Finite Group Databasebeta

← C3×C3⋊D3×#131 #13179500

Subgroups — C3×C3⋊D3×#131 #13179500

Order 1728.

1844 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#41×C3⋊D3C211yesmaximal, characteristic
2C33×#131C211yesmaximal, characteristic, complemented
3C3×#64157C211yesmaximal, characteristic, complemented
4C3×#64116C213yesmaximal, complemented
5C3×M16×C3⋊D3C213yesmaximal, complemented
6C3×#64110C213yesmaximal, complemented
7C2×C3×C8×C3⋊D3C213yesmaximal, complemented
8C3⋊D3×#131C311yesmaximal, characteristic, direct product factor
9C3×D3×#131—34nomaximal
10C33×#41C2211yescharacteristic
11C3×#3832C411yescharacteristic
12C3×Q8×C3⋊D3C411yescharacteristic
13C3×#3830C413yes—
14C3×D4×C3⋊D3C413yes—
15C3×#3828C2213yes—
16C2×C3×C4×C3⋊D3C2213yes—
17C33×M16C2213yescomplemented
18C2×C33×C8C2213yescomplemented
19C3×#3748C2213yescomplemented
20C2×C3×C3⋊(C3⋊C8)C2213yescomplemented
21C3×#3743C2216yescomplemented
22C3×C8×C3⋊D3C2211yescharacteristic
23C3×C8×C3⋊D3C2213yes—
24C3×C8×C3⋊D3C2213yescomplemented
25C3×C8×C3⋊D3C2213yescomplemented
26#41×C3⋊D3C2×C311yescharacteristic
27C3×D3×#41—34no—
28C32×#131—24no—
29C32×#131D314yescomplemented
30C32×#131C2×C311yescharacteristic, complemented
31#64157C2×C311yescharacteristic, complemented
32#64116C2×C313yescomplemented
33M16×C3⋊D3C2×C313yescomplemented
34#64110C2×C313yescomplemented
35C2×C8×C3⋊D3C2×C313yescomplemented
36C3×#948—34no—
37C3×#907—312no—
38C3×D3×M16—312no—
39C3×#901—312no—
40C2×C3×D3×C8—312no—
41C22×C3×C3⋊D3C2×C413yes—
42C33×Q8C2×C411yescharacteristic
43C33×D4C2×C413yes—
44C2×C33×C4C2313yes—
45C3×#638C2×C416yes—
46C2×C3×C3⋊Dic3C2×C413yes—
47C3×C4⋊(C3⋊D3)C2×C413yes—
48C3×C4×C3⋊D3C2×C413yes—
49C3×C4×C3⋊D3C2×C413yes—
50C3×C4×C3⋊D3C2311yescharacteristic
51C3×C4×C3⋊D3C2313yes—
52C3×#634C2×C413yes—
53C33×C8C2311yescharacteristic
54C33×C8C2313yes—
55C3×C3⋊(C3⋊C8)C2311yescharacteristic
56C3×C3⋊(C3⋊C8)C2313yes—
57C2×C3×#131—91no—
58D3×#131—34no—
59C32×#41—24no—
60C32×#41C2×D314yes—
61C32×#41C22×C311yescharacteristic
62#3832C3×C411yescharacteristic
63Q8×C3⋊D3C3×C411yescharacteristic
64#3830C3×C413yes—
65D4×C3⋊D3C3×C413yes—
66#3828C22×C313yes—
67C2×C4×C3⋊D3C22×C313yes—
68C3×#239—34no—
69C3×D3×Q8—34no—
70C3×#237—312no—
71C3×D3×D4—312no—
72C3×#235—312no—
73C2×C3×C4×D3—312no—
74C32×M16—212no—
75C32×M16C2×D3112yescomplemented
76C32×M16C22×C313yescomplemented
77C2×C32×C8—212no—
78C2×C32×C8C2×D3112yescomplemented
79C2×C32×C8C22×C313yescomplemented
80#3748C22×C313yescomplemented
81C2×C3⋊(C3⋊C8)C22×C313yescomplemented
82#3743C22×C316yescomplemented
83C8×C3⋊D3C22×C311yescharacteristic
84C8×C3⋊D3C22×C313yes—
85C8×C3⋊D3C22×C313yescomplemented
86C8×C3⋊D3C22×C313yescomplemented
87C3×#208—312no—
88C2×C3×C3⋊C8—312no—
89C3×ΓL(1,𝔽25)—324no—
90C3×D3×C8—34no—
91C3×D3×C8—312no—
92C3×D3×C8—312no—
93C3×D3×C8—312no—
94C22×C33C22×C413yes—
95C2×C3×C3⋊D3—26no—
