Finite Group Databasebeta

← C3×#10596521 #13181040

Subgroups — C3×#10596521 #13181040

Order 1728.

906 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 170 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×#64320C211yesmaximal, characteristic, complemented
2C3×#64316C211yesmaximal, characteristic, complemented
3C3×#64289C211yesmaximal, characteristic, complemented
4C3×#64227C213yesmaximal, complemented
5C3×#64226C213yesmaximal, complemented
6C3×#64224C213yesmaximal, complemented
7C3×#64223C213yesmaximal, complemented
8#10596521C311yesmaximal, characteristic, direct product factor
9C3×#3832C2211yescharacteristic
10C32×#239—21no—
11C32×D3×Q8—21no—
12C3×#3802C2213yes—
13C3×D3×D12—23no—
14C3×#3799—23no—
15C3×#3798C2213yes—
16C3×#3798C2213yescomplemented
17C3×#3797C2213yescomplemented
18C3×#3788C2213yescomplemented
19C3×#3787C2213yescomplemented
20C3×#3776C2213yescomplemented
21C3×#3775C2211yescharacteristic, complemented
22C3×#3775C2213yescomplemented
23C3×#3775C2216yescomplemented
24C3×#3774C2213yescomplemented
25C3×#1009—61no—
26C3×#1007—61no—
27#64320C2×C311yescharacteristic, complemented
28#64316C2×C311yescharacteristic, complemented
29#64289C2×C311yescharacteristic, complemented
30#64227C2×C313yescomplemented
31#64226C2×C313yescomplemented
32#64224C2×C313yescomplemented
33#64223C2×C313yescomplemented
34C2×C3×D32—23no—
35C33×Q8D411yescharacteristic
36C3×C4⋊(C3⋊D3)D413yes—
37C3×C4×C3⋊D3C2313yes—
38C32×D12—23no—
39C32×C4×D3—23no—
40C32×C4×D3—23no—
41C32×Dic6—23no—
42C3×#627C2313yes—
43C3×#627C2311yescharacteristic
44C3×#626—23no—
45C3×#626—23no—
46C3×#625C2311yescharacteristic
47C3×#625C2313yes—
48C3×#624—23no—
49C3×#622C2313yes—
50C3×#622C2311yescharacteristic
51C3×#577—91nomaximal
52C32×#41—61no—
53C2×C32×Q8—61no—
54#3832C22×C311yescharacteristic
55C3×#239—21no—
56C3×#239—21no—
57C3×#239—41no—
58C3×#239—63no—
59C3×#239—61no—
60C3×#239—61no—
61C3×D3×Q8—21no—
62C3×D3×Q8—21no—
63C3×D3×Q8—41no—
64C3×D3×Q8—63no—
65C3×D3×D4—63no—
66C3×D3×D4—63no—
67C3×#235—63no—
68C3×#235—63no—
69C3×#235—63no—
70C2×C3×D12—63no—
71#3802C22×C313yes—
72D3×D12—23no—
73#3799—23no—
74#3798C22×C313yes—
75#3798C22×C313yescomplemented
76#3797C22×C313yescomplemented
77#3788C22×C313yescomplemented
78#3787C22×C313yescomplemented
79#3776C22×C313yescomplemented
80#3775C22×C311yescharacteristic, complemented
81#3775C22×C313yescomplemented
82#3775C22×C316yescomplemented
83#3774C22×C313yescomplemented
84C2×C3×C3⋊D3C2×D413yes—
85C2×C32×D3—21no—
86C2×C32×D3—23no—
87C3×D32—83no—
88C33×C4C2×D413yes—
89C3×C3⋊Dic3C2411yescharacteristic
90C32×Dic3—23no—
91C32×Dic3—21no—
92C3×2−1+4—91no—
93C3×H(5,𝔽2)—91no—
94#1009—61no—
95#1007—61no—
96C3×#137—93no—
97C3×AGL(1,ℤ/8)—93no—
98C3×#135—93no—
99C2×C3×SD16—93no—
100C3×#131—91no—
101C22×C3×D3—66no—
102C2×D32—23no—
103C32×Q8—21no—
104C32×Q8—21no—
105C32×Q8—41no—
106C32×Q8—41no—
107C32×Q8—63no—
108C32×Q8C3×D411yescharacteristic
109C32×D4—63no—
110C2×C32×C4—63no—
111C2×C32×C4—63no—
