Finite Group Databasebeta

← C3×#10589919 #13153647

Subgroups — C3×#10589919 #13153647

Order 1728.

1026 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×Q8×C9C211yesmaximal, characteristic, complemented
2C2×C3×#3701C211yesmaximal, characteristic, complemented
3C3×#63509C213yesmaximal, complemented
4C3×#63508C213yesmaximal, complemented
5C3×Q8×Dic9C211yesmaximal, characteristic, complemented
6C3×#63492C213yesmaximal, complemented
7C3×C4×#611C213yesmaximal, complemented
8C3×#5817—31nomaximal
9#10589919C311yesmaximal, characteristic, direct product factor
10C2×C3×Q8×C9—24no—
11C2×C3×Q8×C9C2213yes—
12C2×C3×Q8×C9C2211yescharacteristic
13C22×C3×C4×C9C2213yes—
14C2×C3×#611C2213yes—
15C3×#3701—24no—
16C2×C3×D36C2213yes—
17C2×C3×C4×D9C2213yes—
18C3×#3676C2211yescharacteristic
19C3×(C2×D9)⋊C4C2213yes—
20C3×(C2×D9)⋊C4C2216yes—
21C3×C4⋊Dic9C2213yes—
22C3×Dic9⋊C4C2213yes—
23C3×C4×Dic9C2213yes—
24C22×C32×Q8D311yescharacteristic
25C2×C3×#239—31no—
26C3×#947—33no—
27C3×#946—33no—
28C3×Q8×Dic3—31no—
29C3×#930—33no—
30C3×C4×#67—33no—
31C22×Q8×C9—21no—
32C22×Q8×C9C2×C311yescharacteristic, complemented
33C2×#3701C2×C311yescharacteristic, complemented
34#63509C2×C313yescomplemented
35#63508C2×C313yescomplemented
36Q8×Dic9C2×C311yescharacteristic, complemented
37#63492C2×C313yescomplemented
38C4×#611C2×C313yescomplemented
39C23×C3×C9C2311yescharacteristic
40C22×C3×D9C2313yes—
41C3×Q8×C9—26no—
42C3×Q8×C9D414yescomplemented
43C2×C3×C4×C9—26no—
44C2×C3×C4×C9C2313yes—
45C2×C3×C4×C9C2313yes—
46C3×#611—26no—
47C2×C3×Dic9C2311yescharacteristic
48C2×C3×Dic9C2313yes—
49C2×C3×Dic9C2311yescharacteristic
50C3×D36—26no—
51C3×C4×D9—26no—
52C3×#548—91no—
53#5817—31no—
54C2×C32×Q8—24no—
55C2×C32×Q8C2×D313yes—
56C2×C32×Q8C2×D311yescharacteristic
57C22×C32×C4C2×D313yes—
58C2×C3×#67—33no—
59C3×#239—64no—
60C2×C3×D12—33no—
61C2×C3×C4×D3—33no—
62C3×#217—31no—
63C3×(C2×D3)⋊C4—33no—
64C3×(C2×D3)⋊C4—36no—
65C3×C4⋊Dic3—33no—
66C3×Dic3⋊C4—33no—
67C3×C4×Dic3—33no—
68C2×Q8×C9—28no—
69C2×Q8×C9—23no—
70C2×Q8×C9—21no—
71C2×Q8×C9—24no—
72C2×Q8×C9C22×C313yes—
73C2×Q8×C9C22×C311yescharacteristic
74C22×C4×C9—23no—
75C22×C4×C9C22×C313yes—
76C2×#611C22×C313yes—
77#3701—24no—
78C2×D36C22×C313yes—
79C2×C4×D9C22×C313yes—
80#3676C22×C311yescharacteristic
81(C2×D9)⋊C4C22×C313yes—
82(C2×D9)⋊C4C22×C316yes—
83C4⋊Dic9C22×C313yes—
84Dic9⋊C4C22×C313yes—
85C4×Dic9C22×C313yes—
86C22×C3×C9—21no—
87C22×C3×C9—21no—
88C22×C3×C9#4112yescomplemented
89C22×C3×C9C2411yescharacteristic
90C2×C3×D9—23no—
91C2×C3×D9—23no—
92C2×C3×D9—23no—
93C3×C4×C9—23no—
94C3×C4×C9C2×D416yes—
95C3×Dic9—21no—
96C3×Dic9—23no—
97C3×Dic9—21no—
98C2×C3×#41—91no—
99C22×C3×Q8—21no—
100C22×C3×Q8D911yescharacteristic, complemented
101C22×C3×Q8C3×D311yescharacteristic
102C2×#239—31no—
103C3×#124—93no—
104C3×C22⋊Q8—93no—
105C3×#121—93no—
106C3×C4×Q8—91no—
107C3×C4×D4—93no—
108#947—33no—
109#946—33no—
110Q8×Dic3—31no—
111#930—33no—
112C4×#67—33no—
113C23×C32C22×D311yescharacteristic
114C22×C3×D3—33no—
115C32×Q8—26no—
116C32×Q8#6714yes—
117C2×C32×C4—26no—
118C2×C32×C4C22×D313yes—
119C2×C32×C4C22×D313yes—
120C3×#67—66no—
121C2×C3×Dic3—31no—
122C2×C3×Dic3—33no—
123C2×C3×Dic3—31no—
124C3×D12—66no—
125C3×C4×D3—66no—
126C23×C9—21no—
127C23×C9C23×C311yescharacteristic
128C22×D9C23×C313yes—
129Q8×C9—212no—
130Q8×C9—26no—
131Q8×C9—24no—
132Q8×C9C3×D414yescomplemented
133C2×C4×C9—212no—
134C2×C4×C9—26no—
135C2×C4×C9—23no—
136C2×C4×C9—23no—
137C2×C4×C9C23×C313yes—
138C2×C4×C9C23×C313yes—
139#611—26no—
140C2×Dic9C23×C311yescharacteristic
141C2×Dic9C23×C313yes—
142C2×Dic9C23×C311yescharacteristic
143D36—26no—
144C4×D9—26no—
145#548—91noSylow
146C2×C3×C9—22no—
147C2×C3×C9C22×D411yescharacteristic
148C2×C3×C9C2×#4111yescharacteristic
149C2×C3×C92−1+411yescharacteristic
150C3×D9—43no—
151C3×#41—184no—
152C2×C3×Q8—28no—
153C2×C3×Q8—24no—
154C2×C3×Q8—24no—
155C2×C3×Q8—23no—
156C2×C3×Q8—21no—
157C2×C3×Q8C2×D913yes—
158C2×C3×Q8C2×D911yescharacteristic
159C2×C3×Q8C2×C3×D313yes—
160C2×C3×Q8C2×C3×D311yescharacteristic
161C2×C3×D4—93no—
162C2×C3×D4—93no—
163C22×C3×C4—23no—
164C22×C3×C4—93no—
165C22×C3×C4C2×D913yes—
166C22×C3×C4C2×C3×D313yes—