96C2×C3×C3⋊D3C22×C413yes—
97C2×C3×C3⋊D3C22×C411yescharacteristic
98C33×C4C22×C413yes—
99C33×C4C2411yescharacteristic
100C3×C3⋊Dic3C22×C413yes—
101C3×C3⋊Dic3C22×C411yescharacteristic
102C2×C3×#41—91no—
103D3×#41—34no—
104C3×#131—24no—
105C3×#131—91no—
106C3×#131—93no—
107C3×#131—93no—
108C3×#131C3×D314yescomplemented
109C3×#131C3⋊D311yescharacteristic, direct product factor
110C2×C3×M16—93no—
111C22×C3×C8—93no—
112#948—34no—
113#907—312no—
114D3×M16—312no—
115#901—312no—
116C2×D3×C8—312no—
117C22×C3⋊D3C2×C3×C413yes—
118C22×C3×D3—312no—
119C32×Q8—24no—
120C32×Q8C4×D314yes—
121C32×Q8C2×C3×C411yescharacteristic
122C32×D4—212no—
123C32×D4C4×D3112yes—
124C32×D4C2×C3×C413yes—
125C2×C32×C4—212no—
126C2×C32×C4C22×D3112yes—
127C2×C32×C4C23×C313yes—
128#638C2×C3×C416yes—
129C2×C3⋊Dic3C2×C3×C413yes—
130C4⋊(C3⋊D3)C2×C3×C413yes—
131C4×C3⋊D3C2×C3×C413yes—
132C4×C3⋊D3C2×C3×C413yes—
133C4×C3⋊D3C23×C311yescharacteristic
134C4×C3⋊D3C23×C313yes—
135#634C2×C3×C413yes—
136C3×#67—324no—
137C2×C3×Dic3—312no—
138C3×D12—312no—
139C3×C4×D3—312no—
140C3×C4×D3—312no—
141C3×C4×D3—34no—
142C3×C4×D3—312no—
143C3×Dic6—312no—
144C32×C8—24no—
145C32×C8—212no—
146C32×C8C22×D3112yes—
147C32×C8C22×D314yes—
148C32×C8C23×C311yescharacteristic
149C32×C8C23×C313yes—
150C3⋊(C3⋊C8)C23×C311yescharacteristic
151C3⋊(C3⋊C8)C23×C313yes—
152C3×C3⋊C8—34no—
153C3×C3⋊C8—312no—
154C2×#131—91noSylow
155C2×C33—23no—
156C2×C33C23×C411yescharacteristic
157C3×C3⋊D3—23no—
158C3×C3⋊D3#13112yesdirect product factor
159C3×#41—24no—
160C3×#41—91no—
161C3×#41—93no—
162C3×#41—93no—
163C3×#41C2×C3×D314yes—
164C3×#41C2×C3⋊D311yescharacteristic
165C2×C3×Q8—91no—
166C2×C3×D4—93no—
167C22×C3×C4—93no—
168#239—34no—
169D3×Q8—34no—
170#237—312no—
171D3×D4—312no—
172#235—312no—
173C2×C4×D3—312no—
174C3×M16—212no—
175C3×M16—93no—
176C3×M16—96no—
177C3×M16C2×C3×D3112yescomplemented
178C3×M16C2×C3⋊D313yescomplemented
179C2×C3×C8—212no—
180C2×C3×C8—93no—
181C2×C3×C8—93no—
182C2×C3×C8—93no—
183C2×C3×C8—91no—
184C2×C3×C8—93no—
185C2×C3×C8C2×C3×D3112yescomplemented
186C2×C3×C8C2×C3⋊D313yescomplemented
187#208—312no—
188C2×C3⋊C8—312no—
189ΓL(1,𝔽25)—324no—
190D3×C8—34no—
191D3×C8—312no—
192D3×C8—312no—
193D3×C8—312no—
194C22×C32—212no—
195C22×C32C2×C4×D3112yes—
196C22×C32C22×C3×C413yes—
197C2×C3⋊D3—26no—
198C2×C3⋊D3C22×C3×C413yes—
199C2×C3⋊D3C22×C3×C411yescharacteristic
200C2×C3×D3—34no—
201C2×C3×D3—312no—
202C2×C3×D3—624no—
203C32×C4—24no—
204C32×C4—212no—
205C32×C4C2×C4×D3112yes—
206C32×C4C22×C3×C413yes—
207C32×C4C23×D314yes—
208C32×C4C24×C311yescharacteristic
209C3⋊Dic3C22×C3×C413yes—
210C3⋊Dic3C22×C3×C411yescharacteristic
211C3×Dic3—312no—
212C3×Dic3—34no—
213C2×#41—91no—
214#131—91no—
215#131—93no—
216#131—93no—
217#131C3×C3⋊D311yescharacteristic, direct product factor
218C2×M16—93no—
219C22×C8—93no—
