Finite Group Databasebeta

← #13177970

Subgroups — #13177970

Order 1728.

1796 conjugacy classes of subgroups; only the 332 normal subgroups are shown.

normal subgroups only

This group has 1796 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×#1010C211yesmaximal, characteristic, complemented
3#11723989C211yesmaximal, characteristic, complemented
4#11723375C213yesmaximal, complemented
5C2×#88793C213yesmaximal, complemented
6#11723365C213yesmaximal, complemented
7#11723363C213yesmaximal, complemented
8#10596308—31nomaximal
9#10595446—34nomaximal
10C33×#41C2211yescharacteristic
11C3×#3832—21no—
12C3×D4×C3⋊D3—23no—
13C3×#3828C2213yes—
14C2×C3×C4⋊(C3⋊D3)C2213yes—
15C32×D3×Q8—21no—
16C32×#237—23no—
17C32×#235C2213yescomplemented
18C2×C32×Dic6C2213yescomplemented
19#88855C2213yescomplemented
20C2×C3⋊(C3⋊(C3⋊C8))C2213yescomplemented
21#88798C2213yescomplemented
22#88793C2213yescomplemented
23#88793C2216yescomplemented
24#88792C2211yescharacteristic, complemented
25#88789C2213yescomplemented
26C32×2−1+4—31no—
27#64366D311yescharacteristic, complemented
28C3×#1010D314yescomplemented
29C3×#1009—34no—
30#64159—33no—
31#64158—33no—
32#64157—31no—
33#64157—34no—
34#64157—64no—
35#64151—33no—
36C2×#3755—33no—
37#63827—312no—
38C2×#3715—312no—
39#63823—312no—
40#63822—312no—
41C22×C3×C3⋊D3—23no—
42C33×Q8D411yescharacteristic
43C33×D4D413yes—
44C2×C33×C4C2313yes—
45C3×#638—23no—
46C3×C4⋊(C3⋊D3)—23no—
47C3×C4⋊(C3⋊D3)C2313yes—
48C3×C4×C3⋊D3—23no—
49C3×#634C2311yescharacteristic
50C32×#67—23no—
51C2×C32×Dic3—23no—
52C32×D12C2311yescharacteristic
53C32×C4×D3—23no—
54C32×Dic6—23no—
55C32×Dic6C2313yes—
56C3⋊(C3⋊(C3⋊C8))C2311yescharacteristic
57C3⋊(C3⋊(C3⋊C8))C2313yes—
58#5837—94no—
59#5836—91no—
60C32×#41—24no—
61C32×#41—33no—
62C32×#41—63no—
63C32×#41C2×D314yes—
64C32×#41C2×D311yescharacteristic
65C2×C32×Q8—33no—
66C2×C32×Q8—61no—
67#3832—21no—
68D4×C3⋊D3—23no—
69#3828C2×D313yes—
70C2×C4⋊(C3⋊D3)C2×D313yes—
71C3×#239—64no—
72C3×D3×Q8—24no—
73C3×#237—212no—
74C3×D3×D4—612no—
75C3×#235—312no—
76C3×#235C2×D3112yescomplemented
77C2×C3×D12—312no—
78C2×C3×Dic6C2×D3112yescomplemented
79#3756—33no—
80#3755—33no—
81#3755—36no—
82#3754—31no—
83#3753—33no—
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—312no—
91#3715—34no—
92#3715—312no—
93#3715—324no—
94#3713—312no—
95C22×C33C2×D413yes—
96C2×C3×C3⋊D3—23no—
97C2×C3×C3⋊D3—83no—
98C2×C32×D3—21no—
99C33×C4C2×D413yes—
100C33×C4C2411yescharacteristic
101C3×C3⋊Dic3—21no—
102C32×Dic3—23no—
103C3×2−1+4—34no—
104C3×H(5,𝔽2)—91no—
105#1010C3⋊D311yescharacteristic, complemented
106#1009—34no—
107#951—93no—
108#950—912no—
109#950—93no—
110#949—912no—
111#948—94no—
112#948—91no—
113#948—184no—
114#942—912no—
115C2×#215—912no—
116C2×#214—93no—
117#932—93no—
