Finite Group Databasebeta

← C3×#10592904 #13163243

Subgroups — C3×#10592904 #13163243

Order 1728.

1092 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64367C211yesmaximal, characteristic
2C33×#137C211yesmaximal, characteristic, complemented
3C3×#64153C211yesmaximal, characteristic, complemented
4C3×#64151C211yesmaximal, characteristic, complemented
5C3×#63664C211yesmaximal, characteristic, complemented
6C3×#63652C211yesmaximal, characteristic, complemented
7C3×#63651C211yesmaximal, characteristic, complemented
8#10592904C311yesmaximal, characteristic, direct product factor
9C3×#5205—34nomaximal
10C33×#41C2211yescharacteristic
11C2×C33×Q8C2211yescharacteristic
12C3×Q8×C3⋊D3—21no—
13C3×#3830—21no—
14C3×#3830—22no—
15C3×#3828—21no—
16C3×#3828C2211yescharacteristic
17C2×C3×#634—21no—
18C2×C3×#634C2211yescharacteristic
19C33×Q16—21no—
20C33×SD16—21no—
21C33×M16C2211yescharacteristic
22C3×#3756—21no—
23C3×#3755—21no—
24C3×#3750—22no—
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C3×#3748C2211yescharacteristic
27#64367C2×C311yescharacteristic
28C3×#1010—34no—
29C32×#137—24no—
30C32×#137D314yescomplemented
31C32×#137C2×C311yescharacteristic, complemented
32#64153C2×C311yescharacteristic, complemented
33#64151C2×C311yescharacteristic, complemented
34C3×#944—34no—
35C3×#942—34no—
36#63664C2×C311yescharacteristic, complemented
37#63652C2×C311yescharacteristic, complemented
38#63651C2×C311yescharacteristic, complemented
39C3×#837—34no—
40C3×#825—34no—
41C3×#824—34no—
42C33×Q8—21no—
43C33×Q8—21no—
44C33×Q8D411yescharacteristic
45C33×D4—21no—
46C33×D4D411yescharacteristic
47C2×C33×C4—21no—
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C3×#638—22no—
51C3×#638—21no—
52C2×C3×C3⋊Dic3—22no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3D412yes—
55C3×C4⋊(C3⋊D3)D411yescharacteristic
56C3×C4×C3⋊D3—22no—
57C3×C4×C3⋊D3—21no—
58C3×#634—22no—
59C3×#634—21no—
60C3×#634—21no—
61C3×#634D411yescharacteristic
62C33×C8—21no—
63C3×C3⋊(C3⋊C8)—21no—
64C3×#456—91no—
65#5205—34no—
66C32×#41—24no—
67C32×#41C2×D314yes—
68C32×#41C22×C311yescharacteristic
69C2×C32×Q8—24no—
70C2×C32×Q8C2×D314yes—
71C2×C32×Q8C22×C311yescharacteristic
72Q8×C3⋊D3—21no—
73#3830—21no—
74#3830—22no—
75#3828—21no—
76#3828C22×C311yescharacteristic
77C2×#634—21no—
78C2×#634C22×C311yescharacteristic
79C3×D3×Q8—64no—
80C3×#237—68no—
81C3×#237—64no—
82C3×#235—34no—
83C3×#235—64no—
84C2×C3×Dic6—34no—
85C2×C3×Dic6—64no—
86C32×Q16—28no—
87C32×Q16—24no—
88C32×Q16—21no—
89C32×SD16—28no—
90C32×SD16—24no—
91C32×SD16—21no—
92C32×M16—24no—
93C32×M16C2×D314yes—
94C32×M16C22×C311yescharacteristic
95#3756—21no—
96#3755—21no—
97#3750—22no—
98C4×C3⋊Dic3C22×C311yescharacteristic
99#3748C22×C311yescharacteristic
100C3×#216—64no—
101C3×#215—64no—
102C3×Dic3⋊C4—68no—
103C3×C4×Dic3—34no—
104C3×#208—34no—
105C22×C33—21no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—21no—
108C33×C4—21no—
109C33×C4—41no—
110C33×C4C2×D411yescharacteristic
111C33×C4C2×D411yescharacteristic
112C3×C3⋊Dic3—22no—
113C3×C3⋊Dic3—22no—
114C3×C3⋊Dic3—21no—
115C3×2−1+4—91no—
116#1010—34no—
117C3×#137—24no—
118C3×#137—91no—
119C3×#137C3×D314yescomplemented
120C3×#137C3⋊D311yescharacteristic, complemented
121C3×C4⋊Q8—91no—
122#944—34no—
123#942—34no—
124C3×#104—91no—
125C3×#104—91no—
126C3×#101—91no—
127#837—34no—