167C2×#67—33no—
168#239—64no—
169C2×D12—33no—
170C2×C4×D3—33no—
171C3×C4⋊C4—93no—
172C3×C4⋊C4—93no—
173C3×C22⋊C4—91no—
174C3×C22⋊C4—93no—
175C3×C22⋊C4—96no—
176C3×C42—93no—
177#217—31no—
178(C2×D3)⋊C4—33no—
179(C2×D3)⋊C4—36no—
180C4⋊Dic3—33no—
181Dic3⋊C4—33no—
182C4×Dic3—33no—
183C22×C32—21no—
184C22×C32—21no—
185C22×C32#23912yes—
186C22×C32C23×D311yescharacteristic
187C2×C3×D3—63no—
188C2×C3×D3—63no—
189C2×C3×D3—63no—
190C32×C4—23no—
191C32×C4C2×#6716yes—
192C3×Dic3—63no—
193C3×Dic3—61no—
194C3×Dic3—61no—
195C22×C9—22no—
196C22×C9—22no—
197C22×C9—21no—
198C22×C9—21no—
199C22×C9—21no—
200C22×C9—22no—
201C22×C9C3×#4112yescomplemented
202C22×C9C24×C311yescharacteristic
203C2×D9—23no—
204C2×D9—23no—
205C2×D9—23no—
206C4×C9—26no—
207C4×C9—23no—
208C4×C9—26no—
209C4×C9C2×C3×D416yes—
210Dic9—21no—
211Dic9—23no—
212Dic9—21no—
213C2×#41—91no—
214C22×Q8C3×D911yescharacteristic, complemented
215#124—93no—
216C22⋊Q8—93no—
217#121—93no—
218C4×Q8—91no—
219C4×D4—93no—
220C3×C9#54811yescharacteristic, complemented, Sylow
221C23×C3—21no—
222C23×C3—93no—
223C23×C3C22×D911yescharacteristic
224C23×C3C22×C3×D311yescharacteristic
225C22×D3—33no—
226C3×Q8—212no—
227C3×Q8—26no—
228C3×Q8—26no—
229C3×Q8—24no—
230C3×Q8#61114yescomplemented
231C3×Q8C3×#6714yes—
232C3×D4—186no—
233C3×D4—186no—
234C2×C3×C4—212no—
235C2×C3×C4—26no—
236C2×C3×C4—26no—
237C2×C3×C4—23no—
238C2×C3×C4—23no—
239C2×C3×C4—91no—
240C2×C3×C4—93no—
241C2×C3×C4—91no—
242C2×C3×C4—186no—
243C2×C3×C4C22×D913yes—
244C2×C3×C4C22×D913yes—
245C2×C3×C4C22×C3×D313yes—
246C2×C3×C4C22×C3×D313yes—
247#67—66no—
248C2×Dic3—31no—
249C2×Dic3—33no—
250C2×Dic3—31no—
251D12—66no—
252C4×D3—66no—
253C2×C32—22no—
254C2×C32C2×#23911yescharacteristic
255C2×C32#100711yescharacteristic
256C2×C32C22×#6711yescharacteristic
257C3×D3—123no—
258C2×C9—24no—
259C2×C9—22no—
260C2×C9—21no—
261C2×C9—21no—
262C2×C9—21no—
263C2×C9C22×C3×D411yescharacteristic
264C2×C9C2×C3×#4111yescharacteristic
265C2×C9C3×2−1+411yescharacteristic
266D9—43no—
267#41—184no—
268C2×Q8—24no—
269C2×Q8C2×C3×D913yes—
270C2×Q8C2×C3×D911yescharacteristic
271C2×D4—93no—
272C2×D4—93no—
273C22×C4—93no—
274C22×C4C2×C3×D913yes—
275C4⋊C4—93no—
276C4⋊C4—93no—
277C22⋊C4—91no—
278C22⋊C4—93no—
279C22⋊C4—96no—
280C42—93no—
281C22×C3—22no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—22no—
289C22×C3—183no—
290C22×C3—183no—
291C22×C3—183no—
292C22×C3#370112yescomplemented
293C22×C3C23×D911yescharacteristic
294C22×C3C3×#23912yes—
295C22×C3C23×C3×D311yescharacteristic
296C2×D3—63no—
297C2×D3—63no—
298C2×D3—63no—
299C3×C4—26no—
300C3×C4—23no—
301C3×C4—23no—
302C3×C4—26no—
303C3×C4—183no—
304C3×C4—181no—
305C3×C4—181no—
306C3×C4C2×#61116yes—
307C3×C4C2×C3×#6716yes—
308Dic3—63no—
309Dic3—61no—
310Dic3—61no—
311C32#581711yescharacteristic
312C9—21no—
313C9C3×#54811yescharacteristic, complemented
314C23—93no—
315C23C22×C3×D911yescharacteristic
316Q8—26no—
317Q8C3×#61114yescomplemented
318D4—186no—
319D4—186no—
320C2×C4—26no—
321C2×C4—91no—
322C2×C4—93no—
323C2×C4—91no—
324C2×C4—186no—
325C2×C4C22×C3×D913yes—
326C2×C4C22×C3×D913yes—
327C2×C3—24no—
328C2×C3—22no—
329C2×C3—22no—
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—363no—
334C2×C3C2×#370111yescharacteristic
335C2×C3#6371311yescharacteristic
336C2×C3C22×#61111yescharacteristic
337C2×C3C2×C3×#23911yescharacteristic
338C2×C3C3×#100711yescharacteristic
339C2×C3C22×C3×#6711yescharacteristic
340D3—123no—
341C22—21no—
342C22—21no—
343C22—183no—
344C22—183no—
345C22—183no—
346C22C3×#370112yescomplemented
347C22C23×C3×D911yescharacteristic
348C4—23no—
349C4—183no—
350C4—181no—
351C4—181no—
352C4C2×C3×#61116yes—
353C3—21no—
354C3#1058991911yescharacteristic, direct product factor
355C3C3×#581711yescharacteristic
356C2—22no—
357C2—363no—
358C2C2×C3×#370111yescharacteristic
359C2C3×#6371311yescharacteristic
360C2C22×C3×#61111yescharacteristic
subgroupG/H
centerC22×C3C23×D9
commutator, cosocleC22×C9C24×C3
FrattiniC22×C3C23×C3×D3
FittingC22×C3×Q8×C9C2
socleC22×C32C23×D3
hypercenterC22×C3×Q8D9
hypocenterC9C3×#548
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22×C3 ◁ C22×C3×Q8

