Finite Group Databasebeta

← C3×#10592686 #13163027

Subgroups — C3×#10592686 #13163027

Order 1728.

1772 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C33×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#3752C212yesmaximal
4C2×C3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5C3×#63658C212yesmaximal, complemented
6#10592686C311yesmaximal, characteristic, direct product factor
7C3×#4987—34nomaximal
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×C33×C4C2212yes—
10C22×C33×C4C2211yescharacteristic
11C22×C3×C3⋊Dic3C2212yes—
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C2×C3×C4×C3⋊D3—22no—
14C2×C3×C4×C3⋊D3—22no—
15C2×C3×C4×C3⋊D3C414yescomplemented
16C33×C4⋊C4—22no—
17C3×#3752—24no—
18C3×C4⋊(C3⋊Dic3)—22no—
19C22×C4×C3⋊D3C2×C311yescharacteristic
20C22×C3×C4×D3—34no—
21C2×C32×C4⋊C4—24no—
22C2×C32×C4⋊C4D314yescomplemented
23C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
24C2×#3752C2×C312yes—
25C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
26C2×C3×(C2×D3)⋊C4—38no—
27C2×C3×C4⋊Dic3—34no—
28#63658C2×C312yescomplemented
29C3×C3⋊#95—38no—
30C23×C33C2311yescharacteristic
31C22×C3×C3⋊D3—21no—
32C22×C3×C3⋊D3—21no—
33C22×C3×C3⋊D3—21no—
34C22×C3×C3⋊D3—21no—
35C22×C3×C3⋊D3C2×C412yes—
36C22×C3×C3⋊D3D412yes—
37C22×C3×C3⋊D3Q812yes—
38C2×C33×C4—22no—
39C2×C33×C4—22no—
40C2×C33×C4—22no—
41C2×C33×C4C2×C412yes—
42C2×C33×C4D412yes—
43C2×C33×C4D412yes—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—22no—
47C2×C3×C3⋊Dic3—22no—
48C2×C3×C3⋊Dic3—22no—
49C2×C3×C3⋊Dic3C2×C412yes—
50C3×C4×C3⋊D3—28no—
51C3×#380—91no—
52#4987—34no—
53C23×C3⋊D3C22×C311yescharacteristic
54C23×C3×D3—34no—
55C22×C32×C4—24no—
56C22×C32×C4—28no—
57C22×C32×C4C2×D318yes—
58C22×C32×C4C2×D314yes—
59C22×C32×C4C22×C312yes—
60C22×C32×C4C22×C311yescharacteristic
61C22×C3⋊Dic3C22×C312yes—
62C22×C3⋊Dic3C22×C311yescharacteristic
63C2×C4×C3⋊D3—22no—
64C2×C4×C3⋊D3—22no—
65C2×C4×C3⋊D3C3×C414yescomplemented
66C22×C3×Dic3—34no—
67C22×C3×Dic3—38no—
68C2×C3×C4×D3—316no—
69C2×C3×C4×D3—68no—
70C2×C3×C4×D3—68no—
71C32×C4⋊C4—216no—
72C32×C4⋊C4—28no—
73C32×C4⋊C4—22no—
74#3752—24no—
75C4⋊(C3⋊Dic3)—22no—
76C3×(C2×D3)⋊C4—616no—
77C3×C4⋊Dic3—68no—
78C22×C33C22×C411yescharacteristic
79C22×C33C2×D411yescharacteristic
80C22×C33C2×D411yescharacteristic
81C22×C33C2×D411yescharacteristic
82C22×C33C2×Q811yescharacteristic
83C22×C33#4111yescharacteristic
84C22×C33#4111yescharacteristic
85C2×C3×C3⋊D3—22no—
86C2×C3×C3⋊D3—22no—
87C2×C3×C3⋊D3—22no—
88C2×C3×C3⋊D3—22no—
89C2×C3×C3⋊D3—22no—
90C2×C3×C3⋊D3—22no—
91C2×C3×C3⋊D3C4⋊C414yescomplemented
92C33×C4—42no—
93C33×C4C22⋊C414yescomplemented
94C3×C3⋊Dic3—22no—
95C3×C3⋊Dic3—42no—
96C23×C3×C4—91no—
97C22×C4×D3—34no—
98C2×C3×C4⋊C4—24no—
99C2×C3×C4⋊C4—91no—
100C2×C3×C4⋊C4C3×D314yescomplemented
101C2×C3×C4⋊C4C3⋊D311yescharacteristic, complemented
102C2×C3×C22⋊C4—92no—
103C2×(C2×D3)⋊C4—38no—
104C2×C4⋊Dic3—34no—
105C3×#95—92no—
106C3⋊#95—38no—
107C23×C32—24no—
