Finite Group Databasebeta

← #13153747

Subgroups — #13153747

Order 1728.

1104 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2#11722638C211yesmaximal, characteristic, complemented
3#64159⋊C3C213yesmaximal, complemented
4#64158⋊C3C213yesmaximal, complemented
5#11721689C211yesmaximal, characteristic, complemented
6#64141⋊C3C213yesmaximal, complemented
7C2×#3753⋊C3C213yesmaximal, complemented
8#10596306C311yesmaximal, characteristic, complemented
9C3×#5835—31nomaximal
10#41×H(3,𝔽3)—21no—
11#41×H(3,𝔽3)C2211yescharacteristic
12#41×H(3,𝔽3)C2213yescomplemented
13C2×D4×H(3,𝔽3)—23no—
14C2×D4×H(3,𝔽3)C2213yescomplemented
15#88722—21no—
16D4×(C3⋊D3)⋊C3—23no—
17#3828⋊C3C2213yescomplemented
18C2×C4⋊((C3⋊D3)⋊C3)C2213yescomplemented
19#3755⋊C3C2213yescomplemented
20#3756⋊C3C2211yescharacteristic, complemented
21#3753⋊C3C2213yescomplemented
22#3753⋊C3C2216yescomplemented
23#3754⋊C3C2213yescomplemented
24#3748⋊C3C2213yescomplemented
25C2×(C3⋊(C3⋊C8))⋊C3C2213yescomplemented
26C32×H(5,𝔽2)—21no—
27C32×H(5,𝔽2)D311yescharacteristic, complemented
28C32×H(5,𝔽2)C2×C311yescharacteristic, complemented
29#64366C2×C311yescharacteristic, complemented
30C3×#1009—31no—
31#64159C2×C313yescomplemented
32#64158C2×C313yescomplemented
33#64157C2×C311yescharacteristic, complemented
34#64141C2×C313yescomplemented
35C2×#3753C2×C313yescomplemented
36C3×#950—33no—
37C3×#949—33no—
38C3×#948—31no—
39C3×#932—33no—
40C2×C3×#213—33no—
41C23×H(3,𝔽3)—23no—
42C22×(C3⋊D3)⋊C3—23no—
43Q8×H(3,𝔽3)D411yescharacteristic
44Q8×H(3,𝔽3)C2311yescharacteristic
45D4×H(3,𝔽3)—23no—
46D4×H(3,𝔽3)—23no—
47D4×H(3,𝔽3)D413yes—
48D4×H(3,𝔽3)C2313yes—
49C2×C4×H(3,𝔽3)—23no—
50C2×C4×H(3,𝔽3)C2313yes—
51#638⋊C3—23no—
52C4⋊((C3⋊D3)⋊C3)—23no—
53C4⋊((C3⋊D3)⋊C3)C2313yes—
54C4×(C3⋊D3)⋊C3—23no—
55#634⋊C3C2311yescharacteristic
56(C3⋊(C3⋊C8))⋊C3C2311yescharacteristic
57(C3⋊(C3⋊C8))⋊C3C2313yes—
58C3×#576—91no—
59#5835—31no—
60#5835—91no—
61C32×#41—22no—
62C32×#41—21no—
63C32×#41—23no—
64C32×#41—21no—
65C32×#41—21no—
66C32×#41C2×D311yescharacteristic
67C32×#41C2×D313yescomplemented
68C32×#41C22×C311yescharacteristic
69C32×#41C22×C313yescomplemented
70C2×C32×D4—26no—
71C2×C32×D4—23no—
72C2×C32×D4—23no—
73C2×C32×D4—23no—
74C2×C32×D4C2×D313yescomplemented
75C2×C32×D4C22×C313yescomplemented
76#3832—21no—
77D4×C3⋊D3—23no—
78#3828C22×C313yescomplemented
79C2×C4⋊(C3⋊D3)C22×C313yescomplemented
80C3×#239—61no—
81C3×D3×D4—63no—
82C3×#235—33no—
83C2×C3×D12—33no—
84#3756C22×C311yescharacteristic, complemented
85#3755C22×C313yescomplemented
86#3754C22×C313yescomplemented
87#3753C22×C313yescomplemented
88#3753C22×C316yescomplemented
89#3748C22×C313yescomplemented
90C2×C3⋊(C3⋊C8)C22×C313yescomplemented
91C3×#216—31no—
