Finite Group Databasebeta

← C3×#27765 #10646372

Subgroups — C3×#27765 #10646372

Order 768.

736 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2552C211yesmaximal, characteristic, complemented
2C3×#2545C211yesmaximal, characteristic
3C3×#2072C211yesmaximal, characteristic, complemented
4C3×#2071C211yesmaximal, characteristic
5C3×#2043C211yesmaximal, characteristic, complemented
6C3×#2042C211yesmaximal, characteristic
7C3×#1983C211yesmaximal, characteristic
8C3×#1959C211yesmaximal, characteristic
9C3×#1959C211yesmaximal, characteristic
10C3×#1956C211yesmaximal, characteristic, complemented
11C3×#1935C211yesmaximal, characteristic, complemented
12C3×#1934C211yesmaximal, characteristic
13C3×#1803C211yesmaximal, characteristic, complemented
14C3×#1771C211yesmaximal, characteristic, complemented
15C3×#1763C211yesmaximal, characteristic, complemented
16#27765C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#125—21no—
19C2×C3×#124C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#102C2211yescharacteristic
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C2×C3×#97C2211yescharacteristic
36C2×C3×#97C2211yescharacteristic
37C2×C3×C4×C8C2211yescharacteristic
38C3×#398—21no—
39C3×#398—21no—
40C3×#396—21no—
41C3×#396—21no—
42C3×#391C2211yescharacteristic
43C3×#390C2211yescharacteristic
44C3×#389—21no—
45C3×#389C2211yescharacteristic
46C3×#389C2211yescharacteristic
47C3×#388—21no—
48C3×#388—21no—
49C3×#384C2211yescharacteristic
50C3×#384C2211yescharacteristic
51C3×#384C2211yescharacteristic
52C3×C4×C4⋊C4C2211yescharacteristic
53C3×C4×C4⋊C4C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59C3×#336C2211yescharacteristic
60#2552C2×C311yescharacteristic, complemented
61#2545C2×C311yescharacteristic
62#2072C2×C311yescharacteristic, complemented
63#2071C2×C311yescharacteristic
64#2043C2×C311yescharacteristic, complemented
65#2042C2×C311yescharacteristic
66#1983C2×C311yescharacteristic
67#1959C2×C311yescharacteristic
68#1959C2×C311yescharacteristic
69#1956C2×C311yescharacteristic, complemented
70#1935C2×C311yescharacteristic, complemented
71#1934C2×C311yescharacteristic
72#1803C2×C311yescharacteristic, complemented
73#1771C2×C311yescharacteristic, complemented
74#1763C2×C311yescharacteristic, complemented
75C22×C3×Q8C2311yescharacteristic
76C22×C3×Q8C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C22×C3×C8C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4⋊Q8—22no—
83C3×C4⋊Q8—41no—
84C3×#125—44no—
85C3×#124—22no—
86C3×#124—22no—
87C3×#124—22no—
88C3×#124—41no—
89C3×C4×Q8—22no—
90C3×C4×Q8—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4—21no—
93C2×C3×C4⋊C4—21no—
94C2×C3×C4⋊C4—21no—
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×C22⋊C4—21no—
110C2×C3×C22⋊C4—21no—
111C2×C3×C22⋊C4—21no—
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C42—21no—
116C2×C3×C42—21no—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C3×C4⋊C8—22no—
121C3×#103—22no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#103—41no—
125C3×#103—41no—
126C3×#103—41no—
127C3×#103—41no—
128C3×#103—41no—
129C3×#102—22no—
130C3×#102—41no—
131C3×#102—41no—
132C3×#102—41no—
133C3×#97—22no—
134C3×#97—41no—
135C3×C4×C8—41no—
136C3×#95—21no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95—21no—
141C3×#95D411yescharacteristic
142C3×#95D411yescharacteristic
143C2×C4⋊Q8C22×C311yescharacteristic
144C2×#125—21no—
145C2×#124C22×C311yescharacteristic
146C2×#124C22×C311yescharacteristic
147C2×C4×Q8C22×C311yescharacteristic
148C2×C4⋊C8C22×C311yescharacteristic
149C2×#103C22×C311yescharacteristic
150C2×#103C22×C311yescharacteristic
151C2×#103C22×C311yescharacteristic
152C2×#103C22×C311yescharacteristic