118C22×C3⋊D3—23no—
119C22×C3×D3—612no—
120C32×Q8—24no—
121C32×Q8—33no—
122C32×Q8—63no—
123C32×Q8#6714yes—
124C32×Q8#6711yescharacteristic
125C32×D4—212no—
126C32×D4—31no—
127C32×D4—63no—
128C32×D4#67112yes—
129C32×D4#6713yes—
130C2×C32×C4—212no—
131C2×C32×C4—63no—
132C2×C32×C4—63no—
133C2×C32×C4C22×D3112yes—
134C2×C32×C4C22×D313yes—
135#638—23no—
136C4⋊(C3⋊D3)—23no—
137C4⋊(C3⋊D3)C22×D313yes—
138C4×C3⋊D3—23no—
139#634C22×D311yescharacteristic
140C3×#67—212no—
141C3×#67—612no—
142C2×C3×Dic3—212no—
143C3×D12—312no—
144C3×D12—612no—
145C3×D12C22×D314yes—
146C3×C4×D3—212no—
147C3×C4×D3—612no—
148C3×Dic6—212no—
149C3×Dic6—34no—
150C3×Dic6C22×D3112yes—
151C3⋊(C3⋊C8)—31no—
152C3⋊(C3⋊C8)—33no—
153C3⋊(C3⋊C8)—312no—
154C3⋊(C3⋊C8)—34no—
155C3⋊(C3⋊C8)—64no—
156C3⋊(C3⋊C8)—612no—
157#577—271noSylow
158C2×C33—23no—
159C2×C33C22×D411yescharacteristic
160C3×C3⋊D3—43no—
161C32×D3—41no—
162C3×#41—24no—
163C3×#41—312no—
164C3×#41—612no—
165C3×#41—93no—
166C3×#41—181no—
167C3×#41D3214yes—
168C3×#41C2×C3⋊D311yescharacteristic
169C2×C3×Q8—312no—
170C2×C3×Q8—64no—
171C2×C3×D4—93no—
172C2×C3×D4—183no—
173#239—64no—
174D3×Q8—21no—
175#237—23no—
176D3×D4—612no—
177#235—312no—
178#235C2×C3⋊D313yescomplemented
179C2×D12—312no—
180C2×Dic6C2×C3⋊D313yescomplemented
181#216—912no—
182#216—93no—
183#215—912no—
184#215—924no—
185#215—91no—
186#214—94no—
187#214—96no—
188#214—93no—
189#213—912no—
190#213—93no—
191#208—912no—
192#208—93no—
193#208—1812no—
194C2×C3⋊C8—912no—
195C2×C3⋊C8—93no—
196C2×C3⋊C8—1812no—
197C22×C32—212no—
198C22×C32—61no—
199C22×C32C2×#67112yes—
200C22×C32C2×#6713yes—
201C2×C3⋊D3—23no—
202C2×C3⋊D3—83no—
203C2×C3×D3—24no—
204C2×C3×D3—612no—
205C2×C3×D3—2412no—
206C32×C4—24no—
207C32×C4—212no—
208C32×C4—63no—
209C32×C4C2×#67112yes—
210C32×C4C2×#6713yes—
211C32×C4C23×D314yes—
212C32×C4C23×D311yescharacteristic
213C3⋊Dic3—21no—
214C3×Dic3—212no—
215C3×Dic3—64no—
2162−1+4—31no—
217H(5,𝔽2)—91no—
218#137—273no—
219AGL(1,ℤ/8)—273no—
220#135—273no—
221C2×SD16—273no—
222#131—271no—
223C33#57711yescharacteristic, complemented, Sylow
224C23×C3—183no—
225C22×D3—612no—
226C3×Q8—24no—
227C3×Q8—312no—
228C3×Q8—612no—
229C3×Q8—91no—
230C3×Q8#62514yes—
231C3×Q8#63811yescharacteristic
232C3×D4—212no—
233C3×D4—34no—
234C3×D4—612no—
235C3×D4—93no—
236C3×D4—183no—
237C3×D4—183no—
238C3×D4#625112yes—
239C3×D4#63813yes—
240C2×C3×C4—212no—
241C2×C3×C4—612no—
242C2×C3×C4—612no—
243C2×C3×C4—183no—
244C2×C3×C4C2×D32112yes—
245C2×C3×C4C22×C3⋊D313yes—
246#67—23no—
247#67—612no—
248C2×Dic3—23no—
249D12—312no—
250D12—612no—
251D12C22×C3⋊D311yescharacteristic
252C4×D3—23no—
253C4×D3—612no—
254Dic6—23no—
255Dic6—34no—
256Dic6C22×C3⋊D313yes—
257C3⋊C8—94no—
258C3⋊C8—912no—
259C3⋊C8—91no—
260C3⋊C8—93no—
261C3⋊C8—184no—
