Finite Group Databasebeta

← #13153411

Subgroups — #13153411

Order 1728.

965 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11722639C211yesmaximal, characteristic, complemented
2#11722638C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#131C211yesmaximal, characteristic, complemented
4#64118⋊C3C213yesmaximal, complemented
5#64117⋊C3C213yesmaximal, complemented
6#64113⋊C3C213yesmaximal, complemented
7C2×#3744⋊C3C213yesmaximal, complemented
8#10596222C311yesmaximal, characteristic, complemented
9C3×#5751—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11#88722—21no—
12Q8×(C3⋊D3)⋊C3—21no—
13#3830⋊C3—23no—
14D4×(C3⋊D3)⋊C3—23no—
15#3828⋊C3C2213yes—
16#3828⋊C3C2213yescomplemented
17C2×C4⋊((C3⋊D3)⋊C3)C2213yes—
18C2×#634⋊C3C2213yescomplemented
19M16×H(3,𝔽3)C2213yescomplemented
20C2×C8×H(3,𝔽3)C2213yescomplemented
21#3745⋊C3C2213yescomplemented
22#3744⋊C3C2213yescomplemented
23#3744⋊C3C2216yescomplemented
24#3744⋊C3C2211yescharacteristic, complemented
25#3746⋊C3C2213yescomplemented
26#64367C2×C311yescharacteristic, complemented
27#64366C2×C311yescharacteristic, complemented
28C3×#1010—31no—
29C3×#1009—31no—
30C32×#131—21no—
31C32×#131D311yescharacteristic, complemented
32C32×#131C2×C311yescharacteristic, complemented
33#64118C2×C313yescomplemented
34#64117C2×C313yescomplemented
35#64113C2×C313yescomplemented
36C2×#3744C2×C313yescomplemented
37C3×#909—33no—
38C3×#908—33no—
39C3×#904—33no—
40C2×C3×#204—33no—
41C22×(C3⋊D3)⋊C3—23no—
42Q8×H(3,𝔽3)D411yescharacteristic
43D4×H(3,𝔽3)D413yes—
44C2×C4×H(3,𝔽3)C2313yes—
45#638⋊C3—23no—
46#638⋊C3—23no—
47C2×(C3⋊Dic3)⋊C3—23no—
48C4⋊((C3⋊D3)⋊C3)—23no—
49C4⋊((C3⋊D3)⋊C3)C2313yes—
50C4⋊((C3⋊D3)⋊C3)C2311yescharacteristic
51C4×(C3⋊D3)⋊C3—23no—
52C4×(C3⋊D3)⋊C3—23no—
53#634⋊C3—23no—
54#634⋊C3C2313yes—
55#634⋊C3C2311yescharacteristic
56C8×H(3,𝔽3)C2313yes—
57C8×H(3,𝔽3)C2311yescharacteristic
58C3×#577—91no—
59#5751—31no—
60#5751—91no—
61C32×#41—21no—
62C32×#41C2×D311yescharacteristic
63C32×#41C22×C311yescharacteristic
64#3832—21no—
65Q8×C3⋊D3—21no—
66#3830—23no—
67D4×C3⋊D3—23no—
68#3828C22×C313yes—
69#3828C22×C313yescomplemented
70C2×C4⋊(C3⋊D3)C22×C313yes—
71C2×#634C22×C313yescomplemented
72C3×#239—61no—
73C3×D3×Q8—61no—
74C3×#237—63no—
75C3×D3×D4—63no—
76C3×#235—33no—
77C3×#235—33no—
78C2×C3×D12—33no—
79C2×C3×Dic6—33no—
80C32×M16—23no—
81C32×M16C2×D313yescomplemented
82C32×M16C22×C313yescomplemented
83C2×C32×C8—23no—
84C2×C32×C8C2×D313yescomplemented
85C2×C32×C8C22×C313yescomplemented
86#3746C22×C313yescomplemented
87#3745C22×C313yescomplemented
88#3744C22×C313yescomplemented
89#3744C22×C316yescomplemented
90#3744C22×C311yescharacteristic, complemented
91C3×Dic12—33no—
92C3×D24—33no—
93C3×#204—33no—
94C3×#204—36no—
95C3×#204—31no—
96C22×H(3,𝔽3)C2×D413yes—
97C2×(C3⋊D3)⋊C3—21no—
98C2×(C3⋊D3)⋊C3—23no—
99C2×(C3⋊D3)⋊C3—83no—
100C4×H(3,𝔽3)C2×D413yes—
101C4×H(3,𝔽3)C2411yescharacteristic
102(C3⋊Dic3)⋊C3—23no—
103(C3⋊Dic3)⋊C3—21no—
104C3×2−1+4—91no—
