Finite Group Databasebeta

← #13153748

Subgroups — #13153748

Order 1728.

1069 conjugacy classes of subgroups.

The subgroup lattice has 428 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#11722639C211yesmaximal, characteristic, complemented
3#64159⋊C3C213yesmaximal, complemented
4#64160⋊C3C213yesmaximal, complemented
5#11721689C211yesmaximal, characteristic, complemented
6#64141⋊C3C213yesmaximal, complemented
7C2×#3754⋊C3C213yesmaximal, complemented
8#10596307C311yesmaximal, characteristic, complemented
9C3×#5836—31nomaximal
10#41×H(3,𝔽3)—21no—
11#41×H(3,𝔽3)C2213yes—
12#41×H(3,𝔽3)C2211yescharacteristic
13C2×D4×H(3,𝔽3)—23no—
14C2×D4×H(3,𝔽3)C2213yes—
15Q8×(C3⋊D3)⋊C3—21no—
16#3830⋊C3—23no—
17#3828⋊C3C2213yescomplemented
18C2×#634⋊C3C2213yescomplemented
19#3755⋊C3C2211yescharacteristic, complemented
20#3756⋊C3C2213yescomplemented
21#3753⋊C3C2213yescomplemented
22#3754⋊C3C2213yescomplemented
23#3754⋊C3C2216yescomplemented
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#64367C2×C311yescharacteristic, complemented
30C3×#1010—31no—
31#64160C2×C313yescomplemented
32#64159C2×C313yescomplemented
33#64157C2×C311yescharacteristic, complemented
34C2×#3754C2×C313yescomplemented
35#64141C2×C313yescomplemented
36C3×#951—33no—
37C3×#950—33no—
38C3×#948—31no—
39C2×C3×#214—33no—
40C3×#932—33no—
41C23×H(3,𝔽3)—23no—
42Q8×H(3,𝔽3)D411yescharacteristic
43Q8×H(3,𝔽3)C2311yescharacteristic
44D4×H(3,𝔽3)—23no—
45D4×H(3,𝔽3)—23no—
46D4×H(3,𝔽3)D413yes—
47D4×H(3,𝔽3)C2313yes—
48C2×C4×H(3,𝔽3)—23no—
49C2×C4×H(3,𝔽3)C2313yes—
50#638⋊C3—23no—
51C2×(C3⋊Dic3)⋊C3—23no—
52C4⋊((C3⋊D3)⋊C3)C2311yescharacteristic
53C4×(C3⋊D3)⋊C3—23no—
54#634⋊C3—23no—
55#634⋊C3C2313yes—
56(C3⋊(C3⋊C8))⋊C3C2311yescharacteristic
57(C3⋊(C3⋊C8))⋊C3C2313yes—
58C3×#577—91no—
59#5836—31no—
60#5836—91no—
61C32×#41—22no—
62C32×#41—23no—
63C32×#41—21no—
64C32×#41—21no—
65C32×#41—21no—
66C32×#41C2×D313yes—
67C32×#41C2×D311yescharacteristic
68C32×#41C22×C313yes—
69C32×#41C22×C311yescharacteristic
70C2×C32×D4—26no—
71C2×C32×D4—23no—
72C2×C32×D4—23no—
73C2×C32×D4—23no—
74C2×C32×D4C2×D313yes—
75C2×C32×D4C22×C313yes—
76Q8×C3⋊D3—21no—
77#3830—23no—
78#3828C22×C313yescomplemented
79C2×#634C22×C313yescomplemented
80C3×D3×Q8—61no—
81C3×#237—63no—
82C3×#235—33no—
83C2×C3×Dic6—33no—
84#3756C22×C313yescomplemented
85#3755C22×C311yescharacteristic, complemented
86#3754C22×C316yescomplemented
87#3754C22×C313yescomplemented
88#3753C22×C313yescomplemented
