Finite Group Databasebeta

← C22×#5649 #11701861

Subgroups — C22×#5649 #11701861

Order 768.

8670 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#123C211yesmaximal, characteristic, complemented
2C22×#939C211yesmaximal, characteristic
3C22×D4×Dic3C211yesmaximal, characteristic, complemented
4C22×#929C211yesmaximal, characteristic, complemented
5C23×C4×Dic3C211yesmaximal, characteristic, complemented
6C2×#5649C2148yesmaximal, direct product factor
7C22×#898C211yesmaximal, characteristic, complemented
8C22×#889C211yesmaximal, characteristic, complemented
9C22×#887C211yesmaximal, characteristic, complemented
10C22×#892C211yesmaximal, characteristic, complemented
11C22×#879C211yesmaximal, characteristic, complemented
12C22×#885C211yesmaximal, characteristic, complemented
13C22×#878C211yesmaximal, characteristic, complemented
14C22×#67⋊C4C211yesmaximal, characteristic, complemented
15C22×#67⋊C4C211yesmaximal, characteristic, complemented
16C22×#877C211yesmaximal, characteristic, complemented
17C22×#529—31nomaximal, Sylow
18C23×C3×D4C2211yescharacteristic
19C24×C3×C4C2211yescharacteristic
20C23×#67C2211yescharacteristic
21C23×#67C2211yescharacteristic
22C24×Dic3C2211yescharacteristic
23C24×Dic3C2211yescharacteristic
24C24×Dic3C2211yescharacteristic
25C23×C4×D3C2211yescharacteristic
26C23×Dic6C2211yescharacteristic
27C2×C3×#123C22124yescomplemented
28C22×C3×C4⋊C4C2211yescharacteristic
29C22×C3×C4⋊C4C2211yescharacteristic
30C22×C3×C22⋊C4C2211yescharacteristic
31C22×C3×C22⋊C4C2211yescharacteristic
32C22×C3×C22⋊C4C2211yescharacteristic
33C22×#217C2211yescharacteristic
34C22×#217C2211yescharacteristic
35C22×#217C2211yescharacteristic
36C2×#939C22124yes—
37C2×D4×Dic3C22124yescomplemented
38C2×#929C22124yescomplemented
39C22×(C2×D3)⋊C4C2211yescharacteristic
40C22×(C2×D3)⋊C4C2211yescharacteristic
41C22×(C2×D3)⋊C4C2211yescharacteristic
42C22×(C2×D3)⋊C4C2211yescharacteristic
43C22×C4⋊Dic3C2211yescharacteristic
44C22×C4⋊Dic3C2211yescharacteristic
45C22×Dic3⋊C4C2211yescharacteristic
46C22×Dic3⋊C4C2211yescharacteristic
47C22×Dic3⋊C4C2211yescharacteristic
48C22×Dic3⋊C4C2211yescharacteristic
49C22×Dic3⋊C4C2211yescharacteristic
50C22×Dic3⋊C4C2211yescharacteristic
51C22×C4×Dic3C2211yescharacteristic
52C22×C4×Dic3C2211yescharacteristic
53C22×C4×Dic3C2211yescharacteristic
54C22×C4×Dic3C2211yescharacteristic
55C22×C4×Dic3C2211yescharacteristic
56C22×C4×Dic3C2211yescharacteristic
57C22×C4×Dic3C22124yescomplemented
58#5649C221256yesdirect product factor
59C2×#898C22124yescomplemented
60C2×#892C22124yescomplemented
61C2×#889C22124yescomplemented
