Finite Group Databasebeta

← C3×#10592872 #13163212

Subgroups — C3×#10592872 #13163212

Order 1728.

1184 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic
2C2×C33×SD16C211yesmaximal, characteristic, complemented
3C3×#64155C211yesmaximal, characteristic, complemented
4C2×C3×#3756C211yesmaximal, characteristic, complemented
5C3×#63659C211yesmaximal, characteristic, complemented
6C3×#63647C211yesmaximal, characteristic, complemented
7C3×#63643C211yesmaximal, characteristic, complemented
8#10592872C311yesmaximal, characteristic, direct product factor
9C3×#5173—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C22×C3×C3⋊Dic3—21no—
14C3×#3830—22no—
15C3×#3830—22no—
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C33×SD16—22no—
19C2×C33×C8C2211yescharacteristic
20C3×#3756—22no—
21C3×#3752—21no—
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3750—21no—
24C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
25C2×#3830C2×C311yescharacteristic
26C2×C3×#237—34no—
27C2×C32×SD16—24no—
28C2×C32×SD16D314yescomplemented
29C2×C32×SD16C2×C311yescharacteristic, complemented
30#64155C2×C311yescharacteristic, complemented
31C2×#3756C2×C311yescharacteristic, complemented
32C3×#946—34no—
33C2×C3×#216—34no—
34#63659C2×C311yescharacteristic, complemented
35#63647C2×C311yescharacteristic, complemented
36#63643C2×C311yescharacteristic, complemented
37C3×#832—34no—
38C3×(C2×D3)⋊C8—34no—
39C3×#816—34no—
40C23×C33—21no—
41C22×C3×C3⋊D3D411yescharacteristic
42C33×Q8—21no—
43C33×Q8—21no—
44C33×D4—21no—
45C33×D4D412yes—
46C2×C33×C4—21no—
47C2×C33×C4C2311yescharacteristic
48C3×#638—24no—
49C2×C3×C3⋊Dic3—24no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4×C3⋊D3—21no—
54C3×C4×C3⋊D3—21no—
55C3×#634—21no—
56C3×#634D412yes—
57C33×C8—21no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×#452—91no—
60#5173—34no—
61C2×C32×Q8—24no—
62C2×C32×Q8C2×D314yes—
63C2×C32×Q8C22×C311yescharacteristic
64C2×C32×D4—24no—
65C2×C32×D4C2×D314yes—
66C2×C32×D4C22×C311yescharacteristic
67C2×#638—21no—
68C22×C3⋊Dic3—21no—
69#3830—22no—
70#3830—22no—
71C2×C4×C3⋊D3C22×C311yescharacteristic
72C2×#634C22×C311yescharacteristic
73C2×C3×#67—64no—
74C22×C3×Dic3—64no—
75C3×#237—68no—
76C3×#237—68no—
77C2×C3×C4×D3—34no—
78C2×C3×Dic6—34no—
79C32×SD16—216no—
80C32×SD16—28no—
81C32×SD16—22no—
82C2×C32×C8—24no—
83C2×C32×C8C2×D314yes—
84C2×C32×C8C22×C311yescharacteristic
85#3756—22no—
86#3752—21no—
87C4⋊(C3⋊Dic3)C22×C311yescharacteristic
88#3750—21no—
89C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
