Finite Group Databasebeta

← C3×#27656 #10646263

Subgroups — C3×#27656 #10646263

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#484C211yesmaximal, characteristic
2C2×C3×#485C211yesmaximal, characteristic
3C3×#2323C211yesmaximal, characteristic
4C3×#2317C211yesmaximal, characteristic
5C3×#2085C211yesmaximal, characteristic, complemented
6C3×#2083C211yesmaximal, characteristic, complemented
7C3×#2074C211yesmaximal, characteristic, complemented
8C3×#2072C211yesmaximal, characteristic, complemented
9C3×#1949C211yesmaximal, characteristic, complemented
10C3×#1947C211yesmaximal, characteristic, complemented
11C3×#1825C211yesmaximal, characteristic
12C3×#1823C211yesmaximal, characteristic
13C3×#1805C211yesmaximal, characteristic, complemented
14C3×#1803C211yesmaximal, characteristic, complemented
15C3×#1798C211yesmaximal, characteristic
16#27656C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C2×C3×#117C2211yescharacteristic
21C22×C3×C4⋊C4C2211yescharacteristic
22C3×#485—24no—
23C3×#484—24no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×#107C2211yescharacteristic
26C2×C3×#107C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×C3×C22⋊C8C2211yescharacteristic
36C3×#391C2211yescharacteristic
37C3×#389—21no—
38C3×#389—21no—
39C3×#389—21no—
40C3×#389C2211yescharacteristic
41C3×#388—21no—
42C3×#388—21no—
43C3×#387—21no—
44C3×#385—21no—
45C3×#385—21no—
46C3×#384C2211yescharacteristic
47C3×#384C2211yescharacteristic
48C3×#382C2211yescharacteristic
49C3×#382C2211yescharacteristic
50C3×#380C2211yescharacteristic
51C3×C4×C4⋊C4C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60C3×#336C2211yescharacteristic
61C3×#336C2211yescharacteristic
62C2×#484C2×C311yescharacteristic
63C2×#485C2×C311yescharacteristic
64#2323C2×C311yescharacteristic
65#2317C2×C311yescharacteristic
66#2085C2×C311yescharacteristic, complemented
67#2083C2×C311yescharacteristic, complemented
68#2074C2×C311yescharacteristic, complemented
69#2072C2×C311yescharacteristic, complemented
70#1949C2×C311yescharacteristic, complemented
71#1947C2×C311yescharacteristic, complemented
72#1825C2×C311yescharacteristic
73#1823C2×C311yescharacteristic
74#1805C2×C311yescharacteristic, complemented
75#1803C2×C311yescharacteristic, complemented
76#1798C2×C311yescharacteristic
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4—21no—
79C23×C3×C4C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×C22⋊Q8—22no—
85C3×C22⋊Q8D414yes—
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×#117—22no—
89C3×#117—22no—
90C3×#117—22no—
91C3×#117—22no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4D411yescharacteristic
103C2×C3×C4⋊C4D411yescharacteristic
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2311yescharacteristic
106C2×C3×C4⋊C4C2311yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4D411yescharacteristic
115C2×C3×C22⋊C4D411yescharacteristic
116C2×C3×C42—21no—
117C2×C3×C42—21no—
118C2×C3×C42—21no—
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C3×#107—22no—
122C3×#107—22no—
123C3×#107—22no—
124C3×#106—22no—
125C3×C4⋊C8—22no—
126C3×C4⋊C8—22no—
127C3×#103—22no—
128C3×#103—22no—
129C3×#103—22no—
130C3×#103—22no—
131C3×C22⋊C8—22no—
132C3×C22⋊C8—22no—
133C3×#95—21no—
134C3×#95—21no—
135C3×#95—21no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95—21no—
141C3×#95—21no—
142C2×C22⋊Q8C22×C311yescharacteristic
143C2×C4×Q8C22×C311yescharacteristic
144C2×#117C22×C311yescharacteristic
145C2×#117C22×C311yescharacteristic
146C22×C4⋊C4C22×C311yescharacteristic
147#485—24no—
148#484—24no—
149C2×#107C22×C311yescharacteristic
150C2×#107C22×C311yescharacteristic
