31no—
69#3748—34no—
70#3748—64no—
71#3748C2×D311yescharacteristic
72#3746—31no—
73#3745—31no—
74#3744—31no—
75#3744—31no—
76#3743—21no—
77#3743—68no—
78#3743—64no—
79#3743—61no—
80C8×C3⋊D3—21no—
81C8×C3⋊D3—68no—
82C8×C3⋊D3—64no—
83C8×C3⋊D3—61no—
84C3×C4×Dic3C2×D314yes—
85C3×#208—34no—
86C2×C3×C3⋊C8C2×D314yes—
87C3×ΓL(1,𝔽25)—64no—
88C3×D3×C8—64no—
89#3719—34no—
90#3717—34no—
91#3716—34no—
92#3714—34no—
93C22×C33#4111yescharacteristic
94C2×C3⋊(C3⋊D3)—21no—
95C2×C3×C3⋊D3—21no—
96C33×C4C22×C411yescharacteristic
97C33×C4C2×D411yescharacteristic
98C3⋊(C3⋊Dic3)—21no—
99C3×C3⋊Dic3—21no—
100C32×Dic3—41no—
101C3×#131—91no—
102#950—91no—
103#904—94no—
104#901—34no—
105#901—94no—
106#901—91no—
107#901—184no—
108C3×#97—34no—
109#837—91no—
110#825—91no—
111#822—91no—
112#819—94no—
113#815C3⋊D311yescharacteristic, complemented
114#805—94no—
115#805—94no—
116C2×C32×C4—24no—
117C2×C32×C4—61no—
118C2×C32×C4C22×D314yes—
119C2×C32×C4C22×D311yescharacteristic
120#638—21no—
121#638—68no—
122#638—64no—
123#638—61no—
124C4⋊(C3⋊D3)—31no—
125C4⋊(C3⋊D3)—34no—
126C4⋊(C3⋊D3)—64no—
127C4⋊(C3⋊D3)C4×D311yescharacteristic
128C4×C3⋊D3—21no—
129C4×C3⋊D3—68no—
130C4×C3⋊D3—64no—
131C4×C3⋊D3—61no—
132#634—31no—
133#634—34no—
134#634—64no—
135#634C4×D311yescharacteristic
136C3×#67—64no—
137C2×C3×Dic3—24no—
138C3×D12—34no—
139C3×C4×D3—64no—
140C3×Dic6—34no—
141C32×C8—24no—
142C32×C8—24no—
143C32×C8—31no—
144C32×C8—31no—
145C32×C8D1214yes—
146C32×C8D1214yes—
147C32×C8#6711yescharacteristic
148C32×C8#6711yescharacteristic
149C3⋊(C3⋊C8)—21no—
150C3⋊(C3⋊C8)—68no—
151C3⋊(C3⋊C8)—64no—
152C3⋊(C3⋊C8)—61no—
153C3×C3⋊C8—64no—
154C3×C3⋊C8C4×D314yes—
155C3×C3⋊C8C4×D314yes—
156#444—271noSylow
157C2×C33—21no—
158C2×C33C4×D411yescharacteristic
159C3⋊(C3⋊D3)—41no—
160C3×C3⋊D3—41no—
161C3×#41—91no—
162#235—34no—
163#235—94no—
164#235—91no—
165#235—184no—
166C3×M16—91no—
167C3×M16—181no—
168C2×C3×C8—24no—
169C2×C3×C8—34no—
170C2×C3×C8—181no—
171C2×C3×C8D3214yes—
172C2×C3×C8C2×C3⋊D311yescharacteristic
173C3×C42—34no—
174#216—91no—
175#215—91no—
176#214—91no—
177#213—91no—
178C4×Dic3C2×C3⋊D311yescharacteristic
179#208—34no—
180#208—94no—
181#208—91no—
182#208—184no—
183C2×C3⋊C8C2×C3⋊D311yescharacteristic
184Dic12—94no—
185D24—94no—
186#204—94no—
187#204—94no—
188ΓL(1,𝔽25)—64no—
189ΓL(1,𝔽25)—188no—
190ΓL(1,𝔽25)—184no—
191ΓL(1,𝔽25)—181no—
192D3×C8—64no—
193D3×C8—188no—
194D3×C8—184no—
195D3×C8—181no—
196C22×C32—24no—
197C22×C32#23514yes—
198C22×C32#23511yescharacteristic
199C2×C3⋊D3—21no—
200C2×C3⋊D3—68no—
201C2×C3⋊D3—64no—
202C2×C3⋊D3—61no—
203C2×C3×D3—64no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—121no—
207C32×C4C2×C4×D314yes—
208C32×C4C2×C4×D311yescharacteristic
209C32×C4C2×D1214yes—
210C32×C4C2×#6711yescharacteristic
211C3⋊Dic3—21no—
212C3⋊Dic3—68no—
213C3⋊Dic3—64no—
214C3⋊Dic3—61no—
215C3×Dic3—44no—
216C3×Dic3—64no—
217#135—271no—
218#131—91no—
219#131—271no—
220#108—271no—
221#104—271no—
222#104—271no—
223#97—31no—
224C33#44411yescharacteristic, complemented, Sylow
225C3×Q8—91no—
226C3×D4—91no—
227C3×D4—181no—
228C2×C3×C4—24no—
229C2×C3×C4—64no—
230C2×C3×C4—181no—
231C2×C3×C4C2×D3214yes—
232C2×C3×C4C22×C3⋊D311yescharacteristic
233#67—64no—
234#67—188no—