108C23×C32C22×D314yes—
109C23×C32C23×C311yescharacteristic
110C22×C3⋊D3—21no—
111C22×C3⋊D3—21no—
112C22×C3⋊D3—21no—
113C22×C3⋊D3—21no—
114C22×C3⋊D3C2×C3×C412yes—
115C22×C3⋊D3C3×D412yes—
116C22×C3⋊D3C3×Q812yes—
117C22×C3×D3—38no—
118C22×C3×D3—38no—
119C22×C3×D3—38no—
120C22×C3×D3—64no—
121C22×C3×D3—64no—
122C22×C3×D3—64no—
123C22×C3×D3—64no—
124C2×C32×C4—216no—
125C2×C32×C4—28no—
126C2×C32×C4—22no—
127C2×C32×C4—22no—
128C2×C32×C4—22no—
129C2×C32×C4—28no—
130C2×C32×C4—28no—
131C2×C32×C4—28no—
132C2×C32×C4—28no—
133C2×C32×C4—28no—
134C2×C32×C4—48no—
135C2×C32×C4—48no—
136C2×C32×C4C4×D318yes—
137C2×C32×C4D1218yes—
138C2×C32×C4#6718yes—
139C2×C32×C4C2×C3×C412yes—
140C2×C32×C4C3×D412yes—
141C2×C32×C4C3×D412yes—
142C2×C3⋊Dic3—21no—
143C2×C3⋊Dic3—21no—
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3—22no—
146C2×C3⋊Dic3—22no—
147C2×C3⋊Dic3C2×C3×C412yes—
148C4×C3⋊D3—28no—
149C2×C3×Dic3—38no—
150C2×C3×Dic3—64no—
151C2×C3×Dic3—64no—
152C2×C3×Dic3—68no—
153C2×C3×Dic3—68no—
154C2×C3×Dic3—68no—
155C3×C4×D3—632no—
156#380—91noSylow
157C2×C33C2×C22⋊C411yescharacteristic
158C2×C33C2×C4⋊C411yescharacteristic
159C2×C33#11711yescharacteristic
160C2×C33#12111yescharacteristic
161C2×C33#12111yescharacteristic
162C2×C33C22⋊Q811yescharacteristic
163C2×C33C22⋊Q811yescharacteristic
164C3×C3⋊D3—24no—
165C24×C3—91no—
166C23×D3—34no—
167C22×C3×C4—24no—
168C22×C3×C4—28no—
169C22×C3×C4—91no—
170C22×C3×C4—94no—
171C22×C3×C4—92no—
172C22×C3×C4—182no—
173C22×C3×C4—182no—
174C22×C3×C4C2×C3×D318yes—
175C22×C3×C4C2×C3×D314yes—
176C22×C3×C4C2×C3⋊D312yes—
177C22×C3×C4C2×C3⋊D311yescharacteristic
178C22×Dic3—34no—
179C22×Dic3—38no—
180C2×C4×D3—316no—
181C2×C4×D3—68no—
182C2×C4×D3—68no—
183C3×C4⋊C4—216no—
184C3×C4⋊C4—28no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—182no—
187C3×C22⋊C4—184no—
188(C2×D3)⋊C4—616no—
189C4⋊Dic3—68no—
190C22×C32—24no—
191C22×C32—24no—
192C22×C32—24no—
193C22×C32—24no—
194C22×C32—24no—
195C22×C32—24no—
196C22×C32—24no—
197C22×C32C2×C4×D314yes—
198C22×C32C2×D1214yes—
199C22×C32D3×D414yes—
200C22×C32#23714yes—
201C22×C32D3×Q814yes—
202C22×C32#23914yes—
203C22×C32C2×#6714yes—
204C22×C32C22×C3×C411yescharacteristic
205C22×C32C2×C3×D411yescharacteristic
206C22×C32C2×C3×D411yescharacteristic
207C22×C32C2×C3×D411yescharacteristic
208C22×C32C2×C3×Q811yescharacteristic
209C22×C32C3×#4111yescharacteristic
210C22×C32C3×#4111yescharacteristic
211C2×C3⋊D3—22no—
212C2×C3⋊D3—22no—
213C2×C3⋊D3—22no—
214C2×C3⋊D3—22no—
215C2×C3⋊D3—22no—
216C2×C3⋊D3—22no—
217C2×C3⋊D3C3×C4⋊C414yescomplemented
218C2×C3×D3—316no—
219C2×C3×D3—68no—
220C2×C3×D3—68no—
221C2×C3×D3—68no—
222C2×C3×D3—68no—
223C2×C3×D3—68no—
224C2×C3×D3—68no—
225C32×C4—216no—
226C32×C4—416no—
227C32×C4—48no—
228C32×C4—42no—
229C32×C4(C2×D3)⋊C4116yescomplemented
230C32×C4C3×C22⋊C414yescomplemented
231C3⋊Dic3—22no—
232C3⋊Dic3—42no—
233C3×Dic3—68no—
234C3×Dic3—128no—
235C23×C4—91no—
236C2×C4⋊C4—91no—