92C3×#215—33no—
93C3×#214—33no—
94C3×#213—33no—
95C3×#213—36no—
96C3×#208—33no—
97C2×C3×C3⋊C8—33no—
98C22×H(3,𝔽3)—23no—
99C22×H(3,𝔽3)—83no—
100C22×H(3,𝔽3)C2×D413yes—
101C2×(C3⋊D3)⋊C3—23no—
102C2×(C3⋊D3)⋊C3—83no—
103C4×H(3,𝔽3)—21no—
104C4×H(3,𝔽3)C2×D413yes—
105C4×H(3,𝔽3)C2411yescharacteristic
106(C3⋊Dic3)⋊C3—21no—
107C3×H(5,𝔽2)—31no—
108C3×H(5,𝔽2)—31no—
109C3×H(5,𝔽2)—61no—
110C3×H(5,𝔽2)—91no—
111C3×H(5,𝔽2)C3×D311yescharacteristic
112#1009—31no—
113#1009—91no—
114C3×AGL(1,ℤ/8)—93no—
115C3×AGL(1,ℤ/8)—93no—
116C3×#135—93no—
117C2×C3×D8—93no—
118C3×#131—91no—
119#950—33no—
120#950—93no—
121#949—33no—
122#949—93no—
123#948—31no—
124#948—91no—
125#932—33no—
126#932—93no—
127C2×#213—33no—
128C2×#213—93no—
129C23×C32—26no—
130C23×C32—23no—
131C23×C32—23no—
132C22×C3⋊D3—23no—
133C22×C3×D3—63no—
134C32×Q8—21no—
135C32×Q8—21no—
136C32×Q8#6711yescharacteristic
137C32×Q8C3×D411yescharacteristic
138C32×Q8C22×D311yescharacteristic
139C32×Q8C23×C311yescharacteristic
140C32×D4—26no—
141C32×D4—26no—
142C32×D4—23no—
143C32×D4—23no—
144C32×D4—23no—
145C32×D4—23no—
146C32×D4—23no—
147C32×D4—23no—
148C32×D4#6713yes—
149C32×D4C3×D413yes—
150C32×D4C22×D313yes—
151C32×D4C23×C313yes—
152C2×C32×C4—26no—
153C2×C32×C4—23no—
154C2×C32×C4—23no—
155C2×C32×C4—23no—
156C2×C32×C4C22×D313yes—
157C2×C32×C4C23×C313yes—
158#638—23no—
159C4⋊(C3⋊D3)—23no—
160C4⋊(C3⋊D3)C23×C313yes—
161C4×C3⋊D3—23no—
162#634C23×C311yescharacteristic
163C3×#67—63no—
164C3×D12—33no—
165C3×D12—63no—
166C3×C4×D3—63no—
167C3×Dic6—31no—
168C3⋊(C3⋊C8)C23×C311yescharacteristic
169C3⋊(C3⋊C8)C23×C313yes—
170C3×C3⋊C8—31no—
171C3×C3⋊C8—33no—
172#576—91noSylow
173C2×H(3,𝔽3)—23no—
174C2×H(3,𝔽3)—43no—
175C2×H(3,𝔽3)C22×D411yescharacteristic
176(C3⋊D3)⋊C3—43no—
177C3×#41—21no—
178C3×#41—31no—
179C3×#41—31no—
180C3×#41—33no—
181C3×#41—33no—
182C3×#41—61no—
183C3×#41—61no—
184C3×#41—62no—
185C3×#41—61no—
186C3×#41—63no—
187C3×#41—93no—
188C3×#41—181no—
189C3×#41C2×C3×D311yescharacteristic
190C3×#41C2×C3×D313yes—
191C2×C3×D4—23no—
192C2×C3×D4—33no—
193C2×C3×D4—33no—
194C2×C3×D4—63no—
195C2×C3×D4—63no—
196C2×C3×D4—66no—
197C2×C3×D4—63no—
198C2×C3×D4—93no—
199C2×C3×D4—183no—
200C2×C3×D4C2×C3×D313yes—
201#239—61no—
202#239—181no—
203D3×D4—63no—
204D3×D4—183no—
205#235—33no—
206#235—93no—
207C2×D12—33no—
208C2×D12—93no—
209C3×Q16—91no—
210C3×SD16—93no—
211C3×SD16—93no—
212C3×D8—93no—
213C3×D8—96no—
214C3×M16—93no—
215C2×C3×C8—93no—
216#216—31no—
217#216—91no—
218#215—33no—
219#215—93no—
220#214—33no—
221#214—93no—
222#213—33no—
223#213—36no—
224#213—93no—
225#213—96no—
226#208—33no—
227#208—93no—
228C2×C3⋊C8—33no—
229C2×C3⋊C8—93no—
230C22×C32—26no—
231C22×C32—23no—
232C22×C32—23no—
233C22×C32—23no—
234C22×C32—83no—