128#825—34no—
129#824—34no—
130C32×Q8—28no—
131C32×Q8—24no—
132C32×Q8—21no—
133C32×Q8—21no—
134C32×Q8—24no—
135C32×Q8—24no—
136C32×Q8—44no—
137C32×Q8#6714yes—
138C32×Q8C3×D411yescharacteristic
139C32×D4—21no—
140C32×D4—24no—
141C32×D4—24no—
142C32×D4—44no—
143C32×D4#6714yes—
144C32×D4C3×D411yescharacteristic
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—24no—
148C2×C32×C4—24no—
149C2×C32×C4—24no—
150C2×C32×C4—44no—
151C2×C32×C4—44no—
152C2×C32×C4C22×D314yes—
153C2×C32×C4C23×C311yescharacteristic
154#638—22no—
155#638—21no—
156C2×C3⋊Dic3—22no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3C3×D412yes—
159C4⋊(C3⋊D3)C3×D411yescharacteristic
160C4×C3⋊D3—22no—
161C4×C3⋊D3—21no—
162#634—22no—
163#634—21no—
164#634—21no—
165#634C3×D411yescharacteristic
166C3×#67—68no—
167C3×#67—64no—
168C2×C3×Dic3—38no—
169C2×C3×Dic3—68no—
170C2×C3×Dic3—64no—
171C3×D12—34no—
172C3×C4×D3—68no—
173C3×C4×D3—64no—
174C3×Dic6—34no—
175C3×Dic6—68no—
176C3×Dic6—64no—
177C3×Dic6—64no—
178C32×C8—28no—
179C32×C8—24no—
180C32×C8—21no—
181C3⋊(C3⋊C8)—21no—
182C3×C3⋊C8—64no—
183#456—91noSylow
184C2×C33—21no—
185C2×C33—41no—
186C2×C33#12011yescharacteristic
187C3×C3⋊D3—41no—
188C3×#41—24no—
189C3×#41—91no—
190C3×#41—181no—
191C3×#41—182no—
192C3×#41—181no—
193C3×#41C2×C3×D314yes—
194C3×#41C2×C3⋊D311yescharacteristic
195C2×C3×Q8—24no—
196C2×C3×Q8—91no—
197C2×C3×Q8—181no—
198C2×C3×Q8—181no—
199C2×C3×Q8C2×C3×D314yes—
200C2×C3×Q8C2×C3⋊D311yescharacteristic
201D3×Q8—64no—
202#237—68no—
203#237—64no—
204#235—34no—
205#235—64no—
206C2×Dic6—34no—
207C2×Dic6—64no—
208C3×Q16—28no—
209C3×Q16—24no—
210C3×Q16—21no—
211C3×Q16—181no—
212C3×SD16—28no—
213C3×SD16—24no—
214C3×SD16—21no—
215C3×SD16—181no—
216C3×M16—24no—
217C3×M16—91no—
218C3×M16C2×C3×D314yes—
219C3×M16C2×C3⋊D311yescharacteristic
220C3×C4⋊C4—182no—
221C3×C42—91no—
222#216—64no—
223#215—64no—
224Dic3⋊C4—68no—
225C4×Dic3—34no—
226#208—34no—
227C22×C32—21no—
228C22×C32—24no—
229C22×C32—24no—
230C22×C32—44no—
231C22×C32D3×D414yes—
232C22×C32C2×C3×D411yescharacteristic
233C2×C3⋊D3—21no—
234C2×C3×D3—64no—
235C32×C4—21no—
236C32×C4—24no—
237C32×C4—24no—
238C32×C4—24no—
239C32×C4—44no—
240C32×C4—48no—
241C32×C4—44no—
242C32×C4—41no—
243C32×C4D3×D414yes—
244C32×C4C2×#6714yes—
245C32×C4C2×C3×D411yescharacteristic
246C32×C4C2×C3×D411yescharacteristic
247C3⋊Dic3—22no—
248C3⋊Dic3—22no—
249C3⋊Dic3—21no—
250C3×Dic3—68no—
251C3×Dic3—64no—
252C3×Dic3—68no—
2532−1+4—91no—
254#137—91no—
255#137C3×C3⋊D311yescharacteristic, complemented
256C4⋊Q8—91no—
257#104—91no—
258#104—91no—
259#101—91no—
260C33#45611yescharacteristic, complemented, Sylow
261C3×Q8—28no—
262C3×Q8—24no—
263C3×Q8—24no—
264C3×Q8—21no—
265C3×Q8—21no—
266C3×Q8—24no—
267C3×Q8—44no—
268C3×Q8—91no—
269C3×Q8—182no—
270C3×Q8—181no—
271C3×Q8—181no—
272C3×Q8C3×#6714yes—
273C3×Q8#63811yescharacteristic
274C3×D4—24no—
275C3×D4—21no—
276C3×D4—24no—
277C3×D4—44no—
278C3×D4—91no—
279C3×D4—182no—
280C3×D4—181no—
281C3×D4C3×#6714yes—
282C3×D4#63811yescharacteristic
283C2×C3×C4—24no—
284C2×C3×C4—21no—
285C2×C3×C4—24no—
286C2×C3×C4—21no—
287C2×C3×C4—24no—
288C2×C3×C4—44no—
289C2×C3×C4—44no—
290C2×C3×C4—92no—
291C2×C3×C4—182no—
292C2×C3×C4—182no—
293C2×C3×C4—181no—
294C2×C3×C4—181no—
295C2×C3×C4C22×C3×D314yes—
296C2×C3×C4C22×C3⋊D311yescharacteristic
297#67—68no—
