Finite Group Databasebeta

← #13160004

Subgroups — #13160004

Order 1728.

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

normal subgroups only

This group has 884 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
2C32×#901C211yesmaximal, characteristic, complemented
3#11723377C211yesmaximal, characteristic
4C3×C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
5#11722060C211yesmaximal, characteristic, complemented
6#11722047C211yesmaximal, characteristic, complemented
7#11722044C211yesmaximal, characteristic, complemented
8#10592843—31nomaximal
9#10590923—34nomaximal
10#89076C2211yescharacteristic
11C32×#235C2211yescharacteristic
12C2×C33×C8C2211yescharacteristic
13#88855C2211yescharacteristic
14#88850—21no—
15C8×C3⋊(C3⋊D3)—21no—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×#208C2211yescharacteristic
19C32×ΓL(1,𝔽25)—21no—
20C32×D3×C8—21no—
21#88799C411yescharacteristic, complemented
22#88795C411yescharacteristic, complemented
23#88794C411yescharacteristic, complemented
24#88790C411yescharacteristic, complemented
25C32×#131—31no—
26#64159—31no—
27#64110—31no—
28#64110—34no—
29#64110—64no—
30C3×#901D314yescomplemented
31#63829—34no—
32#63664—31no—
33#63652—31no—
34#63649—31no—
35C8×C3⋊Dic3D311yescharacteristic, complemented
36C3×C8×Dic3—34no—
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
45C2×C3×C3⋊Dic3—21no—
46C32×#67—21no—
47C32×D12C2×C411yescharacteristic
48C32×C4×D3—21no—
49C32×Dic6C2×C411yescharacteristic
50C33×C8D411yescharacteristic
51C33×C8D411yescharacteristic
52C3⋊(C3⋊(C3⋊C8))—21no—
53C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
54C3×C3⋊(C3⋊C8)C2×C411yescharacteristic
55C32×C3⋊C8—21no—
56#5144—94no—
57#4700—91no—
58C32×#41—31no—
59#3828—31no—
60#3828—34no—
61#3828—64no—
62C3×#235C2×D314yes—
63C32×M16—31no—
64C32×M16—61no—
65C2×C32×C8—24no—
66C2×C32×C8—61no—
67C2×C32×C8C2×D311yescharacteristic
68C2×C32×C8C2×D314yes—
69#3756—31no—
70#3755—31no—
71#3754—31no—
72#3753—31no—
73C4×C3⋊Dic3C2×D311yescharacteristic
74#3748—31no—
75#3748—34no—
76#3748—64no—
77C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
78#3743—68no—
79#3743—61no—
80#3743—64no—
81C8×C3⋊D3—68no—
82C8×C3⋊D3—61no—
83C8×C3⋊D3—64no—
84C3×C4×Dic3—34no—
85C3×#208C2×D314yes—
86C2×C3×C3⋊C8—34no—
87C3×ΓL(1,𝔽25)—24no—
88C3×D3×C8—24no—
89#3719—34no—
90#3717—34no—
91#3716—34no—
92#3714—34no—
93C22×C33#4111yescharacteristic
94C2×C3⋊(C3⋊D3)—21no—
95C2×C32×D3—21no—
96C33×C4C22×C411yescharacteristic
97C33×C4C2×D411yescharacteristic
98C3⋊(C3⋊Dic3)—21no—
99C3×C3⋊Dic3—22no—
100C32×Dic3—21no—
101C3×#131—34no—
102#950—94no—
103#904—91no—
104#901—94no—
105#901—91no—
106#901—184no—
107#901C3⋊D311yescharacteristic, complemented
108C3×C4×C8—91no—
109#837—94no—
110#825—94no—
111#822—94no—
112#819—91no—
113C8×Dic3—34no—
