Finite Group Databasebeta

← C3×#10596309 #13179589

Subgroups — C3×#10596309 #13179589

Order 1728.

2072 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×2−1+4C211yesmaximal, characteristic, complemented
2C3×#64367C211yesmaximal, characteristic, complemented
3C3×#64160C213yesmaximal, complemented
4C3×#64159C213yesmaximal, complemented
5C3×#64157C211yesmaximal, characteristic, complemented
6C2×C3×#3756C213yesmaximal, complemented
7C3×#64151C213yesmaximal, complemented
8#10596309C311yesmaximal, characteristic, direct product factor
9C3×#5838—34nomaximal
10C33×#41—23no—
11C33×#41C2213yes—
12C33×#41C2211yescharacteristic
13C2×C33×Q8—21no—
14C2×C33×Q8C2213yes—
15C3×Q8×C3⋊D3—21no—
16C3×#3830—23no—
17C3×#3828C2213yes—
18C2×C3×#634C2213yes—
19C3×#3756C2216yes—
20C3×#3756C2213yes—
21C3×#3755C2213yes—
22C3×#3754C2213yes—
23C3×#3753C2211yescharacteristic
24C3×#3748C2213yes—
25C2×C3×C3⋊(C3⋊C8)C2213yes—
26C32×2−1+4—24no—
27C32×2−1+4D314yescomplemented
28C32×2−1+4C2×C311yescharacteristic, complemented
29#64367C2×C311yescharacteristic, complemented
30C3×#1010—34no—
31#64160C2×C313yescomplemented
32#64159C2×C313yescomplemented
33#64157C2×C311yescharacteristic, complemented
34C2×#3756C2×C313yescomplemented
35#64151C2×C313yescomplemented
36C3×#951—312no—
37C3×#950—312no—
38C3×#948—34no—
39C2×C3×#216—312no—
40C3×#942—312no—
41C33×Q8—23no—
42C33×Q8D411yescharacteristic
43C33×Q8C2313yes—
44C33×D4—23no—
45C33×D4D413yes—
46C33×D4C2311yescharacteristic
47C2×C33×C4—23no—
48C2×C33×C4—23no—
49C2×C33×C4C2313yes—
50C3×#638—23no—
51C2×C3×C3⋊Dic3—23no—
52C3×C4⋊(C3⋊D3)C2311yescharacteristic
53C3×C4×C3⋊D3—23no—
54C3×#634—23no—
55C3×#634C2313yes—
56C3×C3⋊(C3⋊C8)C2311yescharacteristic
57C3×C3⋊(C3⋊C8)C2313yes—
58C3×#578—91no—
59#5838—34no—
60C32×#41—224no—
61C32×#41—212no—
62C32×#41—24no—
63C32×#41—212no—
64C32×#41—23no—
65C32×#41C2×D3112yes—
66C32×#41C2×D314yes—
67C32×#41C22×C313yes—
68C32×#41C22×C311yescharacteristic
69C2×C32×Q8—28no—
70C2×C32×Q8—212no—
71C2×C32×Q8—24no—
72C2×C32×Q8—21no—
73C2×C32×Q8C2×D3112yes—
74C2×C32×Q8C22×C313yes—
75Q8×C3⋊D3—21no—
76#3830—23no—
77#3828C22×C313yes—
78C2×#634C22×C313yes—
79C3×D3×Q8—64no—
80C3×#237—612no—
81C3×#235—312no—
82C2×C3×Dic6—312no—
83#3756C22×C316yes—
84#3756C22×C313yes—
85#3755C22×C313yes—
86#3754C22×C313yes—
87#3753C22×C311yescharacteristic
88#3748C22×C313yes—
89C2×C3⋊(C3⋊C8)C22×C313yes—
90C3×#216—324no—
91C3×#216—312no—
92C3×#215—312no—
93C3×#214—312no—
94C3×#213—34no—
95C3×#208—312no—
96C2×C3×C3⋊C8—312no—
97C22×C33—21no—
98C22×C33C2×D413yes—
99C2×C3×C3⋊D3—21no—
100C33×C4—23no—
101C33×C4C2×D413yes—
102C33×C4C2411yescharacteristic
103C3×C3⋊Dic3—23no—
104C3×2−1+4—24no—
105C3×2−1+4—91no—
106C3×2−1+4C3×D314yescomplemented
107C3×2−1+4C3⋊D311yescharacteristic, complemented
108#1010—34no—
109C3×#137—93no—
110C3×#137—93no—
111C3×#135—93no—
112C2×C3×Q16—93no—
113C3×#131—91no—
114#951—312no—
115#950—312no—
116#948—34no—
117C2×#216—312no—
118#942—312no—
119C32×Q8—224no—