90C3×#216—68no—
91C3×(C2×D3)⋊C4—64no—
92C3×C4⋊Dic3—34no—
93C3×Dic3⋊C4—64no—
94C2×C3×C3⋊C8—34no—
95C22×C33—22no—
96C22×C33—41no—
97C22×C33—41no—
98C22×C33C2×D411yescharacteristic
99C2×C3×C3⋊D3—21no—
100C2×C3×C3⋊D3—21no—
101C2×C3×C3⋊D3—21no—
102C33×C4—41no—
103C33×C4C2×D411yescharacteristic
104C33×C4C2×D411yescharacteristic
105C3×C3⋊Dic3—22no—
106C3×C3⋊Dic3—21no—
107C3×C3⋊Dic3—41no—
108C2×C3×#41—91no—
109C2×#237—34no—
110C2×C3×Q16—91no—
111C2×C3×SD16—24no—
112C2×C3×SD16C3×D314yescomplemented
113C2×C3×SD16C3⋊D311yescharacteristic, complemented
114C3×C22⋊Q8—91no—
115#946—34no—
116C2×#216—34no—
117C3×#103—91no—
118C3×#102—91no—
119C3×C22⋊C8—91no—
120#832—34no—
121(C2×D3)⋊C8—34no—
122#816—34no—
123C23×C32—21no—
124C23×C32—24no—
125C23×C32—44no—
126C22×C3⋊D3C3×D411yescharacteristic
127C22×C3×D3—34no—
128C32×Q8—28no—
129C32×Q8—24no—
130C32×Q8—21no—
131C32×Q8—21no—
132C32×Q8—24no—
133C32×Q8—44no—
134C32×D4—21no—
135C32×D4—28no—
136C32×D4—24no—
137C32×D4—44no—
138C32×D4#6718yes—
139C32×D4C3×D412yes—
140C2×C32×C4—21no—
141C2×C32×C4—24no—
142C2×C32×C4—24no—
143C2×C32×C4—44no—
144C2×C32×C4C22×D314yes—
145C2×C32×C4C23×C311yescharacteristic
146#638—24no—
147C2×C3⋊Dic3—24no—
148C2×C3⋊Dic3—21no—
149C2×C3⋊Dic3—21no—
150C2×C3⋊Dic3C3×D411yescharacteristic
151C4×C3⋊D3—21no—
152C4×C3⋊D3—21no—
153#634—21no—
154#634C3×D412yes—
155C3×#67—616no—
156C2×C3×Dic3—34no—
157C2×C3×Dic3—616no—
158C2×C3×Dic3—64no—
159C2×C3×Dic3—64no—
160C3×C4×D3—64no—
161C3×C4×D3—64no—
162C3×Dic6—38no—
163C3×Dic6—64no—
164C32×C8—28no—
165C32×C8—24no—
166C32×C8—21no—
167C3⋊(C3⋊C8)—21no—
168C3×C3⋊C8—64no—
169#452—91noSylow
170C2×C33—42no—
171C2×C33#12011yescharacteristic
172C2×C33#13511yescharacteristic
173C2×C33#13711yescharacteristic
174C3×C3⋊D3—41no—
175C3×#41—182no—
176C3×#41—182no—
177C2×C3×Q8—24no—
178C2×C3×Q8—91no—
179C2×C3×Q8C2×C3×D314yes—
180C2×C3×Q8C2×C3⋊D311yescharacteristic
181C2×C3×D4—24no—
182C2×C3×D4—181no—
183C2×C3×D4C2×C3×D314yes—
184C2×C3×D4C2×C3⋊D311yescharacteristic
185C22×C3×C4—91no—
186C22×C3×C4—181no—
187C2×#67—64no—
188C22×Dic3—64no—
189#237—68no—
190#237—68no—
191C2×C4×D3—34no—
192C2×Dic6—34no—
193C3×Q16—182no—
194C3×SD16—216no—
195C3×SD16—28no—
196C3×SD16—22no—
197C2×C3×C8—24no—
198C2×C3×C8—91no—
199C2×C3×C8C2×C3×D314yes—
200C2×C3×C8C2×C3⋊D311yescharacteristic
201C3×C4⋊C4—91no—
202C3×C4⋊C4—181no—
203C3×C22⋊C4—181no—
204#216—68no—
205(C2×D3)⋊C4—64no—
206C4⋊Dic3—34no—
207Dic3⋊C4—64no—
208C2×C3⋊C8—34no—
209C22×C32—22no—
210C22×C32—24no—
211C22×C32—28no—
