Finite Group Databasebeta

← C3×#10591396 #13158408

Subgroups — C3×#10591396 #13158408

Order 1728.

1058 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#627C211yesmaximal, characteristic
2C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
3C2×C3×#3724C211yesmaximal, characteristic
4C32×C3⋊#95C212yesmaximal
5C3×#63579C212yesmaximal
6C3×#4646—32nomaximal
7#10591396C311yesmaximal, characteristic, direct product factor
8C22×C33×C4C2211yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C22×C3×C3⋊Dic3C2211yescharacteristic
11C22×C32×Dic3C2212yes—
12C22×C32×Dic3C2212yes—
13C2×C3×#627—22no—
14C2×C3×#627—22no—
15C2×C3×#627C414yescomplemented
16C3×C4×C3⋊Dic3—22no—
17C3×#3724—22no—
18C22×C3×Dic6—32no—
19C22×#627C2×C311yescharacteristic
20C2×C4×C3⋊Dic3C2×C311yescharacteristic
21C2×C3×Dic3⋊C4—32no—
22C2×C3×C4×Dic3—32no—
23C2×C3×C4×Dic3—61no—
24C2×#3724C2×C311yescharacteristic
25C32×#95—32no—
26C3×C3⋊#95—22no—
27C3×C3⋊#95—32no—
28C3×C3⋊#95—32no—
29C3×C3⋊#95D312yes—
30C3×C3⋊#95C2×C312yes—
31#63579C2×C312yes—
32C23×C33C2311yescharacteristic
33C2×C33×C4—21no—
34C2×C33×C4—21no—
35C2×C33×C4Q812yescomplemented
36C2×C3×C3⋊Dic3—21no—
37C2×C3×C3⋊Dic3—21no—
38C2×C3×C3⋊Dic3C2×C412yes—
39C2×C3×C3⋊Dic3D412yes—
40C2×C3×C3⋊Dic3D412yes—
41C2×C3×C3⋊Dic3Q812yescomplemented
42C2×C32×Dic3—22no—
43C2×C32×Dic3—22no—
44C2×C32×Dic3—22no—
45C2×C32×Dic3—22no—
46C2×C32×Dic3—22no—
47C2×C32×Dic3C2×C414yes—
48C3×#627—28no—
49C3×#391—91no—
50#4646—32no—
51C22×C32×C4—21no—
52C22×C32×C4—22no—
53C22×C32×C4—32no—
54C22×C32×C4—32no—
55C22×C32×C4—41no—
56C22×C32×C4C2×D312yes—
57C22×C32×C4C22×C311yescharacteristic
58C22×C3⋊Dic3C22×C311yescharacteristic
59C22×C3⋊Dic3C22×C311yescharacteristic
60C22×C3×Dic3—22no—
61C22×C3×Dic3—22no—
62C22×C3×Dic3—32no—
63C22×C3×Dic3—32no—
64C22×C3×Dic3—61no—
65C22×C3×Dic3—61no—
66C22×C3×Dic3C2×D312yes—
67C22×C3×Dic3C2×D312yes—
68C22×C3×Dic3C22×C312yes—
69C22×C3×Dic3C22×C312yes—
70C2×C3×Dic6—38no—
71C2×C3×Dic6—64no—
72C2×C3×Dic6—64no—
73C2×#627—22no—
74C2×#627—22no—
75C2×#627C3×C414yescomplemented
76C4×C3⋊Dic3—22no—
77C3×Dic3⋊C4—64no—
78C3×C4×Dic3—64no—
79C3×C4×Dic3—64no—
80#3724—22no—
81C22×C33C22×C411yescharacteristic
82C22×C33C2×D411yescharacteristic
83C22×C33C2×D411yescharacteristic
84C22×C33C2×Q811yescharacteristic
85C22×C33C2×Q811yescharacteristic
86C22×C33#4111yescharacteristic
87C22×C33#4111yescharacteristic
88C33×C4—22no—
89C3×C3⋊Dic3—22no—
90C3×C3⋊Dic3C22⋊C414yes—
91C32×Dic3—24no—
92C32×Dic3—42no—
93C22×C3×Q8—91no—
94C22×Dic6—32no—
95C2×C3×C4⋊C4—91no—
96C2×C3×C42—91no—
97C2×Dic3⋊C4—32no—
98C2×C4×Dic3—32no—
99C2×C4×Dic3—61no—
100C3×#95—32no—
101C3×#95—32no—
102C3×#95—62no—
103C3×#95—92no—
104C3⋊#95—32no—
105C3⋊#95—32no—
106C3⋊#95C3×D312yes—
107C23×C32—21no—
108C23×C32—22no—
109C23×C32—41no—
110C23×C32C22×D312yes—
111C23×C32C23×C311yescharacteristic
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—24no—
