Finite Group Databasebeta

← C3×#70304 #11788656

Subgroups — C3×#70304 #11788656

Order 1152.

705 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#951C211yesmaximal, characteristic
2C2×C3×#948C211yesmaximal, characteristic
3C32×#417C211yesmaximal, characteristic
4C2×C3×#824C211yesmaximal, characteristic, complemented
5C2×C3×#810C211yesmaximal, characteristic, complemented
6C3×#4558C211yesmaximal, characteristic, complemented
7C3×#4530C211yesmaximal, characteristic, complemented
8C3×#1908—31nomaximal
9#70304C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C32×C4⋊C4C2211yescharacteristic
14C3×#951—22no—
15C3×#951C414yescomplemented
16C3×#948—22no—
17C3×#948—22no—
18C2×C3×#216—21no—
19C2×C3×#214—21no—
20C2×C3×#208—21no—
21C2×C3×#208C2211yescharacteristic
22C2×C3×#208C2211yescharacteristic
23C22×C3×C3⋊C8—21no—
24C22×C3×C3⋊C8C2211yescharacteristic
25C32×#103—21no—
26C32×#102—21no—
27C3×#824—22no—
28C3×#810—21no—
29C3×#810—21no—
30C2×C3×#137—31no—
31C2×C3×#131—31no—
32C2×#951C2×C311yescharacteristic
33C2×#948C2×C311yescharacteristic
34C3×#417—21no—
35C3×#417D311yescharacteristic
36C3×#417C2×C311yescharacteristic
37C2×C3×#103—31no—
38C2×C3×#101—31no—
39C2×#824C2×C311yescharacteristic, complemented
40C2×#810C2×C311yescharacteristic, complemented
41C3×#343—31no—
42C3×#341—31no—
43#4558C2×C311yescharacteristic, complemented
44#4530C2×C311yescharacteristic, complemented
45C32×#41—21no—
46C32×#41—21no—
47C32×#41C2×C412yes—
48C32×#41D412yes—
49C2×C32×Q8D411yescharacteristic
50C2×C32×D4—21no—
51C2×C32×D4D411yescharacteristic
52C22×C32×C4—21no—
53C22×C32×C4—21no—
54C22×C32×C4C2311yescharacteristic
55C22×C3×Dic3—21no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6—21no—
58C2×C3×Dic6—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6C2×C412yes—
62C32×M16—21no—
63C32×M16—21no—
64C2×C32×C8—21no—
65C32×C4⋊C4—21no—
66C32×C4⋊C4—21no—
67C3×#216—24no—
68C3×#214—24no—
69C3×#208—24no—
70C3×#208—21no—
71C3×#208—21no—
72C3×#208C2×C412yes—
73C3×#208D412yes—
74C2×C3×C3⋊C8—24no—
75C2×C3×C3⋊C8—21no—
76C2×C3×C3⋊C8—21no—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—21no—
79C2×C3×C3⋊C8D411yescharacteristic
80C2×C3×C3⋊C8D411yescharacteristic
81#1908—31noSylow
82C2×C3×#41—21no—
83C2×C3×#41C2×D311yescharacteristic
84C2×C3×#41C22×C311yescharacteristic
85C22×C3×Q8—31no—
86C22×Dic6C22×C311yescharacteristic
87C3×#137—34no—
88C3×#137—62no—
89C2×C3×Q16—61no—
90C2×C3×SD16—61no—
91C3×#131—62no—
92C3×#131—62no—
93C2×C3×M16—21no—
94C2×C3×M16—31no—
95C2×C3×M16—31no—
96C2×C3×M16—61no—
97C2×C3×M16C2×D311yescharacteristic
98C2×C3×M16C22×C311yescharacteristic
99C22×C3×C8—31no—
100C22×C3×C8—61no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4C2×D311yescharacteristic
103C2×C3×C4⋊C4C22×C311yescharacteristic
104#951—22no—
105#951C3×C414yescomplemented
106#948—22no—
107#948—22no—
108C2×#216—21no—
109C2×#214—21no—
110C2×#208—21no—