237C2×C4⋊C4C3×C3⋊D311yescharacteristic, complemented
238C2×C22⋊C4—92no—
239#95—92no—
240C33#38011yescharacteristic, complemented, Sylow
241C23×C3—24no—
242C23×C3—92no—
243C23×C3—92no—
244C23×C3—92no—
245C23×C3—181no—
246C23×C3—181no—
247C23×C3—181no—
248C23×C3—181no—
249C23×C3C22×C3×D314yes—
250C23×C3C22×C3⋊D311yescharacteristic
251C22×D3—38no—
252C22×D3—38no—
253C22×D3—38no—
254C22×D3—64no—
255C22×D3—64no—
256C22×D3—64no—
257C22×D3—64no—
258C2×C3×C4—216no—
259C2×C3×C4—28no—
260C2×C3×C4—28no—
261C2×C3×C4—28no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—28no—
266C2×C3×C4—28no—
267C2×C3×C4—28no—
268C2×C3×C4—48no—
269C2×C3×C4—48no—
270C2×C3×C4—92no—
271C2×C3×C4—181no—
272C2×C3×C4—181no—
273C2×C3×C4—188no—
274C2×C3×C4—182no—
275C2×C3×C4—182no—
276C2×C3×C4—182no—
277C2×C3×C4C3×C4×D318yes—
278C2×C3×C4C3×D1218yes—
279C2×C3×C4C3×#6718yes—
280C2×C3×C4C4×C3⋊D312yes—
281C2×C3×C4C4⋊(C3⋊D3)12yes—
282C2×C3×C4#63812yes—
283C2×Dic3—38no—
284C2×Dic3—64no—
285C2×Dic3—64no—
286C2×Dic3—68no—
287C2×Dic3—68no—
288C2×Dic3—68no—
289C4×D3—632no—
290C2×C32—24no—
291C2×C32—24no—
292C2×C32—24no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32—24no—
297C2×C32D3×C4⋊C414yes—
298C2×C32#89214yes—
299C2×C32#89514yes—
300C2×C32#89714yes—
301C2×C32C2×(C2×D3)⋊C414yes—
302C2×C32#93814yes—
303C2×C32#94614yes—
304C2×C32C2×C3×C22⋊C411yescharacteristic
305C2×C32C2×C3×C4⋊C411yescharacteristic
306C2×C32C3×#11711yescharacteristic
307C2×C32C3×#12111yescharacteristic
308C2×C32C3×#12111yescharacteristic
309C2×C32C3×C22⋊Q811yescharacteristic
310C2×C32C3×C22⋊Q811yescharacteristic
311C3⋊D3—24no—
312C3×D3—616no—
313C24—91no—
314C22×C4—91no—
315C22×C4—94no—
316C22×C4—92no—
317C22×C4—182no—
318C22×C4—182no—
319C22×C4C2×C3×C3⋊D312yes—
320C22×C4C2×C3×C3⋊D311yescharacteristic
321C4⋊C4—22no—
322C4⋊C4—182no—
323C22⋊C4—184no—
324C22×C3—24no—
325C22×C3—24no—
326C22×C3—24no—
327C22×C3—24no—
328C22×C3—24no—
329C22×C3—24no—
330C22×C3—24no—
331C22×C3—94no—
332C22×C3—182no—
333C22×C3—182no—
334C22×C3—182no—
335C22×C3—182no—
336C22×C3—182no—
337C22×C3—182no—
338C22×C3C2×C3×C4×D314yes—
339C22×C3C2×C3×D1214yes—
340C22×C3C3×D3×D414yes—
341C22×C3C3×#23714yes—
342C22×C3C3×D3×Q814yes—
343C22×C3C3×#23914yes—
344C22×C3C2×C3×#6714yes—
345C22×C3C2×C4×C3⋊D311yescharacteristic
346C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
347C22×C3D4×C3⋊D311yescharacteristic
348C22×C3#383011yescharacteristic
349C22×C3Q8×C3⋊D311yescharacteristic
350C22×C3#383211yescharacteristic
351C22×C3C2×#63811yescharacteristic
352C2×D3—316no—
353C2×D3—68no—
354C2×D3—68no—
355C2×D3—68no—
356C2×D3—68no—
357C2×D3—68no—
358C2×D3—68no—
359C3×C4—216no—
360C3×C4—416no—
361C3×C4—48no—
362C3×C4—42no—
363C3×C4—182no—
364C3×C4—362no—
365C3×C4C3×(C2×D3)⋊C4116yescomplemented
366C3×C4#375214yescomplemented
367Dic3—68no—
368Dic3—128no—
369C32—24no—
370C32#498714yescomplemented
371C32C3×#38011yescharacteristic, complemented
372C23—92no—
373C23—92no—
374C23—92no—
375C23—181no—