105C3×H(5,𝔽2)—91no—
106#1010—31no—
107#1010—91no—
108#1009—31no—
109#1009—91no—
110C3×#137—93no—
111C3×AGL(1,ℤ/8)—93no—
112C3×#135—93no—
113C2×C3×SD16—93no—
114C3×#131—31no—
115C3×#131—31no—
116C3×#131—61no—
117C3×#131C3×D311yescharacteristic
118#909—33no—
119#909—93no—
120#908—33no—
121#908—93no—
122#904—33no—
123#904—93no—
124C2×#204—33no—
125C2×#204—93no—
126C22×C3⋊D3—23no—
127C22×C3×D3—63no—
128C32×Q8—21no—
129C32×Q8D1211yescharacteristic
130C32×Q8C3×D411yescharacteristic
131C32×D4—23no—
132C32×D4D1213yes—
133C32×D4C3×D413yes—
134C2×C32×C4—23no—
135C2×C32×C4C22×D313yes—
136C2×C32×C4C23×C313yes—
137#638—23no—
138#638—23no—
139C2×C3⋊Dic3—23no—
140C4⋊(C3⋊D3)—23no—
141C4⋊(C3⋊D3)C23×C313yes—
142C4⋊(C3⋊D3)C23×C311yescharacteristic
143C4×C3⋊D3—23no—
144C4×C3⋊D3—23no—
145#634—23no—
146#634C23×C313yes—
147#634C23×C311yescharacteristic
148C3×#67—63no—
149C3×#67—63no—
150C2×C3×Dic3—63no—
151C3×D12—33no—
152C3×D12—31no—
153C3×D12—63no—
154C3×C4×D3—63no—
155C3×C4×D3—63no—
156C3×Dic6—33no—
157C3×Dic6—31no—
158C3×Dic6—63no—
159C32×C8—21no—
160C32×C8—23no—
161C32×C8C22×D311yescharacteristic
162C32×C8C22×D313yes—
163C32×C8C23×C311yescharacteristic
164C32×C8C23×C313yes—
165#577—91noSylow
166C2×H(3,𝔽3)—23no—
167C2×H(3,𝔽3)C22×D411yescharacteristic
168(C3⋊D3)⋊C3—41no—
169(C3⋊D3)⋊C3—43no—
170C3×#41—31no—
171C3×#41—31no—
172C3×#41—61no—
173C3×#41—93no—
174C3×#41—93no—
175C3×#41—183no—
176C3×#41—181no—
177C3×#41C2×C3×D311yescharacteristic
178C2×C3×Q8—93no—
179C2×C3×Q8—181no—
180C2×C3×D4—93no—
181C2×C3×D4—183no—
182#239—61no—
183#239—181no—
184D3×Q8—61no—
185D3×Q8—181no—
186#237—63no—
187#237—183no—
188D3×D4—63no—
189D3×D4—183no—
190#235—33no—
191#235—33no—
192#235—93no—
193#235—93no—
194C2×D12—33no—
195C2×D12—93no—
196C2×Dic6—33no—
197C2×Dic6—93no—
198C3×Q16—93no—
199C3×SD16—93no—
200C3×SD16—96no—
201C3×SD16—91no—
202C3×D8—93no—
203C3×M16—33no—
204C3×M16—33no—
205C3×M16—63no—
206C3×M16C2×C3×D313yes—
207C2×C3×C8—33no—
208C2×C3×C8—33no—
209C2×C3×C8—63no—
210C2×C3×C8C2×C3×D313yes—
211Dic12—33no—
212Dic12—93no—
213D24—33no—
214D24—93no—
215#204—33no—
216#204—36no—
217#204—31no—
218#204—93no—
219#204—96no—
220#204—91no—
221C22×C32—23no—
222C22×C32C2×D1213yes—
223C22×C32C2×C3×D413yes—
224C2×C3⋊D3—21no—
225C2×C3⋊D3—23no—
226C2×C3⋊D3—83no—
227C2×C3×D3—61no—
228C2×C3×D3—63no—
229C2×C3×D3—243no—
230C32×C4—21no—
231C32×C4—23no—
232C32×C4C2×D1213yes—
233C32×C4C2×C3×D413yes—
234C32×C4C23×D311yescharacteristic
235C32×C4C24×C311yescharacteristic
236C3⋊Dic3—23no—
237C3⋊Dic3—21no—
238C3×Dic3—63no—
239C3×Dic3—61no—
2402−1+4—91no—
241H(5,𝔽2)—91no—
242#137—93no—
243AGL(1,ℤ/8)—93no—
244#135—93no—
245C2×SD16—93no—
246#131(C3⋊D3)⋊C311yescharacteristic, complemented
247H(3,𝔽3)#57711yescharacteristic, complemented, Sylow
248C23×C3—183no—
249C22×D3—63no—
250C22×D3—183no—
251C3×Q8—31no—
252C3×Q8—31no—
253C3×Q8—61no—
254C3×Q8—93no—
255C3×Q8—91no—
256C3×Q8—183no—
257C3×Q8C3×D1211yescharacteristic
258C3×D4—33no—