112C4⋊(C3⋊D3)C3×D413yes—
113C4×C3⋊D3C23×C313yes—
114C3×#67—61no—
115C3×#67—63no—
116C3×#67—63no—
117C3×#67—63no—
118C3×D12—23no—
119C3×D12—23no—
120C3×D12—43no—
121C3×D12—63no—
122C3×D12—63no—
123C3×D12—63no—
124C3×D12—63no—
125C3×C4×D3—23no—
126C3×C4×D3—23no—
127C3×C4×D3—23no—
128C3×C4×D3—23no—
129C3×C4×D3—43no—
130C3×C4×D3—43no—
131C3×C4×D3—63no—
132C3×C4×D3—63no—
133C3×C4×D3—66no—
134C3×Dic6—23no—
135C3×Dic6—23no—
136C3×Dic6—43no—
137C3×Dic6—61no—
138C3×Dic6—63no—
139#627C23×C313yes—
140#627C23×C311yescharacteristic
141#626—23no—
142#626—23no—
143#625C23×C311yescharacteristic
144#625C23×C313yes—
145#624—23no—
146#622C23×C313yes—
147#622C23×C311yescharacteristic
148#577—91noSylow
149C2×C33C22×D411yescharacteristic
150C3×C3⋊D3—23no—
151C32×D3—41no—
152C32×D3—43no—
153C3×#41—61no—
154C3×#41—61no—
155C3×#41—93no—
156C3×#41—93no—
157C3×#41—91no—
158C3×#41—121no—
159C3×#41—183no—
160C2×C3×Q8—61no—
161C2×C3×Q8—61no—
162C2×C3×Q8—93no—
163C2×C3×Q8—121no—
164C2×C3×D4—93no—
165C2×C3×D4—183no—
166#239—21no—
167#239—61no—
168#239—61no—
169#239—63no—
170D3×Q8—21no—
171D3×Q8—63no—
172D3×D4—63no—
173D3×D4—63no—
174#235—63no—
175#235—63no—
176#235—63no—
177C2×D12—63no—
178C3×Q16—93no—
179C3×SD16—93no—
180C3×SD16—96no—
181C3×SD16—91no—
182C3×D8—93no—
183C3×M16—93no—
184C2×C3×C8—93no—
185C22×C32—61no—
186C22×C32—63no—
187C2×C3⋊D3C2×C3×D413yes—
188C2×C3×D3—21no—
189C2×C3×D3—23no—
190C2×C3×D3—21no—
191C2×C3×D3—23no—
192C2×C3×D3—41no—
193C2×C3×D3—43no—
194C2×C3×D3—63no—
195C2×C3×D3—63no—
196C2×C3×D3—246no—
197D32—83no—
198C32×C4—23no—
199C32×C4—23no—
200C32×C4—43no—
201C32×C4—43no—
202C32×C4—63no—
203C32×C4—61no—
204C32×C4C2×C3×D413yes—
205C3⋊Dic3C24×C311yescharacteristic
206C3×Dic3—23no—
207C3×Dic3—21no—
208C3×Dic3—23no—
209C3×Dic3—21no—
210C3×Dic3—43no—
211C3×Dic3—41no—
212C3×Dic3—61no—
213C3×Dic3—61no—
2142−1+4—91no—
215H(5,𝔽2)—91no—
216#137—93no—
217AGL(1,ℤ/8)—93no—
218#135—93no—
219C2×SD16—93no—
220#131—91no—
221C33#57711yescharacteristic, complemented, Sylow
222C23×C3—183no—
223C22×D3—66no—
224C3×Q8—21no—
225C3×Q8—21no—
226C3×Q8—41no—
227C3×Q8—41no—
228C3×Q8—63no—
229C3×Q8—63no—
230C3×Q8—93no—
231C3×Q8—91no—
232C3×Q8—123no—
233C3×Q8GO+(4,𝔽2)11yescharacteristic
234C3×D4—63no—
235C3×D4—63no—
236C3×D4—91no—
237C3×D4—93no—
238C3×D4—93no—
239C3×D4—123no—
240C3×D4—183no—
241C3×D4—183no—
242C2×C3×C4—63no—
243C2×C3×C4—63no—
244C2×C3×C4—63no—
245C2×C3×C4—63no—
246C2×C3×C4—93no—
247C2×C3×C4—123no—
248C2×C3×C4—123no—
249C2×C3×C4—183no—
250#67—61no—
251#67—63no—
252#67—63no—
253#67—63no—
254D12—23no—
255D12—63no—
256D12—63no—
257D12—63no—
258D12—63no—
259C4×D3—23no—
260C4×D3—23no—
261C4×D3—63no—
262C4×D3—63no—
263C4×D3—66no—
264Dic6—23no—
265Dic6—61no—
266Dic6—63no—
267C3×C8—91no—
268C3×C8—93no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—41no—
272C2×C32—41no—
273C2×C32—121no—
274C2×C32—123no—
275C2×C32C22×C3×D411yescharacteristic
276C3⋊D3—23no—
277C3×D3—41no—
278C3×D3—43no—
279C3×D3—42no—