120C2×C32×C4—34no—
121C2×C32×C4—42no—
122C2×C32×C4—42no—
123C2×C32×C4—42no—
124C2×C32×C4—42no—
125C2×C32×C4—42no—
126C2×C32×C4—62no—
127C2×C32×C4—62no—
128C2×C32×C4—62no—
129C2×C32×C4—62no—
130C2×C32×C4—62no—
131C2×C32×C4Dic614yescomplemented
132C2×C32×C4C3×Q812yescomplemented
133C2×C3⋊Dic3—21no—
134C2×C3⋊Dic3—21no—
135C2×C3⋊Dic3C2×C3×C412yes—
136C2×C3⋊Dic3C3×D412yes—
137C2×C3⋊Dic3C3×D412yes—
138C2×C3⋊Dic3C3×Q812yescomplemented
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—24no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—24no—
151C2×C3×Dic3—34no—
152C2×C3×Dic3—34no—
153C2×C3×Dic3—34no—
154C2×C3×Dic3—34no—
155C2×C3×Dic3—42no—
156C2×C3×Dic3—42no—
157C2×C3×Dic3—42no—
158C2×C3×Dic3—42no—
159C2×C3×Dic3—62no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—62no—
163C2×C3×Dic3—62no—
164C2×C3×Dic3—62no—
165C2×C3×Dic3—62no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3C4×D314yes—
168C2×C3×Dic3C2×C3×C414yes—
169C3×Dic6—616no—
170#627—28no—
171#391—91noSylow
172C2×C33C2×C22⋊C411yescharacteristic
173C2×C33C4×Q811yescharacteristic
174C2×C33C4×Q811yescharacteristic
175C2×C33C22⋊Q811yescharacteristic
176C2×C33C22⋊Q811yescharacteristic
177C2×C33#12411yescharacteristic
178C2×C33C4⋊Q811yescharacteristic
179C2×C3×Q8—94no—
180C2×C3×Q8—182no—
181C2×C3×Q8—182no—
182C22×C3×C4—21no—
183C22×C3×C4—22no—
184C22×C3×C4—32no—
185C22×C3×C4—32no—
186C22×C3×C4—32no—
187C22×C3×C4—32no—
188C22×C3×C4—41no—
189C22×C3×C4—62no—
190C22×C3×C4—62no—
191C22×C3×C4—91no—
192C22×C3×C4—91no—
193C22×C3×C4D3211yescharacteristic
194C22×C3×C4C2×C3×D312yes—
195C22×Dic3—32no—
196C22×Dic3—32no—
197C22×Dic3—61no—
198C22×Dic3—61no—
199C22×Dic3C2×C3×D312yes—
200C22×Dic3C2×C3×D312yes—
201C2×Dic6—38no—
202C2×Dic6—64no—
203C2×Dic6—64no—
204C3×C4⋊C4—182no—
205C3×C42—182no—
206Dic3⋊C4—64no—
207C4×Dic3—64no—
208C4×Dic3—64no—
209C22×C32—21no—
210C22×C32—21no—
211C22×C32—21no—
212C22×C32—21no—
213C22×C32—21no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—22no—
221C22×C32—22no—
222C22×C32—22no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—41no—
226C22×C32—41no—
227C22×C32—41no—
228C22×C32—41no—
229C22×C32—41no—
230C22×C32C2×Dic612yes—
231C22×C32C2×C4×D312yes—
232C22×C32#23512yes—
233C22×C32D3×D412yes—
234C22×C32D3×D412yes—
235C22×C32#23712yes—
236C22×C32D3×Q812yes—
237C22×C32C22×C3×C411yescharacteristic
238C22×C32C2×C3×D411yescharacteristic
239C22×C32C2×C3×D411yescharacteristic
240C22×C32C2×C3×Q811yescharacteristic
241C22×C32C2×C3×Q811yescharacteristic
242C22×C32C3×#4111yescharacteristic
243C22×C32C3×#4111yescharacteristic
244C32×C4—22no—
245C32×C4—24no—
246C32×C4—24no—
247C32×C4—44no—
248C32×C4—44no—
249C32×C4—64no—
250C32×C4—122no—
251C3⋊Dic3—22no—
252C3⋊Dic3C3×C22⋊C414yes—
253C3×Dic3—24no—
254C3×Dic3—24no—
255C3×Dic3—38no—
256C3×Dic3—44no—
257C3×Dic3—44no—
258C3×Dic3—42no—
259C3×Dic3—42no—
260C3×Dic3—64no—
261C3×Dic3—64no—