151C2×#107C22×C311yescharacteristic
152C2×#106C22×C311yescharacteristic
153C2×C4⋊C8C22×C311yescharacteristic
154C2×C4⋊C8C22×C311yescharacteristic
155C2×#103C22×C311yescharacteristic
156C2×#103C22×C311yescharacteristic
157C2×#103C22×C311yescharacteristic
158C2×#103C22×C311yescharacteristic
159C2×C22⋊C8C22×C311yescharacteristic
160C2×C22⋊C8C22×C311yescharacteristic
161#391C22×C311yescharacteristic
162#389—21no—
163#389—21no—
164#389—21no—
165#389C22×C311yescharacteristic
166#388—21no—
167#388—21no—
168#387—21no—
169#385—21no—
170#385—21no—
171#384C22×C311yescharacteristic
172#384C22×C311yescharacteristic
173#382C22×C311yescharacteristic
174#382C22×C311yescharacteristic
175#380C22×C311yescharacteristic
176C4×C4⋊C4C22×C311yescharacteristic
177#340C22×C311yescharacteristic
178#340C22×C311yescharacteristic
179#340C22×C311yescharacteristic
180#340C22×C311yescharacteristic
181#340C22×C311yescharacteristic
182#340C22×C311yescharacteristic
183#340C22×C311yescharacteristic
184#340C22×C311yescharacteristic
185#336C22×C311yescharacteristic
186#336C22×C311yescharacteristic
187C24×C3C2×D411yescharacteristic
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8C2×D412yes—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—42no—
207C22×C3×C4—81no—
208C22×C3×C4—81no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D412yes—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4#4112yes—
216C22×C3×C4C2411yescharacteristic
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C3×C4⋊C4—24no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—81no—
248C3×C4⋊C4—81no—
249C3×C4⋊C4—81no—
250C3×C4⋊C4—81no—
251C3×C4⋊C4—81no—
252C3×C4⋊C4C2×D412yes—
253C3×C22⋊C4—24no—
254C3×C22⋊C4—22no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—42no—
257C3×C22⋊C4—81no—
258C3×C42—21no—
259C3×C42—21no—
260C3×C42—42no—
261C3×C42—42no—
262C3×C42—81no—
263C3×C42#4112yes—
264C3×C42#4112yes—
265C22×Q8C23×C311yescharacteristic
266C23×C4—21no—
267C23×C4C23×C311yescharacteristic
268C22×C8C23×C311yescharacteristic
269C22×C8C23×C311yescharacteristic
270C22×C8C23×C311yescharacteristic
271C22×C8C23×C311yescharacteristic
272C22⋊Q8—22no—
273C22⋊Q8C3×D414yes—
274C4×Q8—22no—
275C4×Q8—22no—
276#117—22no—
277#117—22no—
278#117—22no—
279#117—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4C3×D411yescharacteristic
291C2×C4⋊C4C3×D411yescharacteristic
292C2×C4⋊C4C23×C311yescharacteristic
293C2×C4⋊C4C23×C311yescharacteristic
294C2×C4⋊C4C23×C311yescharacteristic
295C2×C4⋊C4C23×C311yescharacteristic
296C2×C4⋊C4C23×C311yescharacteristic
297C2×C4⋊C4C23×C311yescharacteristic
298C2×C4⋊C4C23×C311yescharacteristic
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4C3×D411yescharacteristic
303C2×C22⋊C4C3×D411yescharacteristic
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42C23×C311yescharacteristic
308C2×C42C23×C311yescharacteristic
309#107—22no—
310#107—22no—
311#107—22no—
312#106—22no—
313C4⋊C8—22no—
314C4⋊C8—22no—
315#103—22no—
316#103—22no—
317#103—22no—
318#103—22no—
319C22⋊C8—22no—
320C22⋊C8—22no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3#12112yes—
337C23×C3C22×D411yescharacteristic
338C3×Q8—22no—
339C3×Q8—22no—
340C3×Q8—41no—
341C3×Q8—41no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—84no—
383C2×C3×C4#12112yes—
384C2×C3×C4#12312yes—
385C2×C3×C4#12312yes—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C42−1+411yescharacteristic
391C2×C3×C42−1+411yescharacteristic
392C3×C8—41no—
393C3×C8—41no—
394C3×C8—41no—
395C3×C8—41no—
396C24C2×C3×D411yescharacteristic
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8C2×C3×D412yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—42no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4C2×C3×D411yescharacteristic