298#67—64no—
299C2×Dic3—38no—
300C2×Dic3—68no—
301C2×Dic3—64no—
302D12—34no—
303C4×D3—68no—
304C4×D3—64no—
305Dic6—34no—
306Dic6—68no—
307Dic6—64no—
308Dic6—64no—
309C3×C8—28no—
310C3×C8—24no—
311C3×C8—21no—
312C3×C8—181no—
313C3⋊C8—64no—
314C2×C32—21no—
315C2×C32—24no—
316C2×C32—24no—
317C2×C32—44no—
318C2×C32—41no—
319C2×C32—44no—
320C2×C32—84no—
321C2×C32#93714yes—
322C2×C32C3×#12011yescharacteristic
323C3⋊D3—41no—
324C3×D3—124no—
325#41—91no—
326#41—181no—
327#41—182no—
328#41—181no—
329#41C2×C3×C3⋊D311yescharacteristic
330C2×Q8—91no—
331C2×Q8—181no—
332C2×Q8—181no—
333C2×Q8C2×C3×C3⋊D311yescharacteristic
334Q16—21no—
335Q16—181no—
336SD16—21no—
337SD16—181no—
338M16—91no—
339M16C2×C3×C3⋊D311yescharacteristic
340C4⋊C4—182no—
341C42—91no—
342C22×C3—24no—
343C22×C3—21no—
344C22×C3—24no—
345C22×C3—44no—
346C22×C3—181no—
347C22×C3C3×D3×D414yes—
348C22×C3D4×C3⋊D311yescharacteristic
349C2×D3—64no—
350C3×C4—24no—
351C3×C4—21no—
352C3×C4—24no—
353C3×C4—24no—
354C3×C4—44no—
355C3×C4—48no—
356C3×C4—44no—
357C3×C4—41no—
358C3×C4—182no—
359C3×C4—182no—
360C3×C4—181no—
361C3×C4C3×D3×D414yes—
362C3×C4C2×C3×#6714yes—
363C3×C4D4×C3⋊D311yescharacteristic
364C3×C4C2×#63811yescharacteristic
365Dic3—68no—
366Dic3—64no—
367Dic3—68no—
368C32—24no—
369C32#520514yescomplemented
370C32C3×#45611yescharacteristic, complemented
371Q8—21no—
372Q8—21no—
373Q8—91no—
374Q8—182no—
375Q8—181no—
376Q8—181no—
377Q8C3×#63811yescharacteristic
378D4—21no—
379D4—91no—
380D4—182no—
381D4—181no—
382D4C3×#63811yescharacteristic
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—92no—
386C2×C4—182no—
387C2×C4—182no—
388C2×C4—181no—
389C2×C4—181no—
390C2×C4C22×C3×C3⋊D311yescharacteristic
391C8—21no—
392C8—181no—
393C2×C3—24no—
394C2×C3—21no—
395C2×C3—24no—
396C2×C3—44no—
397C2×C3—44no—
398C2×C3—41no—
399C2×C3—84no—
400C2×C3—361no—
401C2×C3C3×#93714yes—
402C2×C3#6414611yescharacteristic
403D3—124no—
404C22—21no—
405C22—181no—
406C22C3×D4×C3⋊D311yescharacteristic
407C4—21no—
408C4—41no—
409C4—182no—
410C4—182no—
411C4—181no—
412C4C3×D4×C3⋊D311yescharacteristic
413C4C2×C3×#63811yescharacteristic
414C3—24no—
415C3C3×#520514yescomplemented
416C3#1059290411yescharacteristic, direct product factor
417C2—21no—
418C2—41no—
419C2—361no—
420C2C3×#6414611yescharacteristic
subgroupG/H
centerC2×C3#64146
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×#137C2
socleC2×C33#120
prime radicalC3×#64367C2
hypercenterC3×#137C3⋊D3
hypocenterC32C3×#456
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#137

Complements

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

NH ≅ G/Ntype
2. C33×#137C2semidirect
3. C3×#64153C2semidirect
4. C3×#64151C2semidirect
5. C3×#63664C2semidirect
6. C3×#63652C2semidirect
7. C3×#63651C2semidirect
8. #10592904C3direct (N × H)
30. C32×#137D3semidirect
31. C32×#137C2×C3semidirect
32. #64153C2×C3semidirect
33. #64151C2×C3semidirect
36. #63664C2×C3semidirect
37. #63652C2×C3semidirect
38. #63651C2×C3semidirect
119. C3×#137C3×D3semidirect
120. C3×#137C3⋊D3semidirect
255. #137C3×C3⋊D3semidirect
260. C33#456semidirect
369. C32#5205semidirect
370. C32C3×#456semidirect
415. C3C3×#5205semidirect
416. C3#10592904direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643672
C33×#1372
C3×#641532
C3×#641512
C3×#636642
C3×#636522
C3×#636512
#105929043
C3×#52053