262C3⋊C8—1812no—
263C2×C32—23no—
264C2×C32—24no—
265C2×C32—212no—
266C2×C32—412no—
267C2×C32—121no—
268C2×C32C22×#6714yes—
269C2×C32C22×#6711yescharacteristic
270C3⋊D3—43no—
271C3×D3—44no—
272C3×D3—1212no—
273#41—33no—
274#41—63no—
275#41—93no—
276#41—181no—
277#41D3×C3⋊D311yescharacteristic
278C2×Q8—33no—
279C2×Q8—61no—
280C2×D4—93no—
281C2×D4—183no—
282Q16—273no—
283SD16—271no—
284SD16—273no—
285SD16—276no—
286D8—273no—
287M16—273no—
288C2×C8—273no—
289C22×C3—212no—
290C22×C3—64no—
291C22×C3—183no—
292C22×C3—723no—
293C22×C3C2×#625112yes—
294C22×C3C2×#63813yes—
295C2×D3—21no—
296C2×D3—612no—
297C2×D3—2412no—
298C3×C4—24no—
299C3×C4—212no—
300C3×C4—612no—
301C3×C4—181no—
302C3×C4C2×#625112yes—
303C3×C4C2×#63813yes—
304C3×C4C22×D3214yes—
305C3×C4C23×C3⋊D311yescharacteristic
306Dic3—23no—
307Dic3—64no—
308C32—24no—
309C32#583611yescharacteristic, complemented
310C32#583714yescomplemented
311C23—183no—
312Q8—33no—
313Q8—63no—
314Q8—91no—
315Q8#629011yescharacteristic
316D4—31no—
317D4—63no—
318D4—93no—
319D4—183no—
320D4—183no—
321D4#629013yes—
322C2×C4—63no—
323C2×C4—63no—
324C2×C4—183no—
325C2×C4C2×D3×C3⋊D313yes—
326C8—271no—
327C8—273no—
328C2×C3—212no—
329C2×C3—23no—
330C2×C3—24no—
331C2×C3—412no—
332C2×C3—124no—
333C2×C3—363no—
334C2×C3C22×#62514yes—
335C2×C3C22×#63811yescharacteristic
336D3—41no—
337D3—1212no—
338C22—61no—
339C22—183no—
340C22—723no—
341C22C2×#629013yes—
342C4—63no—
343C4—181no—
344C4C2×#629013yes—
345C4C22×D3×C3⋊D311yescharacteristic
346C3—24no—
347C3#1059544614yescomplemented
348C3#1059630811yescharacteristic, complemented
349C2—23no—
350C2—121no—
351C2—363no—
352C2C22×#629011yescharacteristic
subgroupG/H
centerC2C22×#6290
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC33×#41C22
socleC2×C33C22×D4
hypercenter#41D3×C3⋊D3
hypocenterC33#577
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×#1010C2semidirect
3. #11723989C2semidirect
4. #11723375C2semidirect
5. C2×#88793C2semidirect
6. #11723365C2semidirect
7. #11723363C2semidirect
17. C32×#235C22semidirect
18. C2×C32×Dic6C22semidirect
19. #88855C22semidirect
20. C2×C3⋊(C3⋊(C3⋊C8))C22semidirect
21. #88798C22semidirect
22. #88793C22semidirect
23. #88793C22semidirect
24. #88792C22semidirect
25. #88789C22semidirect
27. #64366D3semidirect
28. C3×#1010D3semidirect
76. C3×#235C2×D3semidirect
78. C2×C3×Dic6C2×D3semidirect
105. #1010C3⋊D3semidirect
178. #235C2×C3⋊D3semidirect
180. C2×Dic6C2×C3⋊D3semidirect
223. C33#577semidirect
309. C32#5836semidirect
310. C32#5837semidirect
347. C3#10595446semidirect
348. C3#10596308semidirect

Maximal subgroups

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

subgroupindex
C3×#643662
C32×#10102
#117239892
#117233752
C2×#887932
#117233652
#117233632
#105963083
#105954463