120C32×Q8—212no—
121C32×Q8—23no—
122C32×Q8—24no—
123C32×Q8—212no—
124C32×Q8#6714yes—
125C32×Q8C3×D411yescharacteristic
126C32×Q8C22×D3112yes—
127C32×Q8C23×C313yes—
128C32×D4—224no—
129C32×D4—212no—
130C32×D4—23no—
131C32×D4—212no—
132C32×D4—24no—
133C32×D4#67112yes—
134C32×D4C3×D413yes—
135C32×D4C22×D314yes—
136C32×D4C23×C311yescharacteristic
137C2×C32×C4—224no—
138C2×C32×C4—224no—
139C2×C32×C4—212no—
140C2×C32×C4—23no—
141C2×C32×C4—212no—
142C2×C32×C4—23no—
143C2×C32×C4—212no—
144C2×C32×C4C22×D3112yes—
145C2×C32×C4C23×C313yes—
146#638—23no—
147C2×C3⋊Dic3—23no—
148C4⋊(C3⋊D3)C23×C311yescharacteristic
149C4×C3⋊D3—23no—
150#634—23no—
151#634C23×C313yes—
152C3×#67—612no—
153C2×C3×Dic3—612no—
154C3×D12—34no—
155C3×C4×D3—612no—
156C3×Dic6—312no—
157C3×Dic6—612no—
158C3⋊(C3⋊C8)C23×C311yescharacteristic
159C3⋊(C3⋊C8)C23×C313yes—
160C3×C3⋊C8—34no—
161C3×C3⋊C8—312no—
162#578—91noSylow
163C2×C33—23no—
164C2×C33—41no—
165C2×C33C22×D411yescharacteristic
166C3×C3⋊D3—41no—
167C3×#41—224no—
168C3×#41—212no—
169C3×#41—23no—
170C3×#41—212no—
171C3×#41—24no—
172C3×#41—93no—
173C3×#41—183no—
174C3×#41C2×C3×D3112yes—
175C3×#41C2×C3×D314yes—
176C3×#41C2×C3⋊D313yes—
177C3×#41C2×C3⋊D311yescharacteristic
178C2×C3×Q8—28no—
179C2×C3×Q8—24no—
180C2×C3×Q8—21no—
181C2×C3×Q8—212no—
182C2×C3×Q8—93no—
183C2×C3×Q8—181no—
184C2×C3×Q8C2×C3×D3112yes—
185C2×C3×Q8C2×C3⋊D313yes—
186D3×Q8—64no—
187#237—612no—
188#235—312no—
189C2×Dic6—312no—
190C3×Q16—93no—
191C3×Q16—96no—
192C3×SD16—93no—
193C3×SD16—93no—
194C3×D8—91no—
195C3×M16—93no—
196C2×C3×C8—93no—
197#216—324no—
198#216—312no—
199#215—312no—
200#214—312no—
201#213—34no—
202#208—312no—
203C2×C3⋊C8—312no—
204C22×C32—28no—
205C22×C32—24no—
206C22×C32—21no—
207C22×C32—212no—
208C22×C32C2×#67112yes—
209C22×C32C2×C3×D413yes—
210C2×C3⋊D3—21no—
211C2×C3×D3—64no—
212C32×C4—224no—
213C32×C4—212no—
214C32×C4—23no—
215C32×C4—24no—
216C32×C4—212no—
217C32×C4C2×#67112yes—
218C32×C4C2×C3×D413yes—
219C32×C4C23×D314yes—
220C32×C4C24×C311yescharacteristic
221C3⋊Dic3—23no—
222C3×Dic3—612no—
2232−1+4—91no—
2242−1+4C3×C3⋊D311yescharacteristic, complemented
225#137—93no—
226#137—93no—
227#135—93no—
228C2×Q16—93no—
229#131—91no—
230C33#57811yescharacteristic, complemented, Sylow
231C3×Q8—224no—
232C3×Q8—212no—
233C3×Q8—23no—
234C3×Q8—24no—
235C3×Q8—212no—
236C3×Q8—93no—
237C3×Q8—183no—
238C3×Q8C3×#6714yes—
239C3×Q8#63811yescharacteristic
240C3×Q8C22×C3×D3112yes—
241C3×Q8C22×C3⋊D313yes—
242C3×D4—224no—
243C3×D4—212no—
244C3×D4—23no—
245C3×D4—212no—
246C3×D4—24no—
247C3×D4—91no—
248C3×D4—183no—
249C3×D4C3×#67112yes—
250C3×D4#63813yes—
251C3×D4C22×C3×D314yes—
252C3×D4C22×C3⋊D311yescharacteristic
253C2×C3×C4—224no—
254C2×C3×C4—224no—
255C2×C3×C4—212no—
256C2×C3×C4—23no—
257C2×C3×C4—212no—
258C2×C3×C4—23no—
259C2×C3×C4—212no—
260C2×C3×C4—183no—
261C2×C3×C4—183no—
262C2×C3×C4C22×C3×D3112yes—
263C2×C3×C4C22×C3⋊D313yes—
264#67—612no—
265C2×Dic3—612no—
266D12—34no—
267C4×D3—612no—
268Dic6—312no—
269Dic6—612no—