89#3748C22×C313yescomplemented
90C2×C3⋊(C3⋊C8)C22×C313yescomplemented
91C3×#216—33no—
92C3×#215—31no—
93C3×#214—36no—
94C3×#214—33no—
95C3×#213—33no—
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—21no—
102C4×H(3,𝔽3)—21no—
103C4×H(3,𝔽3)C2×D413yes—
104C4×H(3,𝔽3)C2411yescharacteristic
105(C3⋊Dic3)⋊C3—23no—
106C3×2−1+4—91no—
107C3×H(5,𝔽2)—31no—
108C3×H(5,𝔽2)—31no—
109C3×H(5,𝔽2)—61no—
110C3×H(5,𝔽2)C3×D311yescharacteristic
111#1010—31no—
112#1010—91no—
113C3×#137—93no—
114C3×AGL(1,ℤ/8)—93no—
115C3×#135—93no—
116C2×C3×SD16—93no—
117C3×#131—91no—
118#951—33no—
119#951—93no—
120#950—33no—
121#950—93no—
122#948—31no—
123#948—91no—
124C2×#214—33no—
125C2×#214—93no—
126#932—33no—
127#932—93no—
128C23×C32—26no—
129C23×C32—23no—
130C23×C32—23no—
131C32×Q8—21no—
132C32×Q8—21no—
133C32×Q8#6711yescharacteristic
134C32×Q8C3×D411yescharacteristic
135C32×Q8C22×D311yescharacteristic
136C32×Q8C23×C311yescharacteristic
137C32×D4—26no—
138C32×D4—26no—
139C32×D4—23no—
140C32×D4—23no—
141C32×D4—23no—
142C32×D4—23no—
143C32×D4—23no—
144C32×D4—23no—
145C32×D4#6713yes—
146C32×D4C3×D413yes—
147C32×D4C22×D313yes—
148C32×D4C23×C313yes—
149C2×C32×C4—26no—
150C2×C32×C4—23no—
151C2×C32×C4—23no—
152C2×C32×C4—23no—
153C2×C32×C4C22×D313yes—
154C2×C32×C4C23×C313yes—
155#638—23no—
156C2×C3⋊Dic3—23no—
157C4⋊(C3⋊D3)C23×C311yescharacteristic
158C4×C3⋊D3—23no—
159#634—23no—
160#634C23×C313yes—
161C3×#67—63no—
162C2×C3×Dic3—63no—
163C3×D12—31no—
164C3×C4×D3—63no—
165C3×Dic6—33no—
166C3×Dic6—63no—
167C3⋊(C3⋊C8)C23×C311yescharacteristic
168C3⋊(C3⋊C8)C23×C313yes—
169C3×C3⋊C8—31no—
170C3×C3⋊C8—33no—
171#577—91noSylow
172C2×H(3,𝔽3)—23no—
173C2×H(3,𝔽3)—43no—
174C2×H(3,𝔽3)C22×D411yescharacteristic
175(C3⋊D3)⋊C3—41no—
176C3×#41—21no—
177C3×#41—33no—
178C3×#41—33no—
179C3×#41—31no—
180C3×#41—31no—
181C3×#41—61no—
182C3×#41—61no—
183C3×#41—62no—
184C3×#41—63no—
185C3×#41—61no—
186C3×#41—93no—
187C3×#41—183no—
188C3×#41C2×C3×D313yes—
189C3×#41C2×C3×D311yescharacteristic
190C2×C3×Q8—93no—
191C2×C3×Q8—181no—
192C2×C3×D4—23no—
193C2×C3×D4—33no—
194C2×C3×D4—33no—
195C2×C3×D4—63no—
196C2×C3×D4—63no—
197C2×C3×D4—66no—
198C2×C3×D4—63no—
199C2×C3×D4C2×C3×D313yes—
200D3×Q8—61no—
201D3×Q8—181no—
202#237—63no—
203#237—183no—
204#235—33no—
205#235—93no—
206C2×Dic6—33no—
207C2×Dic6—93no—
208C3×Q16—93no—
209C3×SD16—93no—
210C3×SD16—96no—
211C3×SD16—91no—
212C3×D8—93no—
213C3×M16—93no—
214C2×C3×C8—93no—
215#216—33no—
216#216—93no—
217#215—31no—
218#215—91no—