280C3×D3—43no—
281C3×D3—41no—
282C3×D3—83no—
283C3×D3—123no—
284C3×D3—123no—
285#41—61no—
286#41—93no—
287#41—93no—
288#41—91no—
289#41—183no—
290C2×Q8—61no—
291C2×Q8—93no—
292C2×D4—93no—
293C2×D4—183no—
294Q16—93no—
295SD16—93no—
296SD16—96no—
297SD16—91no—
298D8—93no—
299M16—93no—
300C2×C8—93no—
301C22×C3—61no—
302C22×C3—63no—
303C22×C3—61no—
304C22×C3—63no—
305C22×C3—93no—
306C22×C3—121no—
307C22×C3—123no—
308C22×C3—723no—
309C2×D3—21no—
310C2×D3—23no—
311C2×D3—63no—
312C2×D3—63no—
313C2×D3—246no—
314C3×C4—23no—
315C3×C4—23no—
316C3×C4—43no—
317C3×C4—43no—
318C3×C4—63no—
319C3×C4—61no—
320C3×C4—61no—
321C3×C4—63no—
322C3×C4—91no—
323C3×C4—121no—
324C3×C4—123no—
325C3×C4C2×GO+(4,𝔽2)13yes—
326Dic3—23no—
327Dic3—21no—
328Dic3—61no—
329Dic3—61no—
330C32—21no—
331C32—21no—
332C32—41no—
333C32—41no—
334C32C3×#57711yescharacteristic, complemented
335C23—183no—
336Q8—63no—
337Q8—93no—
338Q8—91no—
339Q8C3×GO+(4,𝔽2)11yescharacteristic
340D4—63no—
341D4—91no—
342D4—93no—
343D4—93no—
344D4—183no—
345D4—183no—
346C2×C4—63no—
347C2×C4—63no—
348C2×C4—93no—
349C2×C4—183no—
350C8—91no—
351C8—93no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—41no—
355C2×C3—41no—
356C2×C3—123no—
357C2×C3—121no—
358C2×C3—122no—
359C2×C3—123no—
360C2×C3—121no—
361C2×C3—183no—
362C2×C3—243no—
363C2×C3C22×GO+(4,𝔽2)11yescharacteristic
364D3—43no—
365D3—41no—
366D3—123no—
367D3—123no—
368C22—61no—
369C22—63no—
370C22—93no—
371C22—723no—
372C4—61no—
373C4—63no—
374C4—91no—
375C4C2×C3×GO+(4,𝔽2)13yes—
376C3—21no—
377C3—21no—
378C3—41no—
379C3—41no—
380C3#1059652111yescharacteristic, direct product factor
381C2—123no—
382C2—121no—
383C2—183no—
384C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C3C22×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC33×Q8D4
socleC2×C33C22×D4
hypercenterC3×Q8GO+(4,𝔽2)
hypocenterC32C3×#577
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64320C2semidirect
2. C3×#64316C2semidirect
3. C3×#64289C2semidirect
4. C3×#64227C2semidirect
5. C3×#64226C2semidirect
6. C3×#64224C2semidirect
7. C3×#64223C2semidirect
8. #10596521C3direct (N × H)
16. C3×#3798C22semidirect
17. C3×#3797C22semidirect
18. C3×#3788C22semidirect
19. C3×#3787C22semidirect
20. C3×#3776C22semidirect
21. C3×#3775C22semidirect
22. C3×#3775C22semidirect
23. C3×#3775C22semidirect
24. C3×#3774C22semidirect
27. #64320C2×C3semidirect
28. #64316C2×C3semidirect
29. #64289C2×C3semidirect
30. #64227C2×C3semidirect
31. #64226C2×C3semidirect
32. #64224C2×C3semidirect
33. #64223C2×C3semidirect
75. #3798C22×C3semidirect
76. #3797C22×C3semidirect
77. #3788C22×C3semidirect
78. #3787C22×C3semidirect
79. #3776C22×C3semidirect
80. #3775C22×C3semidirect
81. #3775C22×C3semidirect
82. #3775C22×C3semidirect
83. #3774C22×C3semidirect
221. C33#577semidirect
334. C32C3×#577semidirect
380. C3#10596521direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643202
C3×#643162
C3×#642892
C3×#642272
C3×#642262
C3×#642242
C3×#642232
#105965213
C3×#5779