Finite Group Databasebeta

← C3×#10591116 #13158148

Subgroups — C3×#10591116 #13158148

Order 1728.

730 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3748C211yesmaximal, characteristic
2C32×#926C211yesmaximal, characteristic
3C22×C32×C3⋊C8C211yesmaximal, characteristic
4C3×Dic3⋊(C3⋊C8)C214yesmaximal, complemented
5C3×#5102—31nomaximal
6C3×#4999—31nomaximal
7#10591116C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C3×#3748—22no—
10C2×C3×C3⋊(C3⋊C8)C2212yes—
11C32×#217C412yes—
12C32×C4⋊Dic3C412yes—
13C32×C4×Dic3C2212yes—
14C2×C32×C3⋊C8—22no—
15C2×C32×C3⋊C8C2212yes—
16C22×C32×C8—31no—
17C32×#117—31no—
18C2×#3748C2×C311yescharacteristic
19C3×#926—21no—
20C3×#926D311yescharacteristic
21C3×#926C2×C311yescharacteristic
22C2×C3×#208—31no—
23C2×C3×#208—31no—
24C2×C3×#208—61no—
25C22×C3×C3⋊C8—21no—
26C22×C3×C3⋊C8D311yescharacteristic
27C22×C3×C3⋊C8C2×C311yescharacteristic
28C3×Dic3⋊C8—34no—
29C3×C4⋊(C3⋊C8)—34no—
30Dic3⋊(C3⋊C8)C2×C314yescomplemented
31C23×C33C2×C411yescharacteristic
32C2×C33×C4C2×C411yescharacteristic
33C2×C33×C4D412yes—
34C2×C33×C4Q812yes—
35C2×C33×C4C2311yescharacteristic
36C2×C32×Dic3C2×C414yes—
37C3×C3⋊(C3⋊C8)—22no—
38C32×C3⋊C8—22no—
39C3×#424—91no—
40#5102—31no—
41#4999—31no—
42C22×C32×C4—21no—
43C22×C32×C4—21no—
44C22×C32×C4—21no—
45C22×C32×C4—41no—
46C22×C32×C4C2×D311yescharacteristic
47C22×C32×C4C2×D311yescharacteristic
48C22×C32×C4C22×C311yescharacteristic
49C2×C32×C8—32no—
50C2×C32×C8—62no—
51C32×C4⋊C4—32no—
52C32×C22⋊C4—32no—
53C32×C42—32no—
54#3748—22no—
55C2×C3⋊(C3⋊C8)C22×C312yes—
56C3×#217—22no—
57C3×#217Dic312yes—
58C3×#217C3×C412yes—
59C3×C4⋊Dic3—22no—
60C3×C4⋊Dic3Dic312yes—
61C3×C4⋊Dic3C3×C412yes—
62C3×C4×Dic3—22no—
63C3×C4×Dic3C2×D312yes—
64C3×C4×Dic3C22×C312yes—
65C3×#208—64no—
66C3×#208—62no—
67C3×#208—62no—
68C2×C3×C3⋊C8—24no—
69C2×C3×C3⋊C8—22no—
70C2×C3×C3⋊C8—22no—
71C2×C3×C3⋊C8—22no—
72C2×C3×C3⋊C8—32no—
73C2×C3×C3⋊C8—32no—
74C2×C3×C3⋊C8—62no—
75C2×C3×C3⋊C8C2×D312yes—
76C2×C3×C3⋊C8C22×C312yes—
77C22×C33—22no—
78C22×C33C4⋊C412yes—
79C22×C33C22×C411yescharacteristic
80C33×C4C4⋊C412yes—
81C33×C4C2×D411yescharacteristic
82C33×C4C2×Q811yescharacteristic
83C32×Dic3—24no—
84C2×C3×M16—91no—
85C22×C3×C8—31no—
86C22×C3×C8—31no—
87C22×C3×C8—61no—
88C3×#117—31no—
89C3×#117—31no—
90C3×#117—61no—
91#926C3×D311yescharacteristic
92C2×#208—31no—
93C2×#208—31no—
94C2×#208—61no—
95C22×C3⋊C8C3×D311yescharacteristic
96C3×C4⋊C8—94no—
97Dic3⋊C8—34no—
98C4⋊(C3⋊C8)—34no—
99C23×C32—21no—
100C23×C32—21no—
101C23×C32—21no—
102C23×C32—41no—
103C23×C32C4×D311yescharacteristic
104C23×C32C2×Dic311yescharacteristic
105C23×C32C2×C3×C411yescharacteristic
106C2×C32×C4—21no—
107C2×C32×C4—22no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—21no—
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4—22no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—34no—
119C2×C32×C4—41no—
120C2×C32×C4—42no—
121C2×C32×C4—42no—
122C2×C32×C4—41no—
123C2×C32×C4Dic612yes—
124C2×C32×C4Dic612yes—
125C2×C32×C4C4×D311yescharacteristic
126C2×C32×C4D1212yes—
127C2×C32×C4C2×Dic311yescharacteristic
128C2×C32×C4#6712yes—
129C2×C32×C4C2×C3×C411yescharacteristic
130C2×C32×C4C3×D412yes—