235C22×C32—83no—
236C22×C32—86no—
237C22×C32C2×#6713yes—
238C22×C32C2×C3×D413yes—
239C2×C3⋊D3—23no—
240C2×C3⋊D3—83no—
241C2×C3×D3—63no—
242C2×C3×D3—243no—
243C32×C4—22no—
244C32×C4—21no—
245C32×C4—21no—
246C32×C4—21no—
247C32×C4—23no—
248C32×C4C2×#6713yes—
249C32×C4C2×C3×D413yes—
250C32×C4C23×D311yescharacteristic
251C32×C4C24×C311yescharacteristic
252C3⋊Dic3—21no—
253C3×Dic3—61no—
254H(5,𝔽2)—91no—
255H(5,𝔽2)(C3⋊D3)⋊C311yescharacteristic, complemented
256AGL(1,ℤ/8)—93no—
257AGL(1,ℤ/8)—93no—
258#135—93no—
259C2×D8—93no—
260#131—91no—
261H(3,𝔽3)#57611yescharacteristic, complemented, Sylow
262C23×C3—23no—
263C23×C3—63no—
264C23×C3—63no—
265C23×C3—66no—
266C23×C3—183no—
267C22×D3—63no—
268C22×D3—183no—
269C3×Q8—31no—
270C3×Q8—31no—
271C3×Q8—31no—
272C3×Q8—31no—
273C3×Q8—61no—
274C3×Q8—61no—
275C3×Q8—91no—
276C3×Q8C3×#6711yescharacteristic
277C3×Q8C22×C3×D311yescharacteristic
278C3×D4—23no—
279C3×D4—23no—
280C3×D4—33no—
281C3×D4—33no—
282C3×D4—33no—
283C3×D4—33no—
284C3×D4—63no—
285C3×D4—63no—
286C3×D4—63no—
287C3×D4—63no—
288C3×D4—66no—
289C3×D4—66no—
290C3×D4—63no—
291C3×D4—63no—
292C3×D4—93no—
293C3×D4—183no—
294C3×D4—183no—
295C3×D4C3×#6713yes—
296C3×D4C22×C3×D313yes—
297C2×C3×C4—23no—
298C2×C3×C4—33no—
299C2×C3×C4—33no—
300C2×C3×C4—63no—
301C2×C3×C4—63no—
302C2×C3×C4—66no—
303C2×C3×C4—63no—
304C2×C3×C4—183no—
305C2×C3×C4C22×C3×D313yes—
306#67—63no—
307#67—183no—
308D12—33no—
309D12—63no—
310D12—93no—
311D12—183no—
312C4×D3—63no—
313C4×D3—183no—
314Dic6—31no—
315Dic6—91no—
316C3×C8—91no—
317C3×C8—93no—
318C3⋊C8—31no—
319C3⋊C8—33no—
320C3⋊C8—91no—
321C3⋊C8—93no—
322C2×C32—23no—
323C2×C32—23no—
324C2×C32—21no—
325C2×C32—43no—
326C2×C32—46no—
327C2×C32—43no—
328C2×C32—43no—
329C2×C32C22×#6711yescharacteristic
330C2×C32C22×C3×D411yescharacteristic
331C3⋊D3—43no—
332C3×D3—123no—
333#41—21no—
334#41—93no—
335#41—181no—
336#41C2×(C3⋊D3)⋊C311yescharacteristic
337#41C2×(C3⋊D3)⋊C313yescomplemented
338C2×D4—23no—
339C2×D4—93no—
340C2×D4—183no—
341C2×D4C2×(C3⋊D3)⋊C313yescomplemented
342Q16—91no—
343SD16—93no—
344SD16—93no—
345D8—93no—
346D8—96no—
347M16—93no—
348C2×C8—93no—
349C22×C3—23no—
350C22×C3—33no—
351C22×C3—33no—
352C22×C3—63no—
353C22×C3—63no—
354C22×C3—66no—
355C22×C3—63no—
356C22×C3—83no—
357C22×C3—183no—
358C22×C3—243no—
359C22×C3—243no—
360C22×C3—246no—
361C22×C3—723no—
362C22×C3C2×C3×#6713yes—
363C2×D3—63no—
364C2×D3—183no—
365C2×D3—243no—
366C2×D3—723no—
367C3×C4—21no—
368C3×C4—31no—
369C3×C4—31no—
370C3×C4—33no—
371C3×C4—33no—
372C3×C4—61no—
373C3×C4—61no—
374C3×C4—62no—
375C3×C4—61no—
376C3×C4—63no—
377C3×C4—181no—
378C3×C4C2×C3×#6713yes—
379C3×C4C23×C3×D311yescharacteristic
380Dic3—61no—
381Dic3—181no—
382C32—21no—