62C2×#887C22124yescomplemented
63C2×#885C22124yescomplemented
64C2×#67⋊C4C22124yescomplemented
65C2×#67⋊C4C22124yescomplemented
66C2×#879C22124yescomplemented
67C2×#878C22124yescomplemented
68C2×#877C22124yescomplemented
69C2×#529—348no—
70C22×#126—31no—
71C22×#126—31no—
72C22×#125—31no—
73C22×#124—31no—
74C22×#123—31no—
75C22×#123D311yescharacteristic, complemented
76C22×C22⋊Q8—31no—
77C22×#121—31no—
78C22×C4×Q8—31no—
79C22×C4×D4—31no—
80C22×C4×D4—31no—
81C22×C4×D4—31no—
82C22×#117—31no—
83C22×#117—31no—
84C23×C42—31no—
85C25×C3C2311yescharacteristic
86C25×C3C2311yescharacteristic
87C24×D3C2311yescharacteristic
88C22×C3×D4—28no—
89C22×C3×D4C23112yes—
90C23×C3×C4—28no—
91C23×C3×C4C2311yescharacteristic
92C23×C3×C4C2311yescharacteristic
93C23×C3×C4C2311yescharacteristic
94C23×C3×C4C2311yescharacteristic
95C23×C3×C4C23112yes—
96C23×C3×C4C2311yescharacteristic
97C22×#67—28no—
98C22×#67—28no—
99C22×#67C23112yes—
100C22×#67C23112yes—
101C23×Dic3—28no—
102C23×Dic3—28no—
103C23×Dic3—28no—
104C23×Dic3C2311yescharacteristic
105C23×Dic3C23112yes—
106C23×Dic3C23112yes—
107C23×Dic3C23112yes—
108C23×Dic3C2311yescharacteristic
109C23×Dic3C2311yescharacteristic
110C23×Dic3C2311yescharacteristic
111C23×Dic3C2311yescharacteristic
112C23×Dic3C2311yescharacteristic
113C23×Dic3C2311yescharacteristic
114C22×C4×D3—28no—
115C22×C4×D3C23112yes—
116C22×Dic6—28no—
117C22×Dic6C23112yes—
118C3×#123C23164yescomplemented
119C2×C3×C4⋊C4C23112yes—
120C2×C3×C4⋊C4C23112yes—
121C2×C3×C22⋊C4C23112yes—
122C2×C3×C22⋊C4C23112yes—
123C2×C3×C22⋊C4C23112yes—
124C2×#217C23112yes—
125C2×#217C23112yes—
126C2×#217C23112yes—
127#939C23164yes—
128D4×Dic3C23164yescomplemented
129#929C23164yescomplemented
130C2×(C2×D3)⋊C4C23112yes—
131C2×(C2×D3)⋊C4C23112yes—
132C2×(C2×D3)⋊C4C23112yes—
133C2×(C2×D3)⋊C4C23112yes—
134C2×C4⋊Dic3C23112yes—
135C2×C4⋊Dic3C23112yes—
136C2×Dic3⋊C4C23112yes—
137C2×Dic3⋊C4C23112yes—
138C2×Dic3⋊C4C23112yes—
139C2×Dic3⋊C4C23112yes—
140C2×Dic3⋊C4C23112yes—
141C2×Dic3⋊C4C23112yes—
142C2×C4×Dic3C23112yes—
143C2×C4×Dic3C23112yes—
144C2×C4×Dic3C23112yes—
145C2×C4×Dic3C23112yes—
146C2×C4×Dic3C23112yes—
147C2×C4×Dic3C23112yes—
148C2×C4×Dic3C23164yescomplemented
149#898C23164yescomplemented
150#892C23164yescomplemented
151#889C23164yescomplemented
152#887C23164yescomplemented
153#885C23164yescomplemented
154#67⋊C4C23164yescomplemented
155#67⋊C4C23164yescomplemented
156#879C23164yescomplemented
157#878C23164yescomplemented
158#877C23164yescomplemented
159C23×Q8—31no—
160C23×D4—31no—
161C23×D4—31no—
162C23×D4C2×D311yescharacteristic
163C24×C4—31no—