111C2×#208C22×C311yescharacteristic
112C2×#208C22×C311yescharacteristic
113C22×C3⋊C8—21no—
114C22×C3⋊C8C22×C311yescharacteristic
115C3×#103—22no—
116C3×#103—21no—
117C3×#103—21no—
118C3×#103—61no—
119C3×#103—61no—
120C3×#102—22no—
121C3×#102—21no—
122C3×#102—21no—
123C3×#101—62no—
124#824—22no—
125#810—21no—
126#810—21no—
127C23×C32—21no—
128C23×C32#4111yescharacteristic
129C32×Q8—21no—
130C32×Q8C22⋊C412yes—
131C32×D4—22no—
132C32×D4—22no—
133C32×D4—21no—
134C32×D4C22⋊C412yes—
135C2×C32×C4—21no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4—41no—
139C2×C32×C4—81no—
140C2×C32×C4C22×C411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4#4111yescharacteristic
146C2×C3×Dic3—22no—
147C2×C3×Dic3—41no—
148C2×C3×Dic3—41no—
149C3×Dic6—22no—
150C3×Dic6—22no—
151C3×Dic6—41no—
152C3×Dic6—41no—
153C32×C8—41no—
154C3×C3⋊C8—22no—
155C3×C3⋊C8—21no—
156C3×C3⋊C8—22no—
157C3×C3⋊C8—21no—
158C2×#137—31no—
159C2×#131—31no—
160#417C3×D311yescharacteristic
161C2×#103—31no—
162C2×#101—31no—
163#343—31no—
164#341—31no—
165C3×#41—21no—
166C3×#41—21no—
167C3×#41—21no—
168C3×#41—21no—
169C3×#41—22no—
170C3×#41—22no—
171C3×#41—41no—
172C3×#41—41no—
173C3×#41C4×D312yes—
174C3×#41D1212yes—
175C3×#41C2×C3×C412yes—
176C3×#41C3×D412yes—
177C2×C3×Q8—21no—
178C2×C3×Q8—32no—
179C2×C3×Q8—61no—
180C2×C3×Q8—61no—
181C2×C3×Q8—61no—
182C2×C3×Q8—61no—
183C2×C3×Q8—61no—
184C2×C3×Q8#6711yescharacteristic
185C2×C3×Q8C3×D411yescharacteristic
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—41no—
190C2×C3×D4#6711yescharacteristic
191C2×C3×D4C3×D411yescharacteristic
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—41no—
198C22×C3×C4—41no—
199C22×C3×C4—61no—
200C22×C3×C4C22×D311yescharacteristic
201C22×C3×C4C23×C311yescharacteristic
202C22×Dic3—21no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6—21no—
208C2×Dic6C2×C3×C412yes—
209C3×Q16—64no—
210C3×SD16—64no—
211C3×M16—21no—
212C3×M16—21no—
213C3×M16—21no—
214C3×M16—21no—
215C3×M16—32no—
216C3×M16—32no—
217C3×M16—41no—
218C3×M16—41no—
219C3×M16—64no—
220C3×M16—61no—
221C3×M16—61no—
222C2×C3×C8—22no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8—64no—
228C2×C3×C8—61no—
229C2×C3×C8—61no—
230C2×C3×C8—61no—
231C2×C3×C8—61no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—41no—
238#216—24no—
239#214—24no—
240#208—24no—
241#208—21no—
242#208—21no—
243#208C2×C3×C412yes—
244#208C3×D412yes—
245C2×C3⋊C8—24no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8C3×D411yescharacteristic
251C2×C3⋊C8C3×D411yescharacteristic
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—22no—
255C22×C32—41no—
256C22×C32—41no—
257C22×C32C4×D411yescharacteristic
258C22×C32#12111yescharacteristic
259C22×C32#12311yescharacteristic
260C32×C4—22no—
261C32×C4—81no—
262C32×C4C2×C22⋊C411yescharacteristic
263C32×C4C4×D411yescharacteristic
264C32×C4#12011yescharacteristic
265C32×C4#12111yescharacteristic