220C33C2×#13111yescharacteristic, complemented, Sylow
221C23×C3—93no—
222C22×D3—312no—
223C3×Q8—24no—
224C3×Q8—93no—
225C3×Q8C3×C4×D314yes—
226C3×Q8C4×C3⋊D311yescharacteristic
227C3×D4—212no—
228C3×D4—96no—
229C3×D4—93no—
230C3×D4C3×C4×D3112yes—
231C3×D4C4×C3⋊D313yes—
232C2×C3×C4—212no—
233C2×C3×C4—93no—
234C2×C3×C4—91no—
235C2×C3×C4—93no—
236C2×C3×C4—93no—
237C2×C3×C4—93no—
238C2×C3×C4C22×C3×D3112yes—
239C2×C3×C4C22×C3⋊D313yes—
240#67—324no—
241C2×Dic3—312no—
242D12—312no—
243C4×D3—34no—
244C4×D3—312no—
245C4×D3—312no—
246C4×D3—312no—
247Dic6—312no—
248C3×C8—24no—
249C3×C8—212no—
250C3×C8—91no—
251C3×C8—93no—
252C3×C8C22×C3×D314yes—
253C3×C8C22×C3×D3112yes—
254C3×C8C22×C3⋊D313yes—
255C3×C8C22×C3⋊D311yescharacteristic
256C3⋊C8—34no—
257C3⋊C8—312no—
258C2×C32—23no—
259C2×C32—24no—
260C2×C32—212no—
261C2×C32—412no—
262C2×C32C22×C4×D314yes—
263C2×C32C23×C3×C411yescharacteristic
264C3⋊D3—23no—
265C3⋊D3C3×#13112yesdirect product factor
266C3×D3—38no—
267C3×D3—612no—
268#41—91no—
269#41—93no—
270#41—93no—
271#41C2×C3×C3⋊D311yescharacteristic
272C2×Q8—91no—
273C2×D4—93no—
274C22×C4—93no—
275M16—93no—
276M16—96no—
277M16C2×C3×C3⋊D313yescomplemented
278C2×C8—93no—
279C2×C8—91no—
280C2×C8—93no—
281C2×C8—93no—
282C2×C8—93no—
283C2×C8C2×C3×C3⋊D313yescomplemented
284C22×C3—212no—
285C22×C3—91no—
286C22×C3—93no—
287C22×C3—186no—
288C22×C3C2×C3×C4×D3112yes—
289C22×C3C2×C4×C3⋊D313yes—
290C2×D3—34no—
291C2×D3—312no—
292C2×D3—624no—
293C3×C4—24no—
294C3×C4—212no—
295C3×C4—91no—
296C3×C4—93no—
297C3×C4C2×C3×C4×D3112yes—
298C3×C4C2×C4×C3⋊D313yes—
299C3×C4C23×C3×D314yes—
300C3×C4C23×C3⋊D311yescharacteristic
301Dic3—312no—
302Dic3—34no—
303C32—24no—
304C32D3×#13114yescomplemented
305C32C2×C3×#13111yescharacteristic, complemented
306C23—93no—
307Q8—93no—
308Q8C3×C4×C3⋊D311yescharacteristic
309D4—96no—
310D4—93no—
311D4C3×C4×C3⋊D313yes—
312C2×C4—93no—
313C2×C4—91no—
314C2×C4—93no—
315C2×C4—93no—
316C2×C4—93no—
317C2×C4C22×C3×C3⋊D313yes—
318C8—91no—
319C8—93no—
320C8C22×C3×C3⋊D311yescharacteristic
321C8C22×C3×C3⋊D313yes—
322C2×C3—212no—
323C2×C3—23no—
324C2×C3—24no—
325C2×C3—412no—
326C2×C3—92no—
327C2×C3—183no—
328C2×C3C22×C3×C4×D314yes—
329C2×C3C22×C4×C3⋊D311yescharacteristic
330D3—38no—
331D3—612no—
332C22—91no—
333C22—93no—
334C22—186no—
335C22C2×C3×C4×C3⋊D313yes—
336C4—91no—
337C4—93no—
338C4C2×C3×C4×C3⋊D313yes—
339C4C23×C3×C3⋊D311yescharacteristic
340C3—24no—
341C3C3×D3×#13114yescomplemented
342C3C3⋊D3×#13111yescharacteristic, direct product factor
343C2—23no—
344C2—92no—
345C2—183no—
346C2C22×C3×C4×C3⋊D311yescharacteristic
subgroupG/H
centerC3×C8C22×C3⋊D3
commutatorC2×C32C23×C3×C4
FrattiniC4C23×C3×C3⋊D3
FittingC33×#131C2
socleC2×C33C23×C4
prime radicalC3×#41×C3⋊D3C2
cosocleC32×C4C24×C3
hypercenterC3×#131C3⋊D3
hypocenterC32C2×C3×#131
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C3×C8 ◁ C3×#131