270C3×C8—91no—
271C3×C8—93no—
272C3⋊C8—34no—
273C3⋊C8—312no—
274C2×C32—23no—
275C2×C32—24no—
276C2×C32—212no—
277C2×C32—412no—
278C2×C32—48no—
279C2×C32—41no—
280C2×C32—44no—
281C2×C32C22×#6714yes—
282C2×C32C22×C3×D411yescharacteristic
283C3⋊D3—41no—
284C3×D3—124no—
285#41—23no—
286#41—93no—
287#41—183no—
288#41C2×C3×C3⋊D313yes—
289#41C2×C3×C3⋊D311yescharacteristic
290C2×Q8—21no—
291C2×Q8—93no—
292C2×Q8—181no—
293C2×Q8C2×C3×C3⋊D313yes—
294Q16—93no—
295Q16—96no—
296SD16—93no—
297SD16—93no—
298D8—91no—
299M16—93no—
300C2×C8—93no—
301C22×C3—28no—
302C22×C3—24no—
303C22×C3—21no—
304C22×C3—212no—
305C22×C3—181no—
306C22×C3C2×C3×#67112yes—
307C22×C3C2×#63813yes—
308C2×D3—64no—
309C3×C4—224no—
310C3×C4—212no—
311C3×C4—23no—
312C3×C4—24no—
313C3×C4—212no—
314C3×C4—183no—
315C3×C4C2×C3×#67112yes—
316C3×C4C2×#63813yes—
317C3×C4C23×C3×D314yes—
318C3×C4C23×C3⋊D311yescharacteristic
319Dic3—612no—
320C32—24no—
321C32#583814yescomplemented
322C32C3×#57811yescharacteristic, complemented
323Q8—23no—
324Q8—93no—
325Q8—183no—
326Q8C3×#63811yescharacteristic
327Q8C22×C3×C3⋊D313yes—
328D4—23no—
329D4—91no—
330D4—183no—
331D4C3×#63813yes—
332D4C22×C3×C3⋊D311yescharacteristic
333C2×C4—23no—
334C2×C4—23no—
335C2×C4—183no—
336C2×C4—183no—
337C2×C4C22×C3×C3⋊D313yes—
338C8—91no—
339C8—93no—
340C2×C3—212no—
341C2×C3—23no—
342C2×C3—24no—
343C2×C3—412no—
344C2×C3—48no—
345C2×C3—44no—
346C2×C3—41no—
347C2×C3—361no—
348C2×C3C22×C3×#6714yes—
349C2×C3C22×#63811yescharacteristic
350D3—124no—
351C22—21no—
352C22—181no—
353C22C2×C3×#63813yes—
354C4—23no—
355C4—183no—
356C4C2×C3×#63813yes—
357C4C23×C3×C3⋊D311yescharacteristic
358C3—24no—
359C3C3×#583814yescomplemented
360C3#1059630911yescharacteristic, direct product factor
361C2—23no—
362C2—41no—
363C2—361no—
364C2C22×C3×#63811yescharacteristic
subgroupG/H
centerC2×C3C22×#638
commutator, cosocleC32×C4C24×C3
FrattiniC4C23×C3×C3⋊D3
FittingC33×2−1+4C2
socleC2×C33C22×D4
hypercenterC3×2−1+4C3⋊D3
hypocenterC32C3×#578
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×#41 ◁ C3×2−1+4

Complements

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

NH ≅ G/Ntype
1. C33×2−1+4C2semidirect
2. C3×#64367C2semidirect
3. C3×#64160C2semidirect
4. C3×#64159C2semidirect
5. C3×#64157C2semidirect
6. C2×C3×#3756C2semidirect
7. C3×#64151C2semidirect
8. #10596309C3direct (N × H)
27. C32×2−1+4D3semidirect
28. C32×2−1+4C2×C3semidirect
29. #64367C2×C3semidirect
31. #64160C2×C3semidirect
32. #64159C2×C3semidirect
33. #64157C2×C3semidirect
34. C2×#3756C2×C3semidirect
35. #64151C2×C3semidirect
106. C3×2−1+4C3×D3semidirect
107. C3×2−1+4C3⋊D3semidirect
224. 2−1+4C3×C3⋊D3semidirect
230. C33#578semidirect
321. C32#5838semidirect
322. C32C3×#578semidirect
359. C3C3×#5838semidirect
360. C3#10596309direct (N × H)

Maximal subgroups

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

subgroupindex
C33×2−1+42
C3×#643672
C3×#641602
C3×#641592
C3×#641572
C2×C3×#37562
C3×#641512
#105963093
C3×#58383