153C2×#103C22×C311yescharacteristic
154C2×#103C22×C311yescharacteristic
155C2×#103C22×C311yescharacteristic
156C2×#103C22×C311yescharacteristic
157C2×#102C22×C311yescharacteristic
158C2×#102C22×C311yescharacteristic
159C2×#102C22×C311yescharacteristic
160C2×#102C22×C311yescharacteristic
161C2×#97C22×C311yescharacteristic
162C2×#97C22×C311yescharacteristic
163C2×C4×C8C22×C311yescharacteristic
164#398—21no—
165#398—21no—
166#396—21no—
167#396—21no—
168#391C22×C311yescharacteristic
169#390C22×C311yescharacteristic
170#389—21no—
171#389C22×C311yescharacteristic
172#389C22×C311yescharacteristic
173#388—21no—
174#388—21no—
175#384C22×C311yescharacteristic
176#384C22×C311yescharacteristic
177#384C22×C311yescharacteristic
178C4×C4⋊C4C22×C311yescharacteristic
179C4×C4⋊C4C22×C311yescharacteristic
180#340C22×C311yescharacteristic
181#340C22×C311yescharacteristic
182#340C22×C311yescharacteristic
183#340C22×C311yescharacteristic
184#336C22×C311yescharacteristic
185#336C22×C311yescharacteristic
186C24×C3—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8#4112yes—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2411yescharacteristic
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—
229C2×C3×C8—21no—
230C2×C3×C8#4112yes—
231C3×C4⋊C4—22no—
232C3×C4⋊C4—22no—
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—42no—
244C3×C4⋊C4—42no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—42no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4—42no—
250C3×C4⋊C4—42no—
251C3×C4⋊C4—81no—
252C3×C4⋊C4—81no—
253C3×C4⋊C4—81no—
254C3×C22⋊C4—42no—
255C3×C22⋊C4—42no—
256C3×C22⋊C4—42no—
257C3×C22⋊C4—42no—
258C3×C22⋊C4—42no—
259C3×C22⋊C4—42no—
260C3×C42—21no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—21no—
264C3×C42—42no—
265C3×C42—42no—
266C3×C42#4112yes—
267C3×C42#4112yes—
268C22×Q8C23×C311yescharacteristic
269C22×Q8C23×C311yescharacteristic
270C22×D4C23×C311yescharacteristic
271C22×C8C23×C311yescharacteristic
272C22×C8C23×C311yescharacteristic
273C22×C8C23×C311yescharacteristic
274C22×C8C23×C311yescharacteristic
275C4⋊Q8—22no—
276C4⋊Q8—41no—
277#125—44no—
278#124—22no—
279#124—22no—
280#124—22no—
281#124—41no—
282C4×Q8—22no—
283C4×Q8—22no—
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⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4C3×D411yescharacteristic
296C2×C4⋊C4C3×D411yescharacteristic
297C2×C4⋊C4C23×C311yescharacteristic
298C2×C4⋊C4C23×C311yescharacteristic
299C2×C4⋊C4C23×C311yescharacteristic
300C2×C4⋊C4C23×C311yescharacteristic
301C2×C4⋊C4C23×C311yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42C23×C311yescharacteristic
311C2×C42C23×C311yescharacteristic
312C2×C42C23×C311yescharacteristic
313C4⋊C8—22no—
314#103—22no—
315#103—22no—
316#103—22no—
317#103—41no—
318#103—41no—
319#103—41no—
320#103—41no—
321#103—41no—
322#102—22no—
323#102—41no—
324#102—41no—
325#102—41no—
326#97—22no—
327#97—41no—
328C4×C8—41no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95C3×D411yescharacteristic
335#95C3×D411yescharacteristic
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3C22×D411yescharacteristic
344C3×Q8—22no—
345C3×Q8—22no—
346C3×Q8—41no—
347C3×Q8—41no—
348C3×Q8—41no—
349C3×Q8—41no—
350C3×Q8—41no—
351C3×Q8—41no—
352C3×D4—41no—
353C3×D4—41no—
354C3×D4—41no—
355C3×D4—41no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
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—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4#12412yes—
392C2×C3×C4#12412yes—
393C2×C3×C4AGL(1,ℤ/8)12yes—
394C2×C3×C4#13712yes—
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×C3×C4C2×#4111yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4H(5,𝔽2)11yescharacteristic
