Finite Group Databasebeta

← #13177968

Subgroups — #13177968

Order 1728.

1856 conjugacy classes of subgroups; only the 332 normal subgroups are shown.

normal subgroups only

This group has 1856 conjugacy classes of subgroups — too many to draw. Only its 332 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
1C3×#64366C211yesmaximal, characteristic, complemented
2C32×#1009C211yesmaximal, characteristic, complemented
3#11723989C211yesmaximal, characteristic, complemented
4#11723365C213yesmaximal, complemented
5#11723364C213yesmaximal, complemented
6#11723363C213yesmaximal, complemented
7C2×#88789C213yesmaximal, complemented
8#10596306—31nomaximal
9#10595444—34nomaximal
10C33×#41C2211yescharacteristic
11C3×#3832—21no—
12C3×D4×C3⋊D3—23no—
13C3×#3828C2213yescomplemented
14C2×C3×C4⋊(C3⋊D3)C2213yescomplemented
15C32×#239—21no—
16C32×D3×D4—23no—
17C32×#235C2213yescomplemented
18C2×C32×D12C2213yescomplemented
19#88855C2213yescomplemented
20C2×C3⋊(C3⋊(C3⋊C8))C2213yescomplemented
21#88798C2211yescharacteristic, complemented
22#88793C2213yescomplemented
23#88792C2213yescomplemented
24#88789C2213yescomplemented
25#88789C2216yescomplemented
26C32×H(5,𝔽2)—31no—
27#64366D311yescharacteristic, complemented
28C3×#1009—34no—
29C3×#1009D314yescomplemented
30#64159—33no—
31#64158—33no—
32#64157—31no—
33#64157—34no—
34#64157—64no—
35#64141—33no—
36C2×#3753—33no—
37#63823—312no—
38#63823—312no—
39#63822—312no—
40C2×#3713—312no—
41C22×C3×C3⋊D3—23no—
42C22×C32×D3—23no—
43C33×Q8D411yescharacteristic
44C33×D4D413yes—
45C2×C33×C4C2313yes—
46C3×#638—23no—
47C3×C4⋊(C3⋊D3)—23no—
48C3×C4⋊(C3⋊D3)C2313yes—
49C3×C4×C3⋊D3—23no—
50C3×#634C2311yescharacteristic
51C32×#67—23no—
52C32×D12—23no—
53C32×D12C2313yes—
54C32×C4×D3—23no—
55C32×Dic6C2311yescharacteristic
56C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
57C3⋊(C3⋊(C3⋊C8))C2313yes—
58#5835—94no—
59#5835—91no—
60C32×#41—24no—
61C32×#41—33no—
62C32×#41—61no—
63C32×#41C2×D314yes—
64C32×#41C2×D311yescharacteristic
65C2×C32×D4—33no—
66C2×C32×D4—63no—
67#3832—21no—
68D4×C3⋊D3—23no—
69#3828C2×D313yescomplemented
70C2×C4⋊(C3⋊D3)C2×D313yescomplemented
71C3×#239—24no—
72C3×#239—64no—
73C3×D3×D4—212no—
74C3×D3×D4—612no—
75C3×#235—312no—
76C3×#235C2×D3112yescomplemented
77C2×C3×D12—312no—
78C2×C3×D12C2×D3112yescomplemented
79#3756—31no—
80#3755—33no—
81#3754—33no—
82#3753—33no—
83#3753—36no—
84#3748—33no—
85#3748—312no—
86#3748—612no—
87C2×C3⋊(C3⋊C8)—33no—
88C2×C3⋊(C3⋊C8)—312no—
89C2×C3⋊(C3⋊C8)—612no—
90#3718—34no—
91#3715—312no—
92#3715—312no—
93#3713—312no—
94#3713—324no—
95C22×C33C2×D413yes—
96C2×C3×C3⋊D3—23no—
97C2×C3×C3⋊D3—83no—
98C2×C32×D3—23no—
99C2×C32×D3—83no—
100C33×C4C2×D413yes—
101C33×C4C2411yescharacteristic
102C3×C3⋊Dic3—21no—
103C32×Dic3—21no—
104C3×H(5,𝔽2)—34no—
105C3×H(5,𝔽2)—91no—
106#1009—34no—
107#1009C3⋊D311yescharacteristic, complemented
108#950—912no—
109#950—93no—
110#949—912no—
111#949—93no—