219#214—36no—
220#214—33no—
221#214—96no—
222#214—93no—
223#213—33no—
224#213—93no—
225#208—33no—
226#208—93no—
227C2×C3⋊C8—33no—
228C2×C3⋊C8—93no—
229C22×C32—26no—
230C22×C32—23no—
231C22×C32—23no—
232C22×C32—23no—
233C22×C32—83no—
234C22×C32—83no—
235C22×C32—86no—
236C22×C32C2×#6713yes—
237C22×C32C2×C3×D413yes—
238C2×C3⋊D3—21no—
239C2×C3×D3—61no—
240C32×C4—22no—
241C32×C4—21no—
242C32×C4—21no—
243C32×C4—21no—
244C32×C4—23no—
245C32×C4C2×#6713yes—
246C32×C4C2×C3×D413yes—
247C32×C4C23×D311yescharacteristic
248C32×C4C24×C311yescharacteristic
249C3⋊Dic3—23no—
250C3×Dic3—63no—
2512−1+4—91no—
252H(5,𝔽2)(C3⋊D3)⋊C311yescharacteristic, complemented
253#137—93no—
254AGL(1,ℤ/8)—93no—
255#135—93no—
256C2×SD16—93no—
257#131—91no—
258H(3,𝔽3)#57711yescharacteristic, complemented, Sylow
259C23×C3—23no—
260C23×C3—63no—
261C23×C3—63no—
262C23×C3—66no—
263C3×Q8—31no—
264C3×Q8—31no—
265C3×Q8—31no—
266C3×Q8—31no—
267C3×Q8—61no—
268C3×Q8—61no—
269C3×Q8—93no—
270C3×Q8—183no—
271C3×Q8C3×#6711yescharacteristic
272C3×Q8C22×C3×D311yescharacteristic
273C3×D4—23no—
274C3×D4—23no—
275C3×D4—33no—
276C3×D4—33no—
277C3×D4—33no—
278C3×D4—33no—
279C3×D4—63no—
280C3×D4—63no—
281C3×D4—63no—
282C3×D4—63no—
283C3×D4—66no—
284C3×D4—66no—
285C3×D4—63no—
286C3×D4—63no—
287C3×D4—91no—
288C3×D4—183no—
289C3×D4C3×#6713yes—
290C3×D4C22×C3×D313yes—
291C2×C3×C4—23no—
292C2×C3×C4—33no—
293C2×C3×C4—33no—
294C2×C3×C4—63no—
295C2×C3×C4—63no—
296C2×C3×C4—66no—
297C2×C3×C4—63no—
298C2×C3×C4—183no—
299C2×C3×C4—183no—
300C2×C3×C4C22×C3×D313yes—
301#67—63no—
302#67—183no—
303C2×Dic3—63no—
304C2×Dic3—183no—
305D12—31no—
306D12—91no—
307C4×D3—63no—
308C4×D3—183no—
309Dic6—33no—
310Dic6—63no—
311Dic6—93no—
312Dic6—183no—
313C3×C8—91no—
314C3×C8—93no—
315C3⋊C8—31no—
316C3⋊C8—33no—
317C3⋊C8—91no—
318C3⋊C8—93no—
319C2×C32—23no—
320C2×C32—23no—
321C2×C32—21no—
322C2×C32—43no—
323C2×C32—46no—
324C2×C32—43no—
325C2×C32—43no—
326C2×C32C22×#6711yescharacteristic
327C2×C32C22×C3×D411yescharacteristic
328C3⋊D3—41no—
329C3×D3—121no—
330#41—21no—
331#41—93no—
332#41—183no—
333#41C2×(C3⋊D3)⋊C313yes—
334#41C2×(C3⋊D3)⋊C311yescharacteristic
335C2×Q8—93no—
336C2×Q8—181no—
337C2×D4—23no—
338C2×D4C2×(C3⋊D3)⋊C313yes—
339Q16—93no—
340SD16—91no—
341SD16—93no—
342SD16—96no—
343D8—93no—
344M16—93no—
345C2×C8—93no—
346C22×C3—23no—
347C22×C3—33no—
348C22×C3—33no—
349C22×C3—63no—
350C22×C3—63no—
351C22×C3—66no—
352C22×C3—63no—
353C22×C3—83no—
354C22×C3—181no—