Complements

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

NH ≅ G/Ntype
1. C22×C3×Q8×C9C2semidirect
2. C2×C3×#3701C2semidirect
3. C3×#63509C2semidirect
4. C3×#63508C2semidirect
5. C3×Q8×Dic9C2semidirect
6. C3×#63492C2semidirect
7. C3×C4×#611C2semidirect
9. #10589919C3direct (N × H)
32. C22×Q8×C9C2×C3semidirect
33. C2×#3701C2×C3semidirect
34. #63509C2×C3semidirect
35. #63508C2×C3semidirect
36. Q8×Dic9C2×C3semidirect
37. #63492C2×C3semidirect
38. C4×#611C2×C3semidirect
42. C3×Q8×C9D4semidirect
88. C22×C3×C9#41semidirect
100. C22×C3×Q8D9semidirect
132. Q8×C9C3×D4semidirect
201. C22×C9C3×#41semidirect
214. C22×Q8C3×D9semidirect
220. C3×C9#548semidirect
230. C3×Q8#611semidirect
292. C22×C3#3701semidirect
313. C9C3×#548semidirect
317. Q8C3×#611semidirect
346. C22C3×#3701semidirect
354. C3#10589919direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×Q8×C92
C2×C3×#37012
C3×#635092
C3×#635082
C3×Q8×Dic92
C3×#634922
C3×C4×#6112
C3×#58173
#105899193