212C22×C32—48no—
213C22×C32—41no—
214C22×C32—41no—
215C22×C32—44no—
216C22×C32—44no—
217C22×C32—84no—
218C22×C32—84no—
219C22×C32D3×D414yes—
220C22×C32C2×C3×D411yescharacteristic
221C2×C3⋊D3—21no—
222C2×C3⋊D3—21no—
223C2×C3⋊D3—21no—
224C2×C3×D3—64no—
225C2×C3×D3—64no—
226C2×C3×D3—64no—
227C32×C4—24no—
228C32×C4—24no—
229C32×C4—48no—
230C32×C4—44no—
231C32×C4—41no—
232C32×C4D3×D414yes—
233C32×C4C2×#6714yes—
234C32×C4C2×C3×D411yescharacteristic
235C32×C4C2×C3×D411yescharacteristic
236C3⋊Dic3—22no—
237C3⋊Dic3—21no—
238C3⋊Dic3—41no—
239C3×Dic3—68no—
240C3×Dic3—64no—
241C3×Dic3—124no—
242C2×#41—91no—
243C2×Q16—91no—
244C2×SD16C3×C3⋊D311yescharacteristic, complemented
245C22⋊Q8—91no—
246#103—91no—
247#102—91no—
248C22⋊C8—91no—
249C33#45211yescharacteristic, complemented, Sylow
250C23×C3—24no—
251C23×C3—21no—
252C23×C3—44no—
253C23×C3—91no—
254C22×D3—34no—
255C3×Q8—28no—
256C3×Q8—24no—
257C3×Q8—24no—
258C3×Q8—21no—
259C3×Q8—21no—
260C3×Q8—44no—
261C3×Q8—92no—
262C3×Q8—181no—
263C3×D4—24no—
264C3×D4—21no—
265C3×D4—28no—
266C3×D4—44no—
267C3×D4—184no—
268C3×D4C3×#6718yes—
269C3×D4#63812yes—
270C2×C3×C4—24no—
271C2×C3×C4—21no—
272C2×C3×C4—24no—
273C2×C3×C4—44no—
274C2×C3×C4—91no—
275C2×C3×C4—184no—
276C2×C3×C4—181no—
277C2×C3×C4—181no—
278C2×C3×C4—181no—
279C2×C3×C4—181no—
280C2×C3×C4C22×C3×D314yes—
281C2×C3×C4C22×C3⋊D311yescharacteristic
282#67—616no—
283C2×Dic3—34no—
284C2×Dic3—616no—
285C2×Dic3—64no—
286C2×Dic3—64no—
287C4×D3—64no—
288C4×D3—64no—
289Dic6—38no—
290Dic6—64no—
291C3×C8—28no—
292C3×C8—24no—
293C3×C8—21no—
294C3×C8—181no—
295C3⋊C8—64no—
296C2×C32—24no—
297C2×C32—24no—
298C2×C32—24no—
299C2×C32—42no—
300C2×C32—48no—
301C2×C32—88no—
302C2×C32#91514yes—
303C2×C32#91614yes—
304C2×C32#93714yes—
305C2×C32C3×#12011yescharacteristic
306C2×C32C3×#13511yescharacteristic
307C2×C32C3×#13711yescharacteristic
308C3⋊D3—41no—
309C3×D3—124no—
310#41—182no—
311#41—182no—
312C2×Q8—91no—
313C2×Q8C2×C3×C3⋊D311yescharacteristic
314C2×D4—181no—
315C2×D4C2×C3×C3⋊D311yescharacteristic
316C22×C4—91no—
317C22×C4—181no—
318Q16—182no—
319SD16—22no—
320C2×C8—91no—
321C2×C8C2×C3×C3⋊D311yescharacteristic
322C4⋊C4—91no—
323C4⋊C4—181no—
324C22⋊C4—181no—
325C22×C3—28no—
326C22×C3—22no—
327C22×C3—24no—
328C22×C3—48no—
329C22×C3—44no—
330C22×C3—44no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—84no—
334C22×C3—84no—
335C22×C3—181no—
336C22×C3—181no—
337C22×C3—181no—
338C22×C3C3×D3×D414yes—
339C22×C3D4×C3⋊D311yescharacteristic
340C2×D3—64no—
341C2×D3—64no—
342C2×D3—64no—