355C22×C3—243no—
356C22×C3—243no—
357C22×C3—246no—
358C22×C3C2×C3×#6713yes—
359C2×D3—61no—
360C2×D3—181no—
361C3×C4—21no—
362C3×C4—31no—
363C3×C4—31no—
364C3×C4—33no—
365C3×C4—33no—
366C3×C4—61no—
367C3×C4—61no—
368C3×C4—62no—
369C3×C4—61no—
370C3×C4—63no—
371C3×C4—183no—
372C3×C4C2×C3×#6713yes—
373C3×C4C23×C3×D311yescharacteristic
374Dic3—63no—
375Dic3—183no—
376C32—21no—
377C32#583611yescharacteristic, complemented
378C32C3×#57711yescharacteristic, complemented
379C23—23no—
380Q8—93no—
381Q8—183no—
382Q8#638⋊C311yescharacteristic
383Q8C22×(C3⋊D3)⋊C311yescharacteristic
384D4—23no—
385D4—23no—
386D4—91no—
387D4—183no—
388D4#638⋊C313yes—
389D4C22×(C3⋊D3)⋊C313yes—
390C2×C4—23no—
391C2×C4—183no—
392C2×C4—183no—
393C2×C4C22×(C3⋊D3)⋊C313yes—
394C8—91no—
395C8—93no—
396C2×C3—23no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—43no—
400C2×C3—63no—
401C2×C3—63no—
402C2×C3—61no—
403C2×C3—123no—
404C2×C3—123no—
405C2×C3—123no—
406C2×C3—126no—
407C2×C3—361no—
408C2×C3C22×C3×#6711yescharacteristic
409D3—121no—
410D3—361no—
411C22—23no—
412C22—83no—
413C22—181no—
414C22C2×#638⋊C313yes—
415C4—21no—
416C4—183no—
417C4C2×#638⋊C313yes—
418C4C23×(C3⋊D3)⋊C311yescharacteristic
419C3—31no—
420C3—31no—
421C3—61no—
422C3C3×#583611yescharacteristic
423C2—23no—
424C2—43no—
425C2—361no—
426C2C22×#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×#577
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. #11722639C2semidirect
3. #64159⋊C3C2semidirect
4. #64160⋊C3C2semidirect
5. #11721689C2semidirect
6. #64141⋊C3C2semidirect
7. C2×#3754⋊C3C2semidirect
8. #10596307C3semidirect
17. #3828⋊C3C22semidirect
18. C2×#634⋊C3C22semidirect
19. #3755⋊C3C22semidirect
20. #3756⋊C3C22semidirect
21. #3753⋊C3C22semidirect
22. #3754⋊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. #64367C2×C3semidirect
31. #64160C2×C3semidirect
32. #64159C2×C3semidirect
33. #64157C2×C3semidirect
34. C2×#3754C2×C3semidirect
35. #64141C2×C3semidirect
78. #3828C22×C3semidirect
79. C2×#634C22×C3semidirect
84. #3756C22×C3semidirect
85. #3755C22×C3semidirect
86. #3754C22×C3semidirect
87. #3754C22×C3semidirect
88. #3753C22×C3semidirect
89. #3748C22×C3semidirect
90. C2×C3⋊(C3⋊C8)C22×C3semidirect
252. H(5,𝔽2)(C3⋊D3)⋊C3semidirect
258. H(3,𝔽3)#577semidirect
377. C32#5836semidirect
378. C32C3×#577semidirect

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
#117226392
#64159⋊C32
#64160⋊C32
#117216892
#64141⋊C32
C2×#3754⋊C32
#105963073
C3×#58363