235#67—184no—
236#67—181no—
237C2×Dic3—21no—
238D12—34no—
239D12—94no—
240D12—91no—
241D12—184no—
242C4×D3—64no—
243C4×D3—188no—
244C4×D3—184no—
245C4×D3—181no—
246Dic6—34no—
247Dic6—94no—
248Dic6—91no—
249Dic6—184no—
250C3×C8—24no—
251C3×C8—24no—
252C3×C8—34no—
253C3×C8—34no—
254C3×C8—181no—
255C3×C8#62614yes—
256C3×C8#62614yes—
257C3×C8C4⋊(C3⋊D3)11yescharacteristic
258C3×C8C4⋊(C3⋊D3)11yescharacteristic
259C3⋊C8—64no—
260C3⋊C8—188no—
261C3⋊C8—184no—
262C3⋊C8—181no—
263C3⋊C8C4×C3⋊D311yescharacteristic
264C3⋊C8C4×C3⋊D311yescharacteristic
265C2×C32—21no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—44no—
269C2×C32C4×D1214yes—
270C2×C32C4×#6711yescharacteristic
271C3⋊D3—41no—
272C3⋊D3—128no—
273C3⋊D3—121no—
274C3⋊D3—124no—
275C3×D3—124no—
276#41—91no—
277#41—271no—
278Q16—271no—
279SD16—271no—
280SD16—271no—
281D8—271no—
282M16—91no—
283M16—181no—
284M16—271no—
285M16—541no—
286C2×C8—31no—
287C2×C8—181no—
288C2×C8—541no—
289C2×C8D3×C3⋊D311yescharacteristic
290C42—31no—
291C22×C3—24no—
292C22×C3—181no—
293C22×C3#379814yes—
294C22×C3#382811yescharacteristic
295C2×D3—64no—
296C2×D3—188no—
297C2×D3—181no—
298C2×D3—184no—
299C3×C4—24no—
300C3×C4—24no—
301C3×C4—124no—
302C3×C4—181no—
303C3×C4C4×D3214yes—
304C3×C4C2×#62614yes—
305C3×C4C2×C4×C3⋊D311yescharacteristic
306C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
307Dic3—41no—
308Dic3—64no—
309Dic3—188no—
310Dic3—181no—
311Dic3—184no—
312C32—24no—
313C32#471714yescomplemented
314C32#514511yescharacteristic, complemented
315Q8—91no—
316Q8—271no—
317D4—91no—
318D4—181no—
319D4—271no—
320D4—541no—
321C2×C4—61no—
322C2×C4—181no—
323C2×C4—541no—
324C2×C4C2×D3×C3⋊D311yescharacteristic
325C8—31no—
326C8—31no—
327C8—181no—
328C8—541no—
329C8#629211yescharacteristic
330C8#629211yescharacteristic
331C2×C3—21no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—44no—
335C2×C3—361no—
336C2×C3C4×#62614yes—
337C2×C3C4×C4⋊(C3⋊D3)11yescharacteristic
338D3—124no—
339D3—368no—
340D3—361no—
341D3—364no—
342C22—181no—
343C22—541no—
344C22#8902011yescharacteristic
345C4—121no—
346C4—181no—
347C4—541no—
348C4C4×D3×C3⋊D311yescharacteristic
349C4C2×#629211yescharacteristic
350C3—24no—
351C3#1059092414yescomplemented
352C3#1059241611yescharacteristic, complemented
353C2—21no—
354C2—361no—
355C2—1081no—
356C2C4×#629211yescharacteristic
subgroupG/H
centerC4C2×#6292
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC2×C33C4×D4
prime radical#11723369C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#444
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11723942C2semidirect
2. C3×#64110C2semidirect
4. C32×#815C2semidirect
5. #11722058C2semidirect
6. #11722048C2semidirect
7. #11722046C2semidirect
21. #88800C4semidirect
22. #88797C4semidirect
23. #88796C4semidirect
24. #88791C4semidirect
29. #64110D3semidirect
36. C3×#815D3semidirect
113. #815C3⋊D3semidirect
224. C33#444semidirect
313. C32#4717semidirect
314. C32#5145semidirect
351. C3#10590924semidirect
352. C3#10592416semidirect

Maximal subgroups

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

subgroupindex
#117239422
C3×#641102
#117233692
C32×#8152
#117220582
#117220482
#117220462
#105924163
#105909243