266C3×Dic3—42no—
267C2×#41C2×C3×D311yescharacteristic
268C22×Q8—31no—
269#137—34no—
270#137—62no—
271C2×Q16—61no—
272C2×SD16—61no—
273#131—62no—
274#131—62no—
275C2×M16—31no—
276C2×M16—31no—
277C2×M16—61no—
278C2×M16C2×C3×D311yescharacteristic
279C22×C8—31no—
280C22×C8—61no—
281C2×C4⋊C4C2×C3×D311yescharacteristic
282#103—21no—
283#103—61no—
284#103—61no—
285#102—21no—
286#101—62no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—21no—
290C23×C3—41no—
291C23×C3#23711yescharacteristic
292C23×C3C3×#4111yescharacteristic
293C3×Q8—21no—
294C3×Q8—21no—
295C3×Q8—22no—
296C3×Q8—41no—
297C3×Q8—62no—
298C3×Q8—62no—
299C3×Q8—121no—
300C3×Q8—121no—
301C3×Q8(C2×D3)⋊C412yes—
302C3×Q8C3×C22⋊C412yes—
303C3×D4—22no—
304C3×D4—22no—
305C3×D4—22no—
306C3×D4—22no—
307C3×D4—21no—
308C3×D4—21no—
309C3×D4—22no—
310C3×D4—41no—
311C3×D4—42no—
312C3×D4—42no—
313C3×D4(C2×D3)⋊C412yes—
314C3×D4C3×C22⋊C412yes—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—62no—
334C2×C3×C4—82no—
335C2×C3×C4—81no—
336C2×C3×C4—81no—
337C2×C3×C4—121no—
338C2×C3×C4—121no—
339C2×C3×C4C2×C4×D311yescharacteristic
340C2×C3×C4C2×D1211yescharacteristic
341C2×C3×C4D3×D411yescharacteristic
342C2×C3×C4D3×D411yescharacteristic
343C2×C3×C4#23711yescharacteristic
344C2×C3×C4C2×#6711yescharacteristic
345C2×C3×C4C22×C3×C411yescharacteristic
346C2×C3×C4C2×C3×D411yescharacteristic
347C2×C3×C4C2×C3×D411yescharacteristic
348C2×C3×C4C2×C3×D411yescharacteristic
349C2×C3×C4C2×C3×D411yescharacteristic
350C2×C3×C4C3×#4111yescharacteristic
351C2×Dic3—22no—
352C2×Dic3—41no—
353C2×Dic3—41no—
354Dic6—22no—
355Dic6—22no—
356Dic6—41no—
357Dic6—41no—
358C3×C8—42no—
359C3×C8—41no—
360C3×C8—41no—
361C3×C8—61no—
362C3×C8—61no—
363C3×C8—62no—
364C3×C8—62no—
365C3⋊C8—22no—
366C3⋊C8—21no—
367C3⋊C8—22no—
368C3⋊C8—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—42no—
372C2×C32#38611yescharacteristic
373C2×C32#47111yescharacteristic
374C2×C32#47311yescharacteristic
375#41—21no—
376#41—21no—
377#41C3×C4×D312yes—
378#41C3×D1212yes—
379C2×Q8—32no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×Q8C3×#6711yescharacteristic
386C2×D4—21no—
387C2×D4C3×#6711yescharacteristic
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—61no—
391C22×C4C22×C3×D311yescharacteristic
392Q16—64no—
393SD16—64no—
394M16—21no—
395M16—21no—
396M16—32no—
397M16—32no—
398M16—64no—
399M16—61no—
400M16—61no—
401C2×C8—21no—
402C2×C8—31no—
403C2×C8—31no—
404C2×C8—64no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C2×C8—61no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—22no—
416C22×C3—22no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—42no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—81no—
428C22×C3—81no—
429C22×C3#67⋊C411yescharacteristic
430C22×C3#88611yescharacteristic
431C22×C3#93911yescharacteristic
432C22×C3C3×C4×D411yescharacteristic
433C22×C3C3×#12111yescharacteristic