262C3×Dic3—64no—
263C22×Q8—91no—
264C2×C4⋊C4—91no—
265C2×C42—91no—
266#95—32no—
267#95—92no—
268C33#39111yescharacteristic, complemented, Sylow
269C23×C3—22no—
270C23×C3—21no—
271C23×C3—41no—
272C23×C3C2×D3211yescharacteristic
273C23×C3C22×C3×D312yes—
274C3×Q8—188no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—24no—
283C2×C3×C4—34no—
284C2×C3×C4—34no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—64no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—62no—
295C2×C3×C4—64no—
296C2×C3×C4—62no—
297C2×C3×C4—62no—
298C2×C3×C4—62no—
299C2×C3×C4—62no—
300C2×C3×C4—62no—
301C2×C3×C4—62no—
302C2×C3×C4—92no—
303C2×C3×C4—92no—
304C2×C3×C4—92no—
305C2×C3×C4—92no—
306C2×C3×C4—122no—
307C2×C3×C4—122no—
308C2×C3×C4—122no—
309C2×C3×C4—122no—
310C2×C3×C4—181no—
311C2×C3×C4—181no—
312C2×C3×C4#62712yescomplemented
313C2×C3×C4C3×Dic614yescomplemented
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—34no—
320C2×Dic3—34no—
321C2×Dic3—34no—
322C2×Dic3—34no—
323C2×Dic3—62no—
324C2×Dic3—62no—
325C2×Dic3—62no—
326C2×Dic3—62no—
327C2×Dic3—62no—
328C2×Dic3—62no—
329C2×Dic3—62no—
330C2×Dic3—62no—
331C2×Dic3C3×C4×D314yes—
332Dic6—616no—
333C2×C32—22no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—22no—
337C2×C32—22no—
338C2×C32—22no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—22no—
347C2×C32—41no—
348C2×C32—41no—
349C2×C32—41no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32—41no—
353C2×C32—41no—
354C2×C32C4×Dic612yes—
355C2×C32#87812yes—
356C2×C32D3×C22⋊C412yes—
357C2×C32#88512yes—
358C2×C32#88712yes—
359C2×C32#88812yes—
360C2×C32#89612yes—
361C2×C32C2×C3×C22⋊C411yescharacteristic
362C2×C32C3×C4×Q811yescharacteristic
363C2×C32C3×C4×Q811yescharacteristic
364C2×C32C3×C22⋊Q811yescharacteristic
365C2×C32C3×C22⋊Q811yescharacteristic
366C2×C32C3×#12411yescharacteristic
367C2×C32C3×C4⋊Q811yescharacteristic
368C2×Q8—94no—
369C2×Q8—182no—
370C2×Q8—182no—
371C22×C4—32no—
372C22×C4—32no—
373C22×C4—91no—
374C22×C4—91no—
375C22×C4C3×D3211yescharacteristic
376C4⋊C4—182no—
377C42—182no—
378C22×C3—21no—
379C22×C3—22no—
380C22×C3—21no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—22no—
384C22×C3—21no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3—22no—
388C22×C3—22no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—22no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3#379711yescharacteristic
400C22×C3#379811yescharacteristic
401C22×C3C4×D3211yescharacteristic
402C22×C3#380211yescharacteristic
403C22×C3#380511yescharacteristic
404C22×C3C2×#62711yescharacteristic
405C22×C3#381111yescharacteristic
406C22×C3C2×C3×Dic612yes—
407C22×C3C2×C3×C4×D312yes—
408C22×C3C3×#23512yes—
409C22×C3C3×D3×D412yes—
410C22×C3C3×D3×D412yes—
411C22×C3C3×#23712yes—
412C22×C3C3×D3×Q812yes—
413C3×C4—24no—
414C3×C4—24no—
415C3×C4—22no—
416C3×C4—44no—
417C3×C4—44no—
418C3×C4—64no—
419C3×C4—64no—
420C3×C4—94no—
