Finite Group Databasebeta

← C3×#27784 #10646391

Subgroups — C3×#27784 #10646391

Order 768.

798 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#489C211yesmaximal, characteristic
2C2×C3×#488C211yesmaximal, characteristic
3C3×#2550C211yesmaximal, characteristic, complemented
4C3×#2547C211yesmaximal, characteristic
5C3×#2042C211yesmaximal, characteristic
6C3×#2042C211yesmaximal, characteristic
7C3×#2026C211yesmaximal, characteristic, complemented
8C3×#2024C211yesmaximal, characteristic, complemented
9C3×#1981C211yesmaximal, characteristic, complemented
10C3×#1960C211yesmaximal, characteristic, complemented
11C3×#1935C211yesmaximal, characteristic, complemented
12C3×#1933C211yesmaximal, characteristic, complemented
13C3×#1804C211yesmaximal, characteristic
14C3×#1771C211yesmaximal, characteristic, complemented
15C3×#1762C211yesmaximal, characteristic
16#27784C311yesmaximal, characteristic, direct product factor, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#125C2211yescharacteristic
20C2×C3×#124C2211yescharacteristic
21C2×C3×#124C2211yescharacteristic
22C2×C3×C4×D4C2211yescharacteristic
23C3×#489—24no—
24C3×#488—24no—
25C2×C3×C4⋊C8C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C2×C3×#103C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2211yescharacteristic
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×#102C2211yescharacteristic
35C2×C3×#102C2211yescharacteristic
36C2×C3×#102C2211yescharacteristic
37C2×C3×#102C2211yescharacteristic
38C2×C3×#97C2211yescharacteristic
39C2×C3×#97C2211yescharacteristic
40C2×C3×C4×C8C2211yescharacteristic
41C3×#398—21no—
42C3×#397—21no—
43C3×#397—21no—
44C3×#395—21no—
45C3×#391—21no—
46C3×#391C2211yescharacteristic
47C3×#391C2211yescharacteristic
48C3×#389—21no—
49C3×#389C2211yescharacteristic
50C3×#389C2211yescharacteristic
51C3×#382C2211yescharacteristic
52C3×#380—21no—
53C3×#380—21no—
54C3×C4×C4⋊C4C2211yescharacteristic
55C3×C4×C4⋊C4C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60C3×#336C2211yescharacteristic
61C3×#336C2211yescharacteristic
62C2×#489C2×C311yescharacteristic
63C2×#488C2×C311yescharacteristic
64#2550C2×C311yescharacteristic, complemented
65#2547C2×C311yescharacteristic
66#2042C2×C311yescharacteristic
67#2042C2×C311yescharacteristic
68#2026C2×C311yescharacteristic, complemented
69#2024C2×C311yescharacteristic, complemented
70#1981C2×C311yescharacteristic, complemented
71#1960C2×C311yescharacteristic, complemented
72#1935C2×C311yescharacteristic, complemented
73#1933C2×C311yescharacteristic, complemented
74#1804C2×C311yescharacteristic
75#1771C2×C311yescharacteristic, complemented
76#1762C2×C311yescharacteristic
77C22×C3×Q8—21no—
78C22×C3×Q8C2311yescharacteristic
79C22×C3×Q8C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—21no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4⋊Q8—22no—
87C3×C4⋊Q8—22no—
88C3×C4⋊Q8—22no—
89C3×C4⋊Q8—41no—
90C3×#125—22no—
91C3×#125—41no—
92C3×#124—22no—
93C3×#124—22no—
94C3×#124—41no—
95C3×#124—41no—
96C3×C4×D4—22no—
97C3×C4×D4—22no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—21no—
101C2×C3×C4⋊C4—21no—
102C2×C3×C4⋊C4—21no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—21no—
105C2×C3×C4⋊C4D411yescharacteristic
106C2×C3×C4⋊C4D411yescharacteristic
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2311yescharacteristic
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4—21no—
113C2×C3×C22⋊C4—21no—
114C2×C3×C22⋊C4—21no—
115C2×C3×C22⋊C4—21no—
116C2×C3×C22⋊C4—21no—
117C2×C3×C42—21no—
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2311yescharacteristic
120C2×C3×C42C2311yescharacteristic
121C3×C4⋊C8—22no—
122C3×#103—22no—
123C3×#103—22no—
124C3×#103—22no—
125C3×#103—22no—
126C3×#103—22no—
127C3×#103—22no—
128C3×#103—41no—
129C3×#103—41no—
130C3×#102—22no—
131C3×#102—22no—
132C3×#102—22no—
133C3×#102D414yes—
134C3×#97—22no—
135C3×#97—22no—
136C3×C4×C8—22no—
137C3×#95—21no—
138C3×#95—21no—
139C3×#95—21no—
140C3×#95—21no—
141C3×#95—21no—
142C3×#95—21no—
143C3×#95—21no—
144C3×#95—21no—
145C3×#95—21no—
146C3×#95D411yescharacteristic