114#805—91no—
115#805—91no—
116C32×Q8—31no—
117C32×D4—31no—
118C32×D4—61no—
119C2×C32×C4—24no—
120C2×C32×C4—61no—
121C2×C32×C4C22×D314yes—
122C2×C32×C4C22×D311yescharacteristic
123#638—68no—
124#638—64no—
125#638—61no—
126C2×C3⋊Dic3—21no—
127C4⋊(C3⋊D3)—31no—
128C4⋊(C3⋊D3)—34no—
129C4⋊(C3⋊D3)—64no—
130C4×C3⋊D3—68no—
131C4×C3⋊D3—61no—
132C4×C3⋊D3—64no—
133#634—31no—
134#634—34no—
135#634—64no—
136C3×#67—24no—
137C2×C3×Dic3—64no—
138C3×D12C4×D314yes—
139C3×C4×D3—24no—
140C3×Dic6C4×D314yes—
141C32×C8—24no—
142C32×C8—24no—
143C32×C8—61no—
144C32×C8D1211yescharacteristic
145C32×C8D1211yescharacteristic
146C32×C8#6714yes—
147C32×C8#6714yes—
148C3⋊(C3⋊C8)—68no—
149C3⋊(C3⋊C8)—61no—
150C3⋊(C3⋊C8)—64no—
151C3⋊(C3⋊C8)C4×D311yescharacteristic
152C3⋊(C3⋊C8)C4×D311yescharacteristic
153C3×C3⋊C8—24no—
154C3×C3⋊C8—34no—
155C3×C3⋊C8—34no—
156#443—271noSylow
157C2×C33—21no—
158C2×C33C4×D411yescharacteristic
159C3⋊(C3⋊D3)—41no—
160C32×D3—41no—
161C3×#41—34no—
162#235—94no—
163#235—91no—
164#235—184no—
165#235C2×C3⋊D311yescharacteristic
166C3×M16—34no—
167C3×M16—64no—
168C2×C3×C8—24no—
169C2×C3×C8—64no—
170C2×C3×C8—91no—
171C2×C3×C8D3214yes—
172C2×C3×C8C2×C3⋊D311yescharacteristic
173C3×C42—91no—
174#216—94no—
175#215—94no—
176#214—94no—
177#213—94no—
178C4×Dic3—34no—
179#208—94no—
180#208—91no—
181#208—184no—
182#208C2×C3⋊D311yescharacteristic
183C2×C3⋊C8—34no—
184Dic12—91no—
185D24—91no—
186#204—91no—
187#204—91no—
188ΓL(1,𝔽25)—21no—
189ΓL(1,𝔽25)—188no—
190ΓL(1,𝔽25)—184no—
191ΓL(1,𝔽25)—181no—
192D3×C8—21no—
193D3×C8—188no—
194D3×C8—184no—
195D3×C8—181no—
196C22×C32—24no—
197C22×C32—61no—
198C22×C32#23514yes—
199C22×C32#23511yescharacteristic
200C2×C3⋊D3—68no—
201C2×C3⋊D3—61no—
202C2×C3⋊D3—64no—
203C2×C3×D3—24no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—61no—
207C32×C4C2×C4×D314yes—
208C32×C4C2×C4×D311yescharacteristic
209C32×C4C2×D1211yescharacteristic
210C32×C4C2×#6714yes—
211C3⋊Dic3—22no—
212C3⋊Dic3—68no—
213C3⋊Dic3—61no—
214C3⋊Dic3—64no—
215C3×Dic3—24no—
216C3×Dic3—68no—
217#135—271no—
218#131—31no—
219#131—271no—
220#108—271no—
221#104—271no—
222#104—271no—
223C4×C8—91no—
224C33#44311yescharacteristic, complemented, Sylow
225C3×Q8—34no—
226C3×D4—34no—
227C3×D4—64no—
228C2×C3×C4—24no—
229C2×C3×C4—64no—
230C2×C3×C4—181no—
231C2×C3×C4C2×D3214yes—
232C2×C3×C4C22×C3⋊D311yescharacteristic
233#67—21no—
234#67—188no—
235#67—184no—
236#67—181no—
237C2×Dic3—64no—
238D12—94no—
239D12—91no—
240D12—184no—
241D12C4×C3⋊D311yescharacteristic
242C4×D3—21no—
243C4×D3—188no—
244C4×D3—184no—
245C4×D3—181no—
246Dic6—94no—
247Dic6—91no—
248Dic6—184no—
249Dic6C4×C3⋊D311yescharacteristic
250C3×C8—24no—
251C3×C8—24no—
252C3×C8—64no—
253C3×C8—91no—
254C3×C8—91no—
255C3×C8#62614yes—
256C3×C8#62614yes—
257C3×C8#63811yescharacteristic