376C23—181no—
377C23—181no—
378C23—181no—
379C23C22×C3×C3⋊D311yescharacteristic
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—92no—
384C2×C4—181no—
385C2×C4—181no—
386C2×C4—188no—
387C2×C4—182no—
388C2×C4—182no—
389C2×C4—182no—
390C2×C4C3×C4×C3⋊D312yes—
391C2×C4C3×C4⋊(C3⋊D3)12yes—
392C2×C4C3×#63812yes—
393C2×C3—24no—
394C2×C3—24no—
395C2×C3—24no—
396C2×C3—24no—
397C2×C3—24no—
398C2×C3—24no—
399C2×C3—24no—
400C2×C3—184no—
401C2×C3C3×D3×C4⋊C414yes—
402C2×C3C3×#89214yes—
403C2×C3C3×#89514yes—
404C2×C3C3×#89714yes—
405C2×C3C2×C3×(C2×D3)⋊C414yes—
406C2×C3C3×#93814yes—
407C2×C3C3×#94614yes—
408C2×C3C4⋊C4×C3⋊D311yescharacteristic
409C2×C3#6410111yescharacteristic
410C2×C3#6410411yescharacteristic
411C2×C3#6410611yescharacteristic
412C2×C3C2×#375211yescharacteristic
413C2×C3#6414711yescharacteristic
414C2×C3#6415511yescharacteristic
415D3—616no—
416C22—94no—
417C22—182no—
418C22—182no—
419C22—182no—
420C22—182no—
421C22—182no—
422C22—182no—
423C22C2×C3×C4×C3⋊D311yescharacteristic
424C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
425C22C3×D4×C3⋊D311yescharacteristic
426C22C3×#383011yescharacteristic
427C22C3×Q8×C3⋊D311yescharacteristic
428C22C3×#383211yescharacteristic
429C22C2×C3×#63811yescharacteristic
430C4—42no—
431C4—182no—
432C4—362no—
433C4C3×#375214yescomplemented
434C3—24no—
435C3C3×#498714yescomplemented
436C3#1059268611yescharacteristic, direct product factor
437C2—184no—
438C2C3×C4⋊C4×C3⋊D311yescharacteristic
439C2C3×#6410111yescharacteristic
440C2C3×#6410411yescharacteristic
441C2C3×#6410611yescharacteristic
442C2C2×C3×#375211yescharacteristic
443C2C3×#6414711yescharacteristic
444C2C3×#6415511yescharacteristic
subgroupG/H
centerC23×C3C22×C3⋊D3
commutatorC22×C32C22×C3×C4
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C4⋊C4C2
socleC23×C33C23
prime radicalC23×C3×C3⋊D3C22
cosocleC23×C32C23×C3
hypercenterC2×C3×C4⋊C4C3⋊D3
hypocenterC32C3×#380
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C2×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C33×C4⋊C4C2semidirect
4. C2×C3×C4⋊(C3⋊Dic3)C2semidirect
5. C3×#63658C2semidirect
6. #10592686C3direct (N × H)
15. C2×C3×C4×C3⋊D3C4semidirect
22. C2×C32×C4⋊C4D3semidirect
23. C2×C32×C4⋊C4C2×C3semidirect
25. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
28. #63658C2×C3semidirect
65. C2×C4×C3⋊D3C3×C4semidirect
91. C2×C3×C3⋊D3C4⋊C4semidirect
93. C33×C4C22⋊C4semidirect
100. C2×C3×C4⋊C4C3×D3semidirect
101. C2×C3×C4⋊C4C3⋊D3semidirect
217. C2×C3⋊D3C3×C4⋊C4semidirect
229. C32×C4(C2×D3)⋊C4semidirect
230. C32×C4C3×C22⋊C4semidirect
237. C2×C4⋊C4C3×C3⋊D3semidirect
240. C33#380semidirect
365. C3×C4C3×(C2×D3)⋊C4semidirect
366. C3×C4#3752semidirect
370. C32#4987semidirect
371. C32C3×#380semidirect
433. C4C3×#3752semidirect
435. C3C3×#4987semidirect
436. C3#10592686direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×C33×C4⋊C42
C2×C3×#37522
C2×C3×C4⋊(C3⋊Dic3)2
C3×#636582
#105926863
C3×#49873