259C3×D4—33no—
260C3×D4—63no—
261C3×D4—91no—
262C3×D4—93no—
263C3×D4—183no—
264C3×D4—183no—
265C3×D4—183no—
266C3×D4C3×D1213yes—
267C2×C3×C4—33no—
268C2×C3×C4—33no—
269C2×C3×C4—63no—
270C2×C3×C4—183no—
271C2×C3×C4—183no—
272C2×C3×C4—183no—
273C2×C3×C4C22×C3×D313yes—
274#67—63no—
275#67—63no—
276#67—183no—
277#67—183no—
278C2×Dic3—63no—
279C2×Dic3—183no—
280D12—33no—
281D12—31no—
282D12—63no—
283D12—93no—
284D12—91no—
285D12—183no—
286C4×D3—63no—
287C4×D3—63no—
288C4×D3—183no—
289C4×D3—183no—
290Dic6—33no—
291Dic6—31no—
292Dic6—63no—
293Dic6—93no—
294Dic6—91no—
295Dic6—183no—
296C3×C8—31no—
297C3×C8—33no—
298C3×C8—31no—
299C3×C8—33no—
300C3×C8—61no—
301C3×C8—63no—
302C3×C8C22×C3×D311yescharacteristic
303C3×C8C22×C3×D313yes—
304C2×C32—23no—
305C2×C32—23no—
306C2×C32—21no—
307C2×C32—43no—
308C2×C32C22×D1211yescharacteristic
309C2×C32C22×C3×D411yescharacteristic
310C3⋊D3—41no—
311C3⋊D3—43no—
312C3×D3—121no—
313C3×D3—123no—
314#41—93no—
315#41—93no—
316#41—181no—
317#41—183no—
318#41C2×(C3⋊D3)⋊C311yescharacteristic
319C2×Q8—93no—
320C2×Q8—181no—
321C2×D4—93no—
322C2×D4—183no—
323Q16—93no—
324SD16—93no—
325SD16—96no—
326SD16—91no—
327D8—93no—
328M16C2×(C3⋊D3)⋊C313yescomplemented
329C2×C8C2×(C3⋊D3)⋊C313yescomplemented
330C22×C3—33no—
331C22×C3—33no—
332C22×C3—63no—
333C22×C3—181no—
334C22×C3—183no—
335C22×C3—723no—
336C22×C3C2×C3×D1213yes—
337C2×D3—61no—
338C2×D3—63no—
339C2×D3—181no—
340C2×D3—183no—
341C2×D3—243no—
342C2×D3—723no—
343C3×C4—31no—
344C3×C4—31no—
345C3×C4—33no—
346C3×C4—33no—
347C3×C4—61no—
348C3×C4—63no—
349C3×C4—181no—
350C3×C4—183no—
351C3×C4C2×C3×D1213yes—
352C3×C4C23×C3×D311yescharacteristic
353Dic3—63no—
354Dic3—61no—
355Dic3—183no—
356Dic3—181no—
357C32—21no—
358C32#575111yescharacteristic, complemented
359C32C3×#57711yescharacteristic, complemented
360C23—183no—
361Q8—93no—
362Q8—91no—
363Q8—183no—
364Q8C4⋊((C3⋊D3)⋊C3)11yescharacteristic
365D4—91no—
366D4—93no—
367D4—183no—
368D4—183no—
369D4—183no—
370D4C4⋊((C3⋊D3)⋊C3)13yes—
371C2×C4—183no—
372C2×C4—183no—
373C2×C4—183no—
374C2×C4C22×(C3⋊D3)⋊C313yes—
375C8C22×(C3⋊D3)⋊C311yescharacteristic
376C8C22×(C3⋊D3)⋊C313yes—
377C2×C3—23no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—63no—
381C2×C3—63no—
382C2×C3—61no—
383C2×C3—123no—
384C2×C3—363no—
385C2×C3—361no—
386C2×C3C22×C3×D1211yescharacteristic
387D3—123no—
388D3—121no—
389D3—363no—
390D3—361no—
391C22—181no—
392C22—183no—
393C22—723no—
394C22C2×C4⋊((C3⋊D3)⋊C3)13yes—
395C4—181no—
396C4—183no—
397C4C2×C4⋊((C3⋊D3)⋊C3)13yes—
398C4C23×(C3⋊D3)⋊C311yescharacteristic
399C3—31no—
400C3—31no—
401C3—61no—
402C3C3×#575111yescharacteristic
403C2—23no—
404C2—363no—
405C2—361no—
406C2C22×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
subgroupG/H
centerC2C22×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
FittingH(3,𝔽3)×#131C2
socleC2×C3C22×C3×D12
hypercenter#131(C3⋊D3)⋊C3
hypocenterC32C3×#577
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ #41 ◁ #131