434C22×C3C3×#12311yescharacteristic
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—42no—
442C3×C4—82no—
443C3×C4—81no—
444C3×C4—81no—
445C3×C4—122no—
446C3×C4#67⋊C411yescharacteristic
447C3×C4#88211yescharacteristic
448C3×C4C2×(C2×D3)⋊C411yescharacteristic
449C3×C4#93911yescharacteristic
450C3×C4C2×C3×C22⋊C411yescharacteristic
451C3×C4C3×C4×D411yescharacteristic
452C3×C4C3×#12011yescharacteristic
453C3×C4C3×#12111yescharacteristic
454Dic3—42no—
455C32#190811yescharacteristic, complemented, Sylow
456C23—21no—
457C23C3×#23711yescharacteristic
458Q8—21no—
459Q8—62no—
460Q8—62no—
461Q8—121no—
462Q8—121no—
463Q8C3×(C2×D3)⋊C412yes—
464D4—22no—
465D4—22no—
466D4—21no—
467D4C3×(C2×D3)⋊C412yes—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—62no—
473C2×C4—81no—
474C2×C4—121no—
475C2×C4—121no—
476C2×C4C2×C3×C4×D311yescharacteristic
477C2×C4C2×C3×D1211yescharacteristic
478C2×C4C3×D3×D411yescharacteristic
479C2×C4C3×D3×D411yescharacteristic
480C2×C4C3×#23711yescharacteristic
481C2×C4C2×C3×#6711yescharacteristic
482C8—41no—
483C8—61no—
484C8—61no—
485C8—62no—
486C8—62no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—41no—
495C2×C3—41no—
496C2×C3—42no—
497C2×C3—42no—
498C2×C3—82no—
499C2×C3#495811yescharacteristic
500C2×C3#520011yescharacteristic
501C2×C3#520411yescharacteristic
502C2×C3C3×#38611yescharacteristic
503C2×C3C3×#47111yescharacteristic
504C2×C3C3×#47311yescharacteristic
505C22—21no—
506C22—21no—
507C22—22no—
508C22—41no—
509C22—41no—
510C22C3×#67⋊C411yescharacteristic
511C22C3×#88611yescharacteristic
512C22C3×#93911yescharacteristic
513C4—22no—
514C4—81no—
515C4—122no—
516C4C3×#67⋊C411yescharacteristic
517C4C3×#88211yescharacteristic
518C4C2×C3×(C2×D3)⋊C411yescharacteristic
519C4C3×#93911yescharacteristic
520C3—21no—
521C3#7030411yescharacteristic, direct product factor
522C3C3×#190811yescharacteristic, complemented
523C2—21no—
524C2—21no—
525C2—42no—
526C2C3×#495811yescharacteristic
527C2C3×#520011yescharacteristic
528C2C3×#520411yescharacteristic
subgroupG/H
centerC22×C3#939
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#417C2
socleC22×C32#121
prime radicalC2×C32×#41C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#417D3
hypocenterC3C3×#1908
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#417

Complements

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

NH ≅ G/Ntype
4. C2×C3×#824C2semidirect
5. C2×C3×#810C2semidirect
6. C3×#4558C2semidirect
7. C3×#4530C2semidirect
9. #70304C3direct (N × H)
15. C3×#951C4semidirect
39. C2×#824C2×C3semidirect
40. C2×#810C2×C3semidirect
43. #4558C2×C3semidirect
44. #4530C2×C3semidirect
105. #951C3×C4semidirect
455. C32#1908semidirect
521. C3#70304direct (N × H)
522. C3C3×#1908semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9512
C2×C3×#9482
C32×#4172
C2×C3×#8242
C2×C3×#8102
C3×#45582
C3×#45302
C3×#19083
#703043