421C3×C4—124no—
422C3×C4—124no—
423C3×C4—122no—
424C3×C4—122no—
425C3×C4—182no—
426Dic3—24no—
427Dic3—38no—
428Dic3—42no—
429Dic3—64no—
430Dic3—64no—
431Dic3—64no—
432C32—21no—
433C32—22no—
434C32—41no—
435C32#464612yescomplemented
436C32C3×#39111yescharacteristic, complemented
437C23C2×C3×D3211yescharacteristic
438Q8—188no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—34no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—92no—
448C2×C4—92no—
449C2×C4—92no—
450C2×C4—92no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4C3×#62712yescomplemented
454C2×C3—21no—
455C2×C3—22no—
456C2×C3—22no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—22no—
460C2×C3—22no—
461C2×C3—21no—
462C2×C3—22no—
463C2×C3—21no—
464C2×C3—22no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—22no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3—41no—
471C2×C3—41no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3#6383811yescharacteristic
476C2×C3#6386511yescharacteristic
477C2×C3#6391511yescharacteristic
478C2×C3C4×#62711yescharacteristic
479C2×C3#6391911yescharacteristic
480C2×C3#6392111yescharacteristic
481C2×C3#6398211yescharacteristic
482C2×C3C3×C4×Dic612yes—
483C2×C3C3×#87812yes—
484C2×C3C3×D3×C22⋊C412yes—
485C2×C3C3×#88512yes—
486C2×C3C3×#88712yes—
487C2×C3C3×#88812yes—
488C2×C3C3×#89612yes—
489C22C3×#379711yescharacteristic
490C22C3×#379811yescharacteristic
491C22C3×C4×D3211yescharacteristic
492C22C3×#380211yescharacteristic
493C22C3×#380511yescharacteristic
494C22C2×C3×#62711yescharacteristic
495C22C3×#381111yescharacteristic
496C4—22no—
497C4—64no—
498C4—94no—
499C4—122no—
500C4—182no—
501C3—22no—
502C3—21no—
503C3—41no—
504C3#1059139611yescharacteristic, direct product factor
505C3C3×#464612yescomplemented
506C2C3×#6383811yescharacteristic
507C2C3×#6386511yescharacteristic
508C2C3×#6391511yescharacteristic
509C2C3×C4×#62711yescharacteristic
510C2C3×#6391911yescharacteristic
511C2C3×#6392111yescharacteristic
512C2C3×#6398211yescharacteristic
subgroupG/H
centerC23×C3C2×D32
commutatorC22×C32C22×C3×C4
FrattiniC23C2×C3×D32
FittingC22×C33×C4C22
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC22×C3×C4D32
hypocenterC32C3×#391
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23×C3 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
7. #10591396C3direct (N × H)
15. C2×C3×#627C4semidirect
35. C2×C33×C4Q8semidirect
41. C2×C3×C3⋊Dic3Q8semidirect
75. C2×#627C3×C4semidirect
131. C2×C32×C4Dic6semidirect
132. C2×C32×C4C3×Q8semidirect
138. C2×C3⋊Dic3C3×Q8semidirect
268. C33#391semidirect
312. C2×C3×C4#627semidirect
313. C2×C3×C4C3×Dic6semidirect
435. C32#4646semidirect
436. C32C3×#391semidirect
453. C2×C4C3×#627semidirect
504. C3#10591396direct (N × H)
505. C3C3×#4646semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#6272
C2×C3×C4×C3⋊Dic32
C2×C3×#37242
C32×C3⋊#952
C3×#635792
C3×#46463
#105913963