Complements

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

NH ≅ G/Ntype
2. C33×#131C2semidirect
3. C3×#64157C2semidirect
4. C3×#64116C2semidirect
5. C3×M16×C3⋊D3C2semidirect
6. C3×#64110C2semidirect
7. C2×C3×C8×C3⋊D3C2semidirect
8. C3⋊D3×#131C3direct (N × H)
17. C33×M16C22semidirect
18. C2×C33×C8C22semidirect
19. C3×#3748C22semidirect
20. C2×C3×C3⋊(C3⋊C8)C22semidirect
21. C3×#3743C22semidirect
24. C3×C8×C3⋊D3C22semidirect
25. C3×C8×C3⋊D3C22semidirect
29. C32×#131D3semidirect
30. C32×#131C2×C3semidirect
31. #64157C2×C3semidirect
32. #64116C2×C3semidirect
33. M16×C3⋊D3C2×C3semidirect
34. #64110C2×C3semidirect
35. C2×C8×C3⋊D3C2×C3semidirect
75. C32×M16C2×D3semidirect
76. C32×M16C22×C3semidirect
78. C2×C32×C8C2×D3semidirect
79. C2×C32×C8C22×C3semidirect
80. #3748C22×C3semidirect
81. C2×C3⋊(C3⋊C8)C22×C3semidirect
82. #3743C22×C3semidirect
85. C8×C3⋊D3C22×C3semidirect
86. C8×C3⋊D3C22×C3semidirect
108. C3×#131C3×D3semidirect
109. C3×#131C3⋊D3direct (N × H)
158. C3×C3⋊D3#131direct (N × H)
177. C3×M16C2×C3×D3semidirect
178. C3×M16C2×C3⋊D3semidirect
185. C2×C3×C8C2×C3×D3semidirect
186. C2×C3×C8C2×C3⋊D3semidirect
217. #131C3×C3⋊D3direct (N × H)
220. C33C2×#131semidirect
265. C3⋊D3C3×#131direct (N × H)
277. M16C2×C3×C3⋊D3semidirect
283. C2×C8C2×C3×C3⋊D3semidirect
304. C32D3×#131semidirect
305. C32C2×C3×#131semidirect
341. C3C3×D3×#131semidirect
342. C3C3⋊D3×#131direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#41×C3⋊D32
C33×#1312
C3×#641572
C3×#641162
C3×M16×C3⋊D32
C3×#641102
C2×C3×C8×C3⋊D32
C3⋊D3×#1313
C3×D3×#1313