164C24×C4—31no—
165C24×C4—31no—
166C24×C4—31no—
167C24×C4C2×D311yescharacteristic
168#529—3256no—
169C2×#126—324no—
170C2×#126—324no—
171C2×#125—324no—
172C2×#124—324no—
173C2×#123—324no—
174C2×#123C2×D3124yescomplemented
175C2×C22⋊Q8—324no—
176C2×#121—324no—
177C2×C4×Q8—324no—
178C2×C4×D4—324no—
179C2×C4×D4—324no—
180C2×C4×D4—324no—
181C2×#117—324no—
182C2×#117—324no—
183C22×C4⋊C4—31no—
184C22×C4⋊C4—31no—
185C22×C4⋊C4—31no—
186C22×C4⋊C4—31no—
187C22×C4⋊C4—31no—
188C22×C4⋊C4—31no—
189C22×C4⋊C4—31no—
190C22×C4⋊C4—31no—
191C22×C4⋊C4C2×D311yescharacteristic
192C22×C4⋊C4C2×D311yescharacteristic
193C22×C22⋊C4—31no—
194C22×C22⋊C4—31no—
195C22×C22⋊C4—31no—
196C22×C22⋊C4—31no—
197C22×C22⋊C4—31no—
198C22×C22⋊C4—31no—
199C22×C22⋊C4—31no—
200C22×C22⋊C4C2×D311yescharacteristic
201C22×C22⋊C4C2×D311yescharacteristic
202C22×C22⋊C4C2×D311yescharacteristic
203C22×C42—31no—
204C22×C42—31no—
205C22×C42—31no—
206C22×C42—31no—
207C22×C42—324no—
208C22×C42—31no—
209C22×C42—31no—
210C24×C3—24no—
211C24×C3—24no—
212C24×C3—24no—
213C24×C3—24no—
214C24×C3—24no—
215C24×C3#4118yes—
216C24×C3C2416yes—
217C24×C3C2416yes—
218C24×C3C2411yescharacteristic
219C23×D3—24no—
220C23×D3—24no—
221C23×D3—24no—
222C23×D3C2416yes—
223C2×C3×D4—248no—
224C2×C3×D4C24116yes—
225C22×C3×C4—24no—
226C22×C3×C4—24no—
227C22×C3×C4—248no—
228C22×C3×C4—24no—
229C22×C3×C4—24no—
230C22×C3×C4—24no—
231C22×C3×C4C2416yes—
232C22×C3×C4C2416yes—
233C22×C3×C4C2416yes—
234C22×C3×C4C2416yes—
235C22×C3×C4C24116yes—
236C22×C3×C4C2416yes—
237C2×#67—248no—
238C2×#67—248no—
239C2×#67C24116yes—
240C2×#67C24116yes—
241C22×Dic3—24no—
242C22×Dic3—24no—
243C22×Dic3—24no—
244C22×Dic3—24no—
245C22×Dic3—24no—
246C22×Dic3—248no—
247C22×Dic3—248no—
248C22×Dic3—248no—
249C22×Dic3#4118yes—
250C22×Dic3#4118yes—
251C22×Dic3C24116yes—
252C22×Dic3C24116yes—
253C22×Dic3C24116yes—
254C22×Dic3C2416yes—
255C22×Dic3C2416yes—
256C22×Dic3C2416yes—
257C22×Dic3C2416yes—
258C22×Dic3C2416yes—
259C22×Dic3C2416yes—
260C22×Dic3C2416yes—
261C2×C4×D3—248no—
262C2×C4×D3C24116yes—
263C2×Dic6—248no—
264C2×Dic6C24116yes—
265C3×C4⋊C4C24116yes—
266C3×C4⋊C4C24116yes—
267C3×C22⋊C4C24116yes—
268C3×C22⋊C4C24116yes—
269C3×C22⋊C4C24116yes—
270#217C24116yes—
271#217C24116yes—
272#217C24116yes—
273(C2×D3)⋊C4C24116yes—
274(C2×D3)⋊C4C24116yes—
275(C2×D3)⋊C4C24116yes—
276(C2×D3)⋊C4C24116yes—
277C4⋊Dic3C24116yes—
278C4⋊Dic3C24116yes—
279Dic3⋊C4C24116yes—
280Dic3⋊C4C24116yes—
281Dic3⋊C4C24116yes—
282Dic3⋊C4C24116yes—
283Dic3⋊C4C24116yes—
284Dic3⋊C4C24116yes—
285C4×Dic3C24116yes—
286C4×Dic3C24116yes—
287C4×Dic3C24116yes—
288C4×Dic3C24116yes—