Complements

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

NH ≅ G/Ntype
1. #11722639C2semidirect
2. #11722638C2semidirect
3. H(3,𝔽3)×#131C2semidirect
4. #64118⋊C3C2semidirect
5. #64117⋊C3C2semidirect
6. #64113⋊C3C2semidirect
7. C2×#3744⋊C3C2semidirect
8. #10596222C3semidirect
16. #3828⋊C3C22semidirect
18. C2×#634⋊C3C22semidirect
19. M16×H(3,𝔽3)C22semidirect
20. C2×C8×H(3,𝔽3)C22semidirect
21. #3745⋊C3C22semidirect
22. #3744⋊C3C22semidirect
23. #3744⋊C3C22semidirect
24. #3744⋊C3C22semidirect
25. #3746⋊C3C22semidirect
26. #64367C2×C3semidirect
27. #64366C2×C3semidirect
31. C32×#131D3semidirect
32. C32×#131C2×C3semidirect
33. #64118C2×C3semidirect
34. #64117C2×C3semidirect
35. #64113C2×C3semidirect
36. C2×#3744C2×C3semidirect
69. #3828C22×C3semidirect
71. C2×#634C22×C3semidirect
81. C32×M16C2×D3semidirect
82. C32×M16C22×C3semidirect
84. C2×C32×C8C2×D3semidirect
85. C2×C32×C8C22×C3semidirect
86. #3746C22×C3semidirect
87. #3745C22×C3semidirect
88. #3744C22×C3semidirect
89. #3744C22×C3semidirect
90. #3744C22×C3semidirect
246. #131(C3⋊D3)⋊C3semidirect
247. H(3,𝔽3)#577semidirect
328. M16C2×(C3⋊D3)⋊C3semidirect
329. C2×C8C2×(C3⋊D3)⋊C3semidirect
358. C32#5751semidirect
359. C32C3×#577semidirect

Maximal subgroups

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

subgroupindex
#117226392
#117226382
H(3,𝔽3)×#1312
#64118⋊C32
#64117⋊C32
#64113⋊C32
C2×#3744⋊C32
#105962223
C3×#57513