131C2×C32×C4C3×Q812yes—
132C2×C32×C4C22×D311yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C23×C311yescharacteristic
135C2×C3×Dic3—24no—
136C2×C3×Dic3C2×Dic314yes—
137C2×C3×Dic3C2×C3×C414yes—
138C32×C8—62no—
139C3⋊(C3⋊C8)—22no—
140C3×C3⋊C8—24no—
141C3×C3⋊C8—22no—
142C3×C3⋊C8—22no—
143C3×C3⋊C8—64no—
144C3×C3⋊C8—62no—
145C3×C3⋊C8—62no—
146#424—91noSylow
147C2×C33—22no—
148C2×C33C2×C4⋊C411yescharacteristic
149C2×C33#13112yes—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—41no—
154C22×C3×C4D3211yescharacteristic
155C22×C3×C4C2×C3×D311yescharacteristic
156C22×C3×C4C2×C3×D311yescharacteristic
157C3×M16—182no—
158C2×C3×C8—32no—
159C2×C3×C8—32no—
160C2×C3×C8—62no—
161C2×C3×C8—64no—
162C2×C3×C8—62no—
163C2×C3×C8—62no—
164C2×C3×C8—92no—
165C3×C4⋊C4—32no—
166C3×C4⋊C4—32no—
167C3×C4⋊C4—62no—
168C3×C22⋊C4—32no—
169C3×C22⋊C4—32no—
170C3×C22⋊C4—62no—
171C3×C42—32no—
172C3×C42—32no—
173C3×C42—62no—
174#217C3×Dic312yes—
175C4⋊Dic3C3×Dic312yes—
176C4×Dic3C2×C3×D312yes—
177#208—64no—
178#208—62no—
179#208—62no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—32no—
182C2×C3⋊C8—32no—
183C2×C3⋊C8—62no—
184C2×C3⋊C8C2×C3×D312yes—
185C22×C32—22no—
186C22×C32—24no—
187C22×C32—22no—
188C22×C32—22no—
189C22×C32—22no—
190C22×C32—22no—
191C22×C32—22no—
192C22×C32—21no—
193C22×C32—21no—
194C22×C32—21no—
195C22×C32—42no—
196C22×C32—41no—
197C22×C32—42no—
198C22×C32—42no—
199C22×C32—44no—
200C22×C32Dic3⋊C412yes—
201C22×C32C4⋊Dic312yes—
202C22×C32C3×C4⋊C412yes—
203C22×C32C2×C4×D311yescharacteristic
204C22×C32C22×Dic311yescharacteristic
205C22×C32C22×C3×C411yescharacteristic
206C32×C4—22no—
207C32×C4—22no—
208C32×C4—22no—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—42no—
216C32×C4—41no—
217C32×C4—41no—
218C32×C4—64no—
219C32×C4Dic3⋊C412yes—
220C32×C4C4⋊Dic312yes—
221C32×C4C3×C4⋊C412yes—
222C32×C4C2×Dic611yescharacteristic
223C32×C4C2×Dic611yescharacteristic
224C32×C4C2×D1211yescharacteristic
225C32×C4C2×#6711yescharacteristic
226C32×C4C2×C3×D411yescharacteristic
227C32×C4C2×C3×Q811yescharacteristic
228C3×Dic3—24no—
229C3×Dic3—24no—
230C3×Dic3—44no—
231C2×M16—91no—
232C22×C8—31no—
233#117—31no—
234C4⋊C8—94no—
235C33#42411yescharacteristic, complemented, Sylow
236C23×C3—21no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—41no—
240C23×C3D3×Dic311yescharacteristic
241C23×C3C3×C4×D311yescharacteristic
242C23×C3C2×C3×Dic311yescharacteristic
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—34no—
256C2×C3×C4—34no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—41no—
261C2×C3×C4—64no—
262C2×C3×C4D3×Dic311yescharacteristic
263C2×C3×C4#62612yes—
264C2×C3×C4#62712yes—
265C2×C3×C4C3×Dic612yes—
266C2×C3×C4C3×Dic612yes—
267C2×C3×C4C3×C4×D311yescharacteristic
268C2×C3×C4C3×D1212yes—
269C2×C3×C4C2×C3×Dic311yescharacteristic
270C2×C3×C4C3×#6712yes—
271C2×C3×C4C2×D3211yescharacteristic
272C2×C3×C4C22×C3×D311yescharacteristic
273C2×C3×C4C22×C3×D311yescharacteristic
274C2×Dic3C2×C3×Dic314yes—
275C3×C8—64no—
276C3×C8—62no—
277C3×C8—62no—
278C3×C8—182no—
279C3⋊C8—22no—
280C3⋊C8—64no—
281C3⋊C8—62no—
282C3⋊C8—62no—
283C2×C32—22no—
284C2×C32—24no—
285C2×C32—22no—
286C2×C32—22no—