147C3×#95D411yescharacteristic
148C2×C4⋊Q8C22×C311yescharacteristic
149C2×C4⋊Q8C22×C311yescharacteristic
150C2×#125C22×C311yescharacteristic
151C2×#124C22×C311yescharacteristic
152C2×#124C22×C311yescharacteristic
153C2×C4×D4C22×C311yescharacteristic
154#489—24no—
155#488—24no—
156C2×C4⋊C8C22×C311yescharacteristic
157C2×#103C22×C311yescharacteristic
158C2×#103C22×C311yescharacteristic
159C2×#103C22×C311yescharacteristic
160C2×#103C22×C311yescharacteristic
161C2×#103C22×C311yescharacteristic
162C2×#103C22×C311yescharacteristic
163C2×#103C22×C311yescharacteristic
164C2×#103C22×C311yescharacteristic
165C2×#102C22×C311yescharacteristic
166C2×#102C22×C311yescharacteristic
167C2×#102C22×C311yescharacteristic
168C2×#102C22×C311yescharacteristic
169C2×#97C22×C311yescharacteristic
170C2×#97C22×C311yescharacteristic
171C2×C4×C8C22×C311yescharacteristic
172#398—21no—
173#397—21no—
174#397—21no—
175#395—21no—
176#391—21no—
177#391C22×C311yescharacteristic
178#391C22×C311yescharacteristic
179#389—21no—
180#389C22×C311yescharacteristic
181#389C22×C311yescharacteristic
182#382C22×C311yescharacteristic
183#380—21no—
184#380—21no—
185C4×C4⋊C4C22×C311yescharacteristic
186C4×C4⋊C4C22×C311yescharacteristic
187#340C22×C311yescharacteristic
188#340C22×C311yescharacteristic
189#340C22×C311yescharacteristic
190#340C22×C311yescharacteristic
191#336C22×C311yescharacteristic
192#336C22×C311yescharacteristic
193C24×C3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—21no—
202C2×C3×Q8—21no—
203C2×C3×Q8—21no—
204C2×C3×Q8—21no—
205C2×C3×Q8—21no—
206C2×C3×Q8—42no—
207C2×C3×Q8—42no—
208C2×C3×Q8—81no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4C2×D412yes—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—42no—
223C22×C3×C4—42no—
224C22×C3×C4—42no—
225C22×C3×C4C2×D411yescharacteristic
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2×D411yescharacteristic
231C22×C3×C4C2411yescharacteristic
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8C2×D412yes—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—41no—
253C3×C4⋊C4—41no—
254C3×C4⋊C4—42no—
255C3×C4⋊C4—42no—
256C3×C4⋊C4—42no—
257C3×C4⋊C4—42no—
258C3×C4⋊C4—42no—
259C3×C4⋊C4—42no—
260C3×C4⋊C4—81no—
261C3×C4⋊C4C2×D412yes—
262C3×C4⋊C4#4112yes—
263C3×C22⋊C4—42no—
264C3×C22⋊C4—42no—
265C3×C22⋊C4—42no—
266C3×C22⋊C4—42no—
267C3×C22⋊C4—42no—
268C3×C42—21no—
269C3×C42—21no—
270C3×C42—21no—
271C3×C42—21no—
272C3×C42—81no—
273C3×C42#4112yes—
274C3×C42#4112yes—
275C22×Q8—21no—
276C22×Q8C23×C311yescharacteristic
277C22×Q8C23×C311yescharacteristic
278C22×D4C23×C311yescharacteristic
279C23×C4—21no—
280C22×C8C23×C311yescharacteristic
281C22×C8C23×C311yescharacteristic
282C22×C8C23×C311yescharacteristic
283C22×C8C23×C311yescharacteristic
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—41no—
288#125—22no—
289#125—41no—
290#124—22no—
291#124—22no—
292#124—41no—
293#124—41no—
294C4×D4—22no—
295C4×D4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4C3×D411yescharacteristic
304C2×C4⋊C4C3×D411yescharacteristic
305C2×C4⋊C4C23×C311yescharacteristic
306C2×C4⋊C4C23×C311yescharacteristic
307C2×C4⋊C4C23×C311yescharacteristic
308C2×C4⋊C4C23×C311yescharacteristic
309C2×C4⋊C4C23×C311yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C42—21no—
316C2×C42C23×C311yescharacteristic
317C2×C42C23×C311yescharacteristic
318C2×C42C23×C311yescharacteristic
319C4⋊C8—22no—
320#103—22no—
321#103—22no—
322#103—22no—
323#103—22no—
324#103—22no—
325#103—22no—
326#103—41no—
327#103—41no—
328#102—22no—
329#102—22no—
330#102—22no—
331#102C3×D414yes—
332#97—22no—
333#97—22no—
334C4×C8—22no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95C3×D411yescharacteristic
345#95C3×D411yescharacteristic
346C23×C3—22no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3C22×D411yescharacteristic
354C3×Q8—41no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—84no—
363C3×D4—22no—
364C3×D4—22no—
365C3×D4—41no—
366C3×D4—41no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
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—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—84no—