343C3×C4—24no—
344C3×C4—24no—
345C3×C4—48no—
346C3×C4—44no—
347C3×C4—41no—
348C3×C4—181no—
349C3×C4—182no—
350C3×C4—361no—
351C3×C4C3×D3×D414yes—
352C3×C4C2×C3×#6714yes—
353C3×C4D4×C3⋊D311yescharacteristic
354C3×C4C2×#63811yescharacteristic
355Dic3—68no—
356Dic3—64no—
357Dic3—124no—
358C32—24no—
359C32#517314yescomplemented
360C32C3×#45211yescharacteristic, complemented
361C23—21no—
362C23—91no—
363Q8—21no—
364Q8—21no—
365Q8—92no—
366Q8—181no—
367D4—21no—
368D4—184no—
369D4C3×#63812yes—
370C2×C4—21no—
371C2×C4—91no—
372C2×C4—184no—
373C2×C4—181no—
374C2×C4—181no—
375C2×C4—181no—
376C2×C4—181no—
377C2×C4C22×C3×C3⋊D311yescharacteristic
378C8—21no—
379C8—181no—
380C2×C3—24no—
381C2×C3—24no—
382C2×C3—24no—
383C2×C3—48no—
384C2×C3—42no—
385C2×C3—88no—
386C2×C3—361no—
387C2×C3C3×#91514yes—
388C2×C3C3×#91614yes—
389C2×C3C3×#93714yes—
390C2×C3#6412411yescharacteristic
391C2×C3#6412511yescharacteristic
392C2×C3#6414611yescharacteristic
393D3—124no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—181no—
398C22—181no—
399C22—181no—
400C22C3×D4×C3⋊D311yescharacteristic
401C4—41no—
402C4—181no—
403C4—182no—
404C4—361no—
405C4C3×D4×C3⋊D311yescharacteristic
406C4C2×C3×#63811yescharacteristic
407C3—24no—
408C3C3×#517314yescomplemented
409C3#1059287211yescharacteristic, direct product factor
410C2—42no—
411C2—361no—
412C2C3×#6412411yescharacteristic
413C2C3×#6412511yescharacteristic
414C2C3×#6414611yescharacteristic
subgroupG/H
centerC22×C3D4×C3⋊D3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC2×C33×SD16C2
socleC22×C33C2×D4
prime radicalC2×C3×#3830C2
hypercenterC2×C3×SD16C3⋊D3
hypocenterC32C3×#452
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×SD16

Complements

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

NH ≅ G/Ntype
2. C2×C33×SD16C2semidirect
3. C3×#64155C2semidirect
4. C2×C3×#3756C2semidirect
5. C3×#63659C2semidirect
6. C3×#63647C2semidirect
7. C3×#63643C2semidirect
8. #10592872C3direct (N × H)
28. C2×C32×SD16D3semidirect
29. C2×C32×SD16C2×C3semidirect
30. #64155C2×C3semidirect
31. C2×#3756C2×C3semidirect
34. #63659C2×C3semidirect
35. #63647C2×C3semidirect
36. #63643C2×C3semidirect
112. C2×C3×SD16C3×D3semidirect
113. C2×C3×SD16C3⋊D3semidirect
244. C2×SD16C3×C3⋊D3semidirect
249. C33#452semidirect
359. C32#5173semidirect
360. C32C3×#452semidirect
408. C3C3×#5173semidirect
409. C3#10592872direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C33×SD162
C3×#641552
C2×C3×#37562
C3×#636592
C3×#636472
C3×#636432
#105928723
C3×#51733