287C2×C32—22no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32—22no—
291C2×C32—22no—
292C2×C32—21no—
293C2×C32—41no—
294C2×C32—42no—
295C2×C32—42no—
296C2×C32—42no—
297C2×C32—44no—
298C2×C32#90112yes—
299C2×C32C2×Dic3⋊C411yescharacteristic
300C2×C32C2×C4⋊Dic311yescharacteristic
301C2×C32#94812yes—
302C2×C32C2×C3×C4⋊C411yescharacteristic
303C2×C32C3×#13112yes—
304C22×C4C3×D3211yescharacteristic
305M16—182no—
306C2×C8—32no—
307C2×C8—62no—
308C2×C8—92no—
309C4⋊C4—32no—
310C22⋊C4—32no—
311C42—32no—
312C22×C3—24no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—22no—
320C22×C3—22no—
321C22×C3—22no—
322C22×C3—42no—
323C22×C3—42no—
324C22×C3—44no—
325C22×C3—41no—
326C22×C3—42no—
327C22×C3Dic3⋊Dic312yes—
328C22×C3C3×Dic3⋊C412yes—
329C22×C3C3×C4⋊Dic312yes—
330C22×C3C2×D3×Dic311yescharacteristic
331C22×C3C2×C3×C4×D311yescharacteristic
332C22×C3C22×C3×Dic311yescharacteristic
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—42no—
345C3×C4—64no—
346C3×C4—64no—
347C3×C4—124no—
348C3×C4Dic3⋊Dic312yes—
349C3×C4C3×Dic3⋊C412yes—
350C3×C4C3×C4⋊Dic312yes—
351C3×C4C2×#62611yescharacteristic
352C3×C4C2×#62711yescharacteristic
353C3×C4C2×C3×Dic611yescharacteristic
354C3×C4C2×C3×Dic611yescharacteristic
355C3×C4C2×C3×D1211yescharacteristic
356C3×C4C2×C3×#6711yescharacteristic
357Dic3—24no—
358C32—21no—
359C32—21no—
360C32—21no—
361C32—41no—
362C32#499911yescharacteristic, complemented
363C32#510211yescharacteristic, complemented
364C32C3×#42411yescharacteristic, complemented
365C23C3×D3×Dic311yescharacteristic
366C2×C4—34no—
367C2×C4C3×D3×Dic311yescharacteristic
368C2×C4C3×#62612yes—
369C2×C4C3×#62712yes—
370C2×C4C2×C3×D3211yescharacteristic
371C8—62no—
372C8—182no—
373C2×C3—24no—
374C2×C3—22no—
375C2×C3—22no—
376C2×C3—22no—
377C2×C3—21no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—22no—
381C2×C3—22no—
382C2×C3—22no—
383C2×C3—42no—
384C2×C3—42no—
385C2×C3—44no—
386C2×C3—41no—
387C2×C3—42no—
388C2×C3#6381412yes—
389C2×C3C2×Dic3⋊Dic311yescharacteristic
390C2×C3C3×#90112yes—
391C2×C3C2×C3×Dic3⋊C411yescharacteristic
392C2×C3C2×C3×C4⋊Dic311yescharacteristic
393C2×C3C3×#94812yes—
394C22—22no—
395C22C3×Dic3⋊Dic312yes—
396C22C2×C3×D3×Dic311yescharacteristic
397C4—64no—
398C4C3×Dic3⋊Dic312yes—
399C4C2×C3×#62611yescharacteristic
400C4C2×C3×#62711yescharacteristic
401C3—21no—
402C3—21no—
403C3—21no—
404C3—41no—
405C3#1059111611yescharacteristic, direct product factor
406C3C3×#499911yescharacteristic, complemented
407C3C3×#510211yescharacteristic, complemented
408C2—22no—
409C2C3×#6381412yes—
410C2C2×C3×Dic3⋊Dic311yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC22×C33×C4C22
socleC22×C33C22×C4
prime radicalC23×C33C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#424
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×Dic3⋊(C3⋊C8)C2semidirect
7. #10591116C3direct (N × H)
30. Dic3⋊(C3⋊C8)C2×C3semidirect
235. C33#424semidirect
362. C32#4999semidirect
363. C32#5102semidirect
364. C32C3×#424semidirect
405. C3#10591116direct (N × H)
406. C3C3×#4999semidirect
407. C3C3×#5102semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#37482
C32×#9262
C22×C32×C3⋊C82
C3×Dic3⋊(C3⋊C8)2
C3×#51023
C3×#49993
#105911163