419C22×C4C2×C3×D411yescharacteristic
420C22×C4C2×C3×D411yescharacteristic
421C22×C4C2×C3×D412yes—
422C22×C4C2×C3×D411yescharacteristic
423C22×C4C2×C3×D411yescharacteristic
424C22×C4C3×#4112yes—
425C22×C4C24×C311yescharacteristic
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4C2×C3×D412yes—
462C22⋊C4—24no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—42no—
466C22⋊C4—81no—
467C42—21no—
468C42—21no—
469C42—42no—
470C42—42no—
471C42—81no—
472C42C3×#4112yes—
473C42C3×#4112yes—
474C22×C3—22no—
475C22×C3—22no—
476C22×C3—22no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3—41no—
480C22×C3—41no—
481C22×C3C2×#12111yescharacteristic
482C22×C3C2×#12311yescharacteristic
483C22×C3#53611yescharacteristic
484C22×C3#57611yescharacteristic
485C22×C3#57711yescharacteristic
486C22×C3#57811yescharacteristic
487C22×C3#57811yescharacteristic
488C3×C4—22no—
489C3×C4—22no—
490C3×C4—22no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—42no—
494C3×C4—42no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4#54711yescharacteristic
502C3×C4#54711yescharacteristic
503C3×C4#55211yescharacteristic
504C3×C4#55211yescharacteristic
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C3×#12112yes—
512C23C22×C3×D411yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—41no—
516Q8—41no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—84no—
558C2×C4C3×#12112yes—
559C2×C4C3×#12312yes—
560C2×C4C3×#12312yes—
561C2×C4C22×C3×D411yescharacteristic
562C2×C4C2×C3×#4111yescharacteristic
563C2×C4C2×C3×#4111yescharacteristic
564C2×C4C2×C3×#4111yescharacteristic
565C2×C4C3×2−1+411yescharacteristic
566C2×C4C3×2−1+411yescharacteristic
567C8—41no—
568C8—41no—
569C8—41no—
570C8—41no—
571C2×C3—42no—
572C2×C3#248611yescharacteristic
573C2×C3#304411yescharacteristic
574C2×C3#304511yescharacteristic
575C2×C3#309711yescharacteristic
576C2×C3#309811yescharacteristic
577C2×C3#324211yescharacteristic
578C2×C3#324311yescharacteristic
579C22—22no—
580C22—22no—
581C22—22no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22C2×C3×#12111yescharacteristic
587C22C2×C3×#12311yescharacteristic
588C22C3×#53611yescharacteristic
589C22C3×#57611yescharacteristic
590C22C3×#57711yescharacteristic
591C22C3×#57811yescharacteristic
592C22C3×#57811yescharacteristic
593C4—22no—
594C4—22no—
595C4—22no—
596C4—42no—
597C4—42no—
598C4—41no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4C3×#54711yescharacteristic
607C4C3×#54711yescharacteristic
608C4C3×#55211yescharacteristic
609C4C3×#55211yescharacteristic
610C3#2765611yescharacteristic, direct product factor, Sylow
611C2—42no—
612C2C3×#248611yescharacteristic
613C2C3×#304411yescharacteristic
614C2C3×#304511yescharacteristic
615C2C3×#309711yescharacteristic
616C2C3×#309811yescharacteristic
617C2C3×#324211yescharacteristic
618C2C3×#324311yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
5. C3×#2085C2semidirect
6. C3×#2083C2semidirect
7. C3×#2074C2semidirect
8. C3×#2072C2semidirect
9. C3×#1949C2semidirect
10. C3×#1947C2semidirect
13. C3×#1805C2semidirect
14. C3×#1803C2semidirect
16. #27656C3direct (N × H)
66. #2085C2×C3semidirect
67. #2083C2×C3semidirect
68. #2074C2×C3semidirect
69. #2072C2×C3semidirect
70. #1949C2×C3semidirect
71. #1947C2×C3semidirect
74. #1805C2×C3semidirect
75. #1803C2×C3semidirect
610. C3#27656direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4842
C2×C3×#4852
C3×#23232
C3×#23172
C3×#20852
C3×#20832
C3×#20742
C3×#20722
C3×#19492
C3×#19472
C3×#18252
C3×#18232
C3×#18052
C3×#18032
C3×#17982
#276563