405C2×C3×C4#12412yes—
406C2×C3×C4#12412yes—
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C4C2×#4111yescharacteristic
411C2×C3×C4H(5,𝔽2)11yescharacteristic
412C2×C3×C42−1+411yescharacteristic
413C3×C8—22no—
414C3×C8—41no—
415C3×C8—41no—
416C3×C8—41no—
417C24—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—42no—
431C2×Q8—42no—
432C2×Q8—81no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4C2×C3×D412yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4C2×C3×D411yescharacteristic
450C22×C4C2×C3×D411yescharacteristic
451C22×C4C2×C3×D411yescharacteristic
452C22×C4C2×C3×D411yescharacteristic
453C22×C4C2×C3×D411yescharacteristic
454C22×C4C2×C3×D411yescharacteristic
455C22×C4C24×C311yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8C2×C3×D412yes—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4C2×C3×D412yes—
486C4⋊C4C3×#4112yes—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—81no—
497C42C3×#4112yes—
498C42C3×#4112yes—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3C2×#12411yescharacteristic
507C22×C3D4211yescharacteristic
508C22×C3#54811yescharacteristic
509C22×C3#57511yescharacteristic
510C22×C3#57511yescharacteristic
511C22×C3#57711yescharacteristic
512C22×C3#57811yescharacteristic
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—42no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—42no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—81no—
522C3×C4—81no—
523C3×C4—81no—
524C3×C4—81no—
525C3×C4—81no—
526C3×C4#53211yescharacteristic
527C3×C4#54211yescharacteristic
528C3×C4#54711yescharacteristic
529C3×C4#55511yescharacteristic
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C22×C3×D411yescharacteristic
538Q8—41no—
539Q8—41no—
540Q8—41no—
541Q8—41no—
542Q8—41no—
543Q8—41no—
544Q8—41no—
545Q8—41no—
546Q8—84no—
547D4—22no—
548D4—22no—
549D4—41no—
550D4—41no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—84no—
589C2×C4C3×#12412yes—
590C2×C4C3×#12412yes—
591C2×C4C22×C3×D411yescharacteristic
592C2×C4C2×C3×#4111yescharacteristic
593C2×C4C2×C3×#4111yescharacteristic
594C2×C4C2×C3×#4111yescharacteristic
595C2×C4C3×H(5,𝔽2)11yescharacteristic
596C2×C4C3×2−1+411yescharacteristic
597C8—22no—
598C8—41no—
599C8—41no—
600C8—41no—
601C2×C3—82no—
602C2×C3#249111yescharacteristic
603C2×C3#313711yescharacteristic
604C2×C3#314011yescharacteristic
605C2×C3#326711yescharacteristic
606C2×C3#326911yescharacteristic
607C2×C3#337611yescharacteristic
608C2×C3#337711yescharacteristic
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22C2×C3×#12411yescharacteristic
617C22C3×D4211yescharacteristic
618C22C3×#54811yescharacteristic
619C22C3×#57511yescharacteristic
620C22C3×#57511yescharacteristic
621C22C3×#57711yescharacteristic
622C22C3×#57811yescharacteristic
623C4—22no—
624C4—22no—
625C4—41no—
626C4—41no—
627C4—42no—
628C4—41no—
629C4—41no—
630C4—42no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—81no—
636C4C3×#53211yescharacteristic
637C4C3×#54211yescharacteristic
638C4C3×#54711yescharacteristic
639C4C3×#55511yescharacteristic
640C3#2778411yescharacteristic, direct product factor, Sylow
641C2—82no—
642C2C3×#249111yescharacteristic
643C2C3×#313711yescharacteristic
644C2C3×#314011yescharacteristic
645C2C3×#326711yescharacteristic
646C2C3×#326911yescharacteristic
647C2C3×#337611yescharacteristic
648C2C3×#337711yescharacteristic
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
3. C3×#2550C2semidirect
7. C3×#2026C2semidirect
8. C3×#2024C2semidirect
9. C3×#1981C2semidirect
10. C3×#1960C2semidirect
11. C3×#1935C2semidirect
12. C3×#1933C2semidirect
14. C3×#1771C2semidirect
16. #27784C3direct (N × H)
64. #2550C2×C3semidirect
68. #2026C2×C3semidirect
69. #2024C2×C3semidirect
70. #1981C2×C3semidirect
71. #1960C2×C3semidirect
72. #1935C2×C3semidirect
73. #1933C2×C3semidirect
75. #1771C2×C3semidirect
640. C3#27784direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4892
C2×C3×#4882
C3×#25502
C3×#25472
C3×#20422
C3×#20262
C3×#20242
C3×#19812
C3×#19602
C3×#19352
C3×#19332
C3×#18042
C3×#17712
C3×#17622
#277843