400C2×C3×C4H(5,𝔽2)11yescharacteristic
401C3×C8—22no—
402C3×C8—41no—
403C3×C8—41no—
404C3×C8—41no—
405C24—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8C3×#4112yes—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4C2×C3×D411yescharacteristic
432C22×C4C2×C3×D411yescharacteristic
433C22×C4C2×C3×D411yescharacteristic
434C22×C4C2×C3×D411yescharacteristic
435C22×C4C2×C3×D411yescharacteristic
436C22×C4C2×C3×D411yescharacteristic
437C22×C4C24×C311yescharacteristic
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8C3×#4112yes—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—42no—
484C42—42no—
485C42C3×#4112yes—
486C42C3×#4112yes—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3C2×#12411yescharacteristic
495C22×C3#54611yescharacteristic
496C22×C3#55011yescharacteristic
497C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
498C22×C3C2×#13711yescharacteristic
499C22×C3#57711yescharacteristic
500C22×C3#57811yescharacteristic
501C3×C4—22no—
502C3×C4—22no—
503C3×C4—22no—
504C3×C4—22no—
505C3×C4—42no—
506C3×C4—41no—
507C3×C4—41no—
508C3×C4—81no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4#52911yescharacteristic
515C3×C4#54011yescharacteristic
516C3×C4#55311yescharacteristic
517C3×C4#55311yescharacteristic
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23C22×C3×D411yescharacteristic
526Q8—22no—
527Q8—22no—
528Q8—41no—
529Q8—41no—
530Q8—41no—
531Q8—41no—
532Q8—41no—
533Q8—41no—
534D4—41no—
535D4—41no—
536D4—41no—
537D4—41no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4C3×#12412yes—
574C2×C4C3×#12412yes—
575C2×C4C3×AGL(1,ℤ/8)12yes—
576C2×C4C3×#13712yes—
577C2×C4C2×C3×#4111yescharacteristic
578C2×C4C2×C3×#4111yescharacteristic
579C2×C4C2×C3×#4111yescharacteristic
580C2×C4C2×C3×#4111yescharacteristic
581C2×C4C3×H(5,𝔽2)11yescharacteristic
582C2×C4C3×H(5,𝔽2)11yescharacteristic
583C8—22no—
584C8—41no—
585C8—41no—
586C8—41no—
587C2×C3—161no—
588C2×C3#249011yescharacteristic
589C2×C3#313711yescharacteristic
590C2×C3#314111yescharacteristic
591C2×C3#330811yescharacteristic
592C2×C3#331611yescharacteristic
593C2×C3#339711yescharacteristic
594C2×C3#339811yescharacteristic
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22C2×C3×#12411yescharacteristic
603C22C3×#54611yescharacteristic
604C22C3×#55011yescharacteristic
605C22C2×C3×AGL(1,ℤ/8)11yescharacteristic
606C22C2×C3×#13711yescharacteristic
607C22C3×#57711yescharacteristic
608C22C3×#57811yescharacteristic
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—42no—
614C4—41no—
615C4—41no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4C3×#52911yescharacteristic
623C4C3×#54011yescharacteristic
624C4C3×#55311yescharacteristic
625C4C3×#55311yescharacteristic
626C3#2776511yescharacteristic, direct product factor, Sylow
627C2—161no—
628C2C3×#249011yescharacteristic
629C2C3×#313711yescharacteristic
630C2C3×#314111yescharacteristic
631C2C3×#330811yescharacteristic
632C2C3×#331611yescharacteristic
633C2C3×#339711yescharacteristic
634C2C3×#339811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2552C2semidirect
3. C3×#2072C2semidirect
5. C3×#2043C2semidirect
10. C3×#1956C2semidirect
11. C3×#1935C2semidirect
13. C3×#1803C2semidirect
14. C3×#1771C2semidirect
15. C3×#1763C2semidirect
16. #27765C3direct (N × H)
60. #2552C2×C3semidirect
62. #2072C2×C3semidirect
64. #2043C2×C3semidirect
69. #1956C2×C3semidirect
70. #1935C2×C3semidirect
72. #1803C2×C3semidirect
73. #1771C2×C3semidirect
74. #1763C2×C3semidirect
626. C3#27765direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25522
C3×#25452
C3×#20722
C3×#20712
C3×#20432
C3×#20422
C3×#19832
C3×#19592
C3×#19562
C3×#19352
C3×#19342
C3×#18032
C3×#17712
C3×#17632
#277653