289C4×Dic3C24116yes—
290C4×Dic3C24116yes—
291C25—31no—
292C25C22×D311yescharacteristic
293C25C22×D311yescharacteristic
294C22×Q8—312no—
295C22×Q8—68no—
296C22×D4—28no—
297C22×D4—312no—
298C22×D4—312no—
299C22×D4—68no—
300C22×D4—68no—
301C22×D4C22×D3112yes—
302C23×C4—28no—
303C23×C4—31no—
304C23×C4—31no—
305C23×C4—312no—
306C23×C4—312no—
307C23×C4—312no—
308C23×C4—31no—
309C23×C4—31no—
310C23×C4—31no—
311C23×C4—31no—
312C23×C4—31no—
313C23×C4—312no—
314C23×C4—68no—
315C23×C4—68no—
316C23×C4—68no—
317C23×C4—68no—
318C23×C4C22×D3112yes—
319C23×C4C22×D311yescharacteristic
320C23×C4C22×D311yescharacteristic
321C23×C4C22×D311yescharacteristic
322C23×C4C22×D311yescharacteristic
323C23×C4C22×D311yescharacteristic
324#126—364no—
325#126—364no—
326#125—364no—
327#124—364no—
328#123—364no—
329#123C22×D3164yescomplemented
330C22⋊Q8—364no—
331#121—364no—
332C4×Q8—364no—
333C4×D4—364no—
334C4×D4—364no—
335C4×D4—364no—
336#117—364no—
337#117—364no—
338C2×C4⋊C4—312no—
339C2×C4⋊C4—312no—
340C2×C4⋊C4—312no—
341C2×C4⋊C4—312no—
342C2×C4⋊C4—312no—
343C2×C4⋊C4—312no—
344C2×C4⋊C4—312no—
345C2×C4⋊C4—312no—
346C2×C4⋊C4C22×D3112yes—
347C2×C4⋊C4C22×D3112yes—
348C2×C22⋊C4—312no—
349C2×C22⋊C4—312no—
350C2×C22⋊C4—312no—
351C2×C22⋊C4—312no—
352C2×C22⋊C4—312no—
353C2×C22⋊C4—312no—
354C2×C22⋊C4—312no—
355C2×C22⋊C4C22×D3112yes—
356C2×C22⋊C4C22×D3112yes—
357C2×C22⋊C4C22×D3112yes—
358C2×C42—312no—
359C2×C42—312no—
360C2×C42—312no—
361C2×C42—312no—
362C2×C42—312no—
363C2×C42—312no—
364C2×C42—364no—
365C23×C3—232no—
366C23×C3—212no—
367C23×C3—212no—
368C23×C3—212no—
369C23×C3—212no—
370C23×C3—212no—
371C23×C3—416no—
372C23×C3C2×#41124yes—
373C23×C3C2×#4114yes—
374C23×C3C2×#4114yes—
375C23×C3C2×#4114yes—
376C23×C3C2514yes—
377C23×C3C2514yes—
378C23×C3C2513yes—
379C22×D3—212no—
380C22×D3—212no—
381C22×D3—212no—
382C22×D3—416no—
383C22×D3C2514yes—
384C3×D4—232no—
385C2×C3×C4—212no—
386C2×C3×C4—212no—
387C2×C3×C4—232no—
388C2×C3×C4—212no—
389C2×C3×C4—212no—
390C2×C3×C4—212no—
391C2×C3×C4C2514yes—
392C2×C3×C4C2514yes—
393C2×C3×C4C2514yes—
394C2×C3×C4C2514yes—
395C2×C3×C4C2514yes—
396#67—232no—
397#67—232no—
398C2×Dic3—212no—
399C2×Dic3—212no—
400C2×Dic3—212no—
401C2×Dic3—212no—
402C2×Dic3—212no—
403C2×Dic3—232no—
404C2×Dic3—232no—
405C2×Dic3—232no—
406C2×Dic3C2×#41124yes—
407C2×Dic3C2×#41124yes—
408C2×Dic3C2514yes—
409C2×Dic3C2514yes—
410C2×Dic3C2514yes—
411C2×Dic3C2514yes—
412C2×Dic3C2514yes—
413C2×Dic3C2514yes—
414C2×Dic3C2514yes—
415C4×D3—232no—
416Dic6—232no—
417C24—24no—
418C24—24no—
419C24—24no—
420C24—24no—
421C24—24no—
422C24—36no—
423C24—64no—
424C24—64no—