258C3×C8#63811yescharacteristic
259C3⋊C8—21no—
260C3⋊C8—34no—
261C3⋊C8—34no—
262C3⋊C8—188no—
263C3⋊C8—184no—
264C3⋊C8—181no—
265C2×C32—21no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—44no—
269C2×C32—121no—
270C2×C32C4×D1211yescharacteristic
271C2×C32C4×#6714yes—
272C3⋊D3—128no—
273C3⋊D3—124no—
274C3⋊D3—121no—
275C3×D3—44no—
276#41—31no—
277#41—271no—
278Q16—271no—
279SD16—271no—
280SD16—271no—
281D8—271no—
282M16—31no—
283M16—61no—
284M16—271no—
285M16—541no—
286C2×C8—61no—
287C2×C8—91no—
288C2×C8—541no—
289C2×C8D3×C3⋊D311yescharacteristic
290C42—91no—
291C22×C3—24no—
292C22×C3—64no—
293C22×C3#379814yes—
294C22×C3#382811yescharacteristic
295C2×D3—21no—
296C2×D3—188no—
297C2×D3—181no—
298C2×D3—184no—
299C3×C4—24no—
300C3×C4—24no—
301C3×C4—64no—
302C3×C4—182no—
303C3×C4C4×D3214yes—
304C3×C4C2×#62614yes—
305C3×C4C2×C4×C3⋊D311yescharacteristic
306C3×C4C2×#63811yescharacteristic
307Dic3—21no—
308Dic3—68no—
309Dic3—188no—
310Dic3—181no—
311Dic3—184no—
312C32—24no—
313C32#470011yescharacteristic, complemented
314C32#514414yescomplemented
315Q8—31no—
316Q8—271no—
317D4—31no—
318D4—61no—
319D4—271no—
320D4—541no—
321C2×C4—61no—
322C2×C4—181no—
323C2×C4—541no—
324C2×C4C2×D3×C3⋊D311yescharacteristic
325C8—61no—
326C8—91no—
327C8—91no—
328C8—541no—
329C8#629111yescharacteristic
330C8#629111yescharacteristic
331C2×C3—21no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—44no—
335C2×C3—124no—
336C2×C3C4×#62614yes—
337C2×C3C4×#63811yescharacteristic
338D3—41no—
339D3—368no—
340D3—361no—
341D3—364no—
342C22—61no—
343C22—541no—
344C22#8902011yescharacteristic
345C4—61no—
346C4—182no—
347C4—541no—
348C4C4×D3×C3⋊D311yescharacteristic
349C4C2×#629111yescharacteristic
350C3—24no—
351C3#1059092314yescomplemented
352C3#1059284311yescharacteristic, complemented
353C2—21no—
354C2—121no—
355C2—1081no—
356C2C4×#629111yescharacteristic
subgroupG/H
centerC8#6291
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC2×C33C4×D4
prime radical#11723377C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#443
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C8 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11723942C2semidirect
2. C32×#901C2semidirect
4. C3×C8×C3⋊Dic3C2semidirect
5. #11722060C2semidirect
6. #11722047C2semidirect
7. #11722044C2semidirect
21. #88799C4semidirect
22. #88795C4semidirect
23. #88794C4semidirect
24. #88790C4semidirect
30. C3×#901D3semidirect
35. C8×C3⋊Dic3D3semidirect
107. #901C3⋊D3semidirect
224. C33#443semidirect
313. C32#4700semidirect
314. C32#5144semidirect
351. C3#10590923semidirect
352. C3#10592843semidirect

Maximal subgroups

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

subgroupindex
#117239422
C32×#9012
#117233772
C3×C8×C3⋊Dic32
#117220602
#117220472
#117220442
#105928433
#105909233