383C32#583511yescharacteristic, complemented
384C32C3×#57611yescharacteristic, complemented
385C23—23no—
386C23—183no—
387Q8—91no—
388Q8#638⋊C311yescharacteristic
389Q8C22×(C3⋊D3)⋊C311yescharacteristic
390D4—23no—
391D4—23no—
392D4—93no—
393D4—183no—
394D4—183no—
395D4#638⋊C313yes—
396D4C22×(C3⋊D3)⋊C313yes—
397C2×C4—23no—
398C2×C4—183no—
399C2×C4C22×(C3⋊D3)⋊C313yes—
400C8—91no—
401C8—93no—
402C2×C3—23no—
403C2×C3—31no—
404C2×C3—31no—
405C2×C3—43no—
406C2×C3—63no—
407C2×C3—63no—
408C2×C3—61no—
409C2×C3—123no—
410C2×C3—123no—
411C2×C3—123no—
412C2×C3—126no—
413C2×C3—363no—
414C2×C3C22×C3×#6711yescharacteristic
415D3—123no—
416D3—363no—
417C22—23no—
418C22—83no—
419C22—183no—
420C22—723no—
421C22C2×#638⋊C313yes—
422C4—21no—
423C4—181no—
424C4C2×#638⋊C313yes—
425C4C23×(C3⋊D3)⋊C311yescharacteristic
426C3—31no—
427C3—31no—
428C3—61no—
429C3C3×#583511yescharacteristic
430C2—23no—
431C2—43no—
432C2—363no—
433C2C22×#638⋊C311yescharacteristic
subgroupG/H
centerC2C22×#638⋊C3
commutator, cosocleC32×C4C24×C3
FrattiniC3×C4C23×C3×D3
FittingH(3,𝔽3)×H(5,𝔽2)C2
socleC2×C3C22×C3×#67
hypercenterH(5,𝔽2)(C3⋊D3)⋊C3
hypocenterC32C3×#576
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ #41 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×H(5,𝔽2)C2semidirect
2. #11722638C2semidirect
3. #64159⋊C3C2semidirect
4. #64158⋊C3C2semidirect
5. #11721689C2semidirect
6. #64141⋊C3C2semidirect
7. C2×#3753⋊C3C2semidirect
8. #10596306C3semidirect
12. #41×H(3,𝔽3)C22semidirect
14. C2×D4×H(3,𝔽3)C22semidirect
17. #3828⋊C3C22semidirect
18. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
19. #3755⋊C3C22semidirect
20. #3756⋊C3C22semidirect
21. #3753⋊C3C22semidirect
22. #3753⋊C3C22semidirect
23. #3754⋊C3C22semidirect
24. #3748⋊C3C22semidirect
25. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
27. C32×H(5,𝔽2)D3semidirect
28. C32×H(5,𝔽2)C2×C3semidirect
29. #64366C2×C3semidirect
31. #64159C2×C3semidirect
32. #64158C2×C3semidirect
33. #64157C2×C3semidirect
34. #64141C2×C3semidirect
35. C2×#3753C2×C3semidirect
67. C32×#41C2×D3semidirect
69. C32×#41C22×C3semidirect
74. C2×C32×D4C2×D3semidirect
75. C2×C32×D4C22×C3semidirect
78. #3828C22×C3semidirect
79. C2×C4⋊(C3⋊D3)C22×C3semidirect
84. #3756C22×C3semidirect
85. #3755C22×C3semidirect
86. #3754C22×C3semidirect
87. #3753C22×C3semidirect
88. #3753C22×C3semidirect
89. #3748C22×C3semidirect
90. C2×C3⋊(C3⋊C8)C22×C3semidirect
255. H(5,𝔽2)(C3⋊D3)⋊C3semidirect
261. H(3,𝔽3)#576semidirect
337. #41C2×(C3⋊D3)⋊C3semidirect
341. C2×D4C2×(C3⋊D3)⋊C3semidirect
383. C32#5835semidirect
384. C32C3×#576semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×H(5,𝔽2)2
#117226382
#64159⋊C32
#64158⋊C32
#117216892
#64141⋊C32
C2×#3753⋊C32
#105963063
C3×#58353