112#948—94no—
113#948—91no—
114#948—184no—
115#932—912no—
116#932—93no—
117C2×#213—912no—
118C2×#213—93no—
119C23×C32—63no—
120C22×C3⋊D3—23no—
121C22×C3×D3—212no—
122C22×C3×D3—612no—
123C32×Q8—24no—
124C32×Q8—31no—
125C32×Q8#6714yes—
126C32×Q8#6711yescharacteristic
127C32×D4—212no—
128C32×D4—33no—
129C32×D4—63no—
130C32×D4—63no—
131C32×D4#67112yes—
132C32×D4#6713yes—
133C2×C32×C4—212no—
134C2×C32×C4—63no—
135C2×C32×C4C22×D3112yes—
136C2×C32×C4C22×D313yes—
137#638—23no—
138C4⋊(C3⋊D3)—23no—
139C4⋊(C3⋊D3)C22×D313yes—
140C4×C3⋊D3—23no—
141#634C22×D311yescharacteristic
142C3×#67—212no—
143C3×#67—612no—
144C3×D12—212no—
145C3×D12—312no—
146C3×D12—612no—
147C3×D12C22×D3112yes—
148C3×C4×D3—212no—
149C3×C4×D3—612no—
150C3×Dic6—34no—
151C3×Dic6C22×D314yes—
152C3⋊(C3⋊C8)—31no—
153C3⋊(C3⋊C8)—33no—
154C3⋊(C3⋊C8)—34no—
155C3⋊(C3⋊C8)—312no—
156C3⋊(C3⋊C8)—64no—
157C3⋊(C3⋊C8)—612no—
158#576—271noSylow
159C2×C33—23no—
160C2×C33C22×D411yescharacteristic
161C3×C3⋊D3—43no—
162C32×D3—43no—
163C3×#41—24no—
164C3×#41—312no—
165C3×#41—64no—
166C3×#41—93no—
167C3×#41—181no—
168C3×#41D3214yes—
169C3×#41C2×C3⋊D311yescharacteristic
170C2×C3×D4—312no—
171C2×C3×D4—612no—
172C2×C3×D4—93no—
173C2×C3×D4—183no—
174#239—21no—
175#239—64no—
176D3×D4—23no—
177D3×D4—612no—
178#235—312no—
179#235C2×C3⋊D313yescomplemented
180C2×D12—312no—
181C2×D12C2×C3⋊D313yescomplemented
182#216—94no—
183#216—91no—
184#215—912no—
185#215—93no—
186#214—912no—
187#214—93no—
188#213—912no—
189#213—924no—
190#213—93no—
191#213—96no—
192#208—912no—
193#208—93no—
194#208—1812no—
195C2×C3⋊C8—912no—
196C2×C3⋊C8—93no—
197C2×C3⋊C8—1812no—
198C22×C32—212no—
199C22×C32—63no—
200C22×C32—243no—
201C22×C32C2×#67112yes—
202C22×C32C2×#6713yes—
203C2×C3⋊D3—23no—
204C2×C3⋊D3—83no—
205C2×C3×D3—212no—
206C2×C3×D3—612no—
207C2×C3×D3—812no—
208C2×C3×D3—2412no—
209C32×C4—24no—
210C32×C4—212no—
211C32×C4—61no—
212C32×C4C2×#67112yes—
213C32×C4C2×#6713yes—
214C32×C4C23×D314yes—
215C32×C4C23×D311yescharacteristic
216C3⋊Dic3—21no—
217C3×Dic3—24no—
218C3×Dic3—64no—
219H(5,𝔽2)—31no—
220H(5,𝔽2)—91no—
221AGL(1,ℤ/8)—273no—
222AGL(1,ℤ/8)—273no—
223#135—273no—
224C2×D8—273no—
225#131—271no—
226C33#57611yescharacteristic, complemented, Sylow
227C23×C3—612no—
228C23×C3—183no—
229C22×D3—23no—
230C22×D3—612no—
231C3×Q8—24no—
232C3×Q8—34no—
233C3×Q8—91no—
234C3×Q8#62514yes—
235C3×Q8#63811yescharacteristic
236C3×D4—212no—
237C3×D4—312no—
238C3×D4—612no—
239C3×D4—612no—
240C3×D4—93no—
241C3×D4—183no—
242C3×D4—183no—
243C3×D4#625112yes—
244C3×D4#63813yes—
245C2×C3×C4—212no—
246C2×C3×C4—612no—
247C2×C3×C4—183no—
248C2×C3×C4C2×D32112yes—
249C2×C3×C4C22×C3⋊D313yes—
250#67—23no—
251#67—612no—
252D12—23no—
253D12—312no—
254D12—612no—
255D12C22×C3⋊D313yes—
256C4×D3—23no—
257C4×D3—612no—
258Dic6—34no—
259Dic6C22×C3⋊D311yescharacteristic
260C3⋊C8—94no—
261C3⋊C8—912no—
262C3⋊C8—91no—
263C3⋊C8—93no—
264C3⋊C8—184no—
265C3⋊C8—1812no—
266C2×C32—23no—