425C24—64no—
426C24#23718yes—
427C24C23×D316yes—
428C24C23×D316yes—
429C24C23×D311yescharacteristic
430C2×Q8—316no—
431C2×Q8—648no—
432C2×D4—248no—
433C2×D4—316no—
434C2×D4—316no—
435C2×D4—648no—
436C2×D4—648no—
437C2×D4C23×D3116yes—
438C22×C4—248no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—38no—
445C22×C4—38no—
446C22×C4—36no—
447C22×C4—36no—
448C22×C4—36no—
449C22×C4—36no—
450C22×C4—316no—
451C22×C4—316no—
452C22×C4—36no—
453C22×C4—36no—
454C22×C4—36no—
455C22×C4—316no—
456C22×C4—316no—
457C22×C4—648no—
458C22×C4—648no—
459C22×C4—648no—
460C22×C4—648no—
461C22×C4—64no—
462C22×C4—64no—
463C22×C4—64no—
464C22×C4—64no—
465C22×C4—64no—
466C22×C4C23×D316yes—
467C22×C4C23×D316yes—
468C22×C4C23×D3116yes—
469C22×C4C23×D316yes—
470C22×C4C23×D316yes—
471C22×C4C23×D316yes—
472C4⋊C4—316no—
473C4⋊C4—316no—
474C4⋊C4—316no—
475C4⋊C4—316no—
476C4⋊C4—316no—
477C4⋊C4—316no—
478C4⋊C4—316no—
479C4⋊C4—316no—
480C4⋊C4C23×D3116yes—
481C4⋊C4C23×D3116yes—
482C22⋊C4—316no—
483C22⋊C4—316no—
484C22⋊C4—316no—
485C22⋊C4—316no—
486C22⋊C4—316no—
487C22⋊C4—316no—
488C22⋊C4—316no—
489C22⋊C4C23×D3116yes—
490C22⋊C4C23×D3116yes—
491C22⋊C4C23×D3116yes—
492C42—316no—
493C42—316no—
494C42—316no—
495C42—316no—
496C42—316no—
497C42—316no—
498C22×C3—24no—
499C22×C3—24no—
500C22×C3—248no—
501C22×C3—24no—
502C22×C3—24no—
503C22×C3—24no—
504C22×C3—424no—
505C22×C3#529116yescomplemented
506C22×C3C22×#4118yes—
507C22×C3C22×#4116yes—
508C22×C3C22×#4116yes—
509C22×C3C22×#4116yes—
510C22×C3C2611yescharacteristic
511C2×D3—24no—
512C2×D3—24no—
513C2×D3—24no—
514C2×D3—424no—
515C3×C4—24no—
516C3×C4—24no—
517C3×C4—24no—
518C3×C4—24no—
519C3×C4—24no—
520Dic3—24no—
521Dic3—24no—
522Dic3—24no—
523Dic3—24no—
524Dic3—24no—
525Dic3C22×#4118yes—
526Dic3C22×#4118yes—
527C23—232no—
528C23—212no—
529C23—212no—
530C23—212no—
531C23—212no—
532C23—212no—
533C23—34no—
534C23—416no—
535C23—612no—
536C23—612no—
537C23—612no—
538C23—1216no—
539C23C2×#237124yes—
540C23C2×#23714yes—
541C23D3×#4114yes—
542C23D3×#4114yes—
543C23C24×D314yes—
544C23C24×D314yes—
545C23C24×D313yes—
546Q8—632no—
547D4—232no—
548D4—632no—
549D4—632no—
550C2×C4—232no—
551C2×C4—212no—
552C2×C4—212no—
553C2×C4—212no—
554C2×C4—212no—
555C2×C4—212no—
556C2×C4—324no—
557C2×C4—324no—
558C2×C4—34no—
559C2×C4—34no—
560C2×C4—34no—
561C2×C4—34no—
562C2×C4—34no—
563C2×C4—34no—
564C2×C4—34no—
565C2×C4—632no—
566C2×C4—632no—
567C2×C4—632no—
568C2×C4—632no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4C24×D314yes—
575C2×C4C24×D314yes—
576C2×C4C24×D314yes—
577C2×C4C24×D314yes—
578C2×C4C24×D314yes—
579C2×C3—28no—