267C2×C32—24no—
268C2×C32—212no—
269C2×C32—412no—
270C2×C32—123no—
271C2×C32C22×#6714yes—
272C2×C32C22×#6711yescharacteristic
273C3⋊D3—43no—
274C3×D3—412no—
275C3×D3—1212no—
276#41—33no—
277#41—61no—
278#41—93no—
279#41—181no—
280#41D3×C3⋊D311yescharacteristic
281C2×D4—33no—
282C2×D4—63no—
283C2×D4—93no—
284C2×D4—183no—
285Q16—271no—
286SD16—273no—
287SD16—273no—
288D8—273no—
289D8—276no—
290M16—273no—
291C2×C8—273no—
292C22×C3—212no—
293C22×C3—612no—
294C22×C3—183no—
295C22×C3—2412no—
296C22×C3—723no—
297C22×C3C2×#625112yes—
298C22×C3C2×#63813yes—
299C2×D3—23no—
300C2×D3—612no—
301C2×D3—83no—
302C2×D3—2412no—
303C3×C4—24no—
304C3×C4—212no—
305C3×C4—64no—
306C3×C4—181no—
307C3×C4C2×#625112yes—
308C3×C4C2×#63813yes—
309C3×C4C22×D3214yes—
310C3×C4C23×C3⋊D311yescharacteristic
311Dic3—21no—
312Dic3—64no—
313C32—24no—
314C32#583514yescomplemented
315C32#583511yescharacteristic, complemented
316C23—63no—
317C23—183no—
318Q8—31no—
319Q8—91no—
320Q8#629011yescharacteristic
321D4—33no—
322D4—63no—
323D4—63no—
324D4—93no—
325D4—183no—
326D4—183no—
327D4#629013yes—
328C2×C4—63no—
329C2×C4—183no—
330C2×C4C2×D3×C3⋊D313yes—
331C8—271no—
332C8—273no—
333C2×C3—212no—
334C2×C3—23no—
335C2×C3—24no—
336C2×C3—412no—
337C2×C3—1212no—
338C2×C3—363no—
339C2×C3C22×#62514yes—
340C2×C3C22×#63811yescharacteristic
341D3—43no—
342D3—1212no—
343C22—63no—
344C22—183no—
345C22—243no—
346C22—723no—
347C22C2×#629013yes—
348C4—61no—
349C4—181no—
350C4C2×#629013yes—
351C4C22×D3×C3⋊D311yescharacteristic
352C3—24no—
353C3#1059544414yescomplemented
354C3#1059630611yescharacteristic, complemented
355C2—23no—
356C2—123no—
357C2—363no—
358C2C22×#629011yescharacteristic
subgroupG/H
centerC2C22×#6290
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
hypercenter#41D3×C3⋊D3
hypocenterC33#576
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C3×#64366C2semidirect
2. C32×#1009C2semidirect
3. #11723989C2semidirect
4. #11723365C2semidirect
5. #11723364C2semidirect
6. #11723363C2semidirect
7. C2×#88789C2semidirect
13. C3×#3828C22semidirect
14. C2×C3×C4⋊(C3⋊D3)C22semidirect
17. C32×#235C22semidirect
18. C2×C32×D12C22semidirect
19. #88855C22semidirect
20. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
21. #88798C22semidirect
22. #88793C22semidirect
23. #88792C22semidirect
24. #88789C22semidirect
25. #88789C22semidirect
27. #64366D3semidirect
29. C3×#1009D3semidirect
69. #3828C2×D3semidirect
70. C2×C4⋊(C3⋊D3)C2×D3semidirect
76. C3×#235C2×D3semidirect
78. C2×C3×D12C2×D3semidirect
107. #1009C3⋊D3semidirect
179. #235C2×C3⋊D3semidirect
181. C2×D12C2×C3⋊D3semidirect
226. C33#576semidirect
314. C32#5835semidirect
315. C32#5835semidirect
353. C3#10595444semidirect
354. C3#10596306semidirect

Maximal subgroups

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

subgroupindex
C3×#643662
C32×#10092
#117239892
#117233652
#117233642
#117233632
C2×#887892
#105963063
#105954443