580C2×C3—44no—
581C2×C3C2×#529112yescomplemented
582C2×C3C23×#4111yescharacteristic
583C2×C3C23×#4111yescharacteristic
584C2×C3C23×#4111yescharacteristic
585D3—44no—
586C22—24no—
587C22—24no—
588C22—248no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—424no—
593C22—64no—
594C22—64no—
595C22—64no—
596C22—1224no—
597C22#5649116yesdirect product factor
598C22C22×#23718yes—
599C22C22×#23716yes—
600C22C2×D3×#4116yes—
601C22C2×D3×#4116yes—
602C22C25×D311yescharacteristic
603C4—24no—
604C4—24no—
605C4—24no—
606C4—24no—
607C4—24no—
608C4—38no—
609C4—38no—
610C4—64no—
611C4—64no—
612C4—64no—
613C4—64no—
614C4—64no—
615C3C22×#52911yescharacteristic, complemented, Sylow
616C2—28no—
617C2—44no—
618C2—124no—
619C2C2×#5649112yesdirect product factor
620C2C23×#23711yescharacteristic
621C2C22×D3×#4111yescharacteristic
622C2C22×D3×#4111yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×#123C2
socleC24×C3C24
prime radicalC22×#939C2
hypercenterC22×#123D3
hypocenterC3C22×#529
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
1. C22×C3×#123C2semidirect
3. C22×D4×Dic3C2semidirect
4. C22×#929C2semidirect
5. C23×C4×Dic3C2semidirect
6. C2×#5649C2direct (N × H)
7. C22×#898C2semidirect
8. C22×#889C2semidirect
9. C22×#887C2semidirect
10. C22×#892C2semidirect
11. C22×#879C2semidirect
12. C22×#885C2semidirect
13. C22×#878C2semidirect
14. C22×#67⋊C4C2semidirect
15. C22×#67⋊C4C2semidirect
16. C22×#877C2semidirect
27. C2×C3×#123C22semidirect
37. C2×D4×Dic3C22semidirect
38. C2×#929C22semidirect
57. C22×C4×Dic3C22semidirect
58. #5649C22direct (N × H)
59. C2×#898C22semidirect
60. C2×#892C22semidirect
61. C2×#889C22semidirect
62. C2×#887C22semidirect
63. C2×#885C22semidirect
64. C2×#67⋊C4C22semidirect
65. C2×#67⋊C4C22semidirect
66. C2×#879C22semidirect
67. C2×#878C22semidirect
68. C2×#877C22semidirect
75. C22×#123D3semidirect
118. C3×#123C23semidirect
128. D4×Dic3C23semidirect
129. #929C23semidirect
148. C2×C4×Dic3C23semidirect
149. #898C23semidirect
150. #892C23semidirect
151. #889C23semidirect
152. #887C23semidirect
153. #885C23semidirect
154. #67⋊C4C23semidirect
155. #67⋊C4C23semidirect
156. #879C23semidirect
157. #878C23semidirect
158. #877C23semidirect
174. C2×#123C2×D3semidirect
329. #123C22×D3semidirect
505. C22×C3#529semidirect
581. C2×C3C2×#529semidirect
597. C22#5649direct (N × H)
615. C3C22×#529semidirect
619. C2C2×#5649direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1232
C22×#9392
C22×D4×Dic32
C22×#9292
C23×C4×Dic32
C2×#56492
C22×#8982
C22×#8892
C22×#8872
C22×#8922
C22×#8792
C22×#8852
C22×#8782
C22×#67⋊C42
C22×#8772
C22×#5293