Finite Group Databasebeta

← C22×#5528 #11701862

Subgroups — C22×#5528 #11701862

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×#117C211yesmaximal, characteristic, complemented
2C22×C4×#67C211yesmaximal, characteristic, complemented
3C22×C4×#67C211yesmaximal, characteristic, complemented
4C2×#5528C2148yesmaximal, direct product factor
5C22×#894C211yesmaximal, characteristic, complemented
6C22×#896C211yesmaximal, characteristic, complemented
7C22×#898C211yesmaximal, characteristic, complemented
8C22×#889C211yesmaximal, characteristic, complemented
9C22×#883C211yesmaximal, characteristic, complemented
10C22×#884C211yesmaximal, characteristic
11C22×#879C211yesmaximal, characteristic, complemented
12C22×#885C211yesmaximal, characteristic, complemented
13C22×C4×D12C211yesmaximal, characteristic, complemented
14C22×#872C211yesmaximal, characteristic, complemented
15C22×C42×D3C211yesmaximal, characteristic, complemented
16C22×C4×Dic6C211yesmaximal, characteristic, complemented
17C22×#529—31nomaximal, Sylow
18C24×C3×C4C2211yescharacteristic
19C23×#67C2211yescharacteristic
20C23×#67C2211yescharacteristic
21C23×D12C2211yescharacteristic
22C23×C4×D3C2211yescharacteristic
23C23×C4×D3C2211yescharacteristic
24C23×C4×D3C2211yescharacteristic
25C23×C4×D3C2211yescharacteristic
26C23×Dic6C2211yescharacteristic
27C2×C3×#117C22124yescomplemented
28C22×C3×C4⋊C4C2211yescharacteristic
29C22×C3×C4⋊C4C2211yescharacteristic
30C22×C3×C22⋊C4C2211yescharacteristic
31C22×C3×C22⋊C4C2211yescharacteristic
32C22×C3×C42C2211yescharacteristic
33C22×C3×C42C2211yescharacteristic
34C22×#217C2211yescharacteristic
35C22×#217C2211yescharacteristic
36C2×C4×#67C22124yescomplemented
37C2×C4×#67C22124yescomplemented
38C22×(C2×D3)⋊C4C2211yescharacteristic
39C22×(C2×D3)⋊C4C2211yescharacteristic
40C22×(C2×D3)⋊C4C2211yescharacteristic
41C22×(C2×D3)⋊C4C2211yescharacteristic
42C22×(C2×D3)⋊C4C2211yescharacteristic
43C22×(C2×D3)⋊C4C2211yescharacteristic
44C22×C4⋊Dic3C2211yescharacteristic
45C22×C4⋊Dic3C2211yescharacteristic
46C22×Dic3⋊C4C2211yescharacteristic
47C22×Dic3⋊C4C2211yescharacteristic
48C22×Dic3⋊C4C2211yescharacteristic
49C22×Dic3⋊C4C2211yescharacteristic
50C22×Dic3⋊C4C2211yescharacteristic
51C22×Dic3⋊C4C2211yescharacteristic
52C22×C4×Dic3C2211yescharacteristic
53C22×C4×Dic3C2211yescharacteristic
54C22×C4×Dic3C2211yescharacteristic
55C22×C4×Dic3C2211yescharacteristic
56#5528C221256yesdirect product factor
57C2×#898C22124yescomplemented
58C2×#896C22124yescomplemented
59C2×#894C22124yescomplemented
60C2×#889C22124yescomplemented
61C2×#885C22124yescomplemented
62C2×#884C22124yes—
63C2×#883C22124yescomplemented
64C2×#879C22124yescomplemented
65C2×C4×D12C22124yescomplemented
66C2×#872C22124yescomplemented
67C2×C42×D3C22124yescomplemented
68C2×C4×Dic6C22124yescomplemented
69C2×#529—348no—
70C22×#126—31no—
71C22×#126—31no—
72C22×#125—31no—
73C22×#124—31no—
74C22×#123—31no—
75C22×#123—31no—
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×#117D311yescharacteristic, complemented
84C23×C42—31no—
85C25×C3C2311yescharacteristic
86C24×D3C2311yescharacteristic
87C24×D3C2311yescharacteristic
88C23×C3×C4—28no—
89C23×C3×C4C2311yescharacteristic
90C23×C3×C4C2311yescharacteristic
91C23×C3×C4C2311yescharacteristic
92C23×C3×C4C2311yescharacteristic
93C23×C3×C4C2311yescharacteristic
94C23×C3×C4C23112yes—
95C23×C3×C4C2311yescharacteristic
96C22×#67—28no—
97C22×#67—28no—
98C22×#67C23112yes—
99C22×#67C23112yes—
100C23×Dic3C2311yescharacteristic
101C23×Dic3C2311yescharacteristic
102C23×Dic3C2311yescharacteristic
103C23×Dic3C2311yescharacteristic
104C23×Dic3C2311yescharacteristic
105C23×Dic3C2311yescharacteristic
106C22×D12—28no—
107C22×D12C23112yes—
108C22×C4×D3—28no—
109C22×C4×D3—28no—
110C22×C4×D3—28no—
111C22×C4×D3—28no—
112C22×C4×D3C23112yes—
113C22×C4×D3C23112yes—
114C22×C4×D3C23112yes—
115C22×C4×D3C23112yes—
116C22×Dic6—28no—
117C22×Dic6C23112yes—
118C3×#117C23164yescomplemented
119C2×C3×C4⋊C4C23112yes—
120C2×C3×C4⋊C4C23112yes—
121C2×C3×C22⋊C4C23112yes—
122C2×C3×C22⋊C4C23112yes—
123C2×C3×C42C23112yes—
124C2×C3×C42C23112yes—
125C2×#217C23112yes—
126C2×#217C23112yes—
127C4×#67C23164yescomplemented
128C4×#67C23164yescomplemented
129C2×(C2×D3)⋊C4C23112yes—
130C2×(C2×D3)⋊C4C23112yes—
131C2×(C2×D3)⋊C4C23112yes—
132C2×(C2×D3)⋊C4C23112yes—
133C2×(C2×D3)⋊C4C23112yes—
134C2×(C2×D3)⋊C4C23112yes—
135C2×C4⋊Dic3C23112yes—
136C2×C4⋊Dic3C23112yes—
137C2×Dic3⋊C4C23112yes—
138C2×Dic3⋊C4C23112yes—
139C2×Dic3⋊C4C23112yes—
140C2×Dic3⋊C4C23112yes—
141C2×Dic3⋊C4C23112yes—
142C2×Dic3⋊C4C23112yes—
143C2×C4×Dic3C23112yes—
144C2×C4×Dic3C23112yes—
145C2×C4×Dic3C23112yes—
146C2×C4×Dic3C23112yes—
147#898C23164yescomplemented
148#896C23164yescomplemented
149#894C23164yescomplemented
150#889C23164yescomplemented
151#885C23164yescomplemented
152#884C23164yes—
153#883C23164yescomplemented
154#879C23164yescomplemented
155C4×D12C23164yescomplemented
156#872C23164yescomplemented
157C42×D3C23164yescomplemented
158C4×Dic6C23164yescomplemented
159C23×Q8—31no—
160C23×D4—31no—
161C23×D4—31no—
162C23×D4—31no—
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×#123—324no—
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×#117C2×D3124yescomplemented
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⋊C4—31no—
201C22×C22⋊C4C2×D311yescharacteristic
202C22×C22⋊C4C2×D311yescharacteristic
203C22×C42—31no—
204C22×C42—31no—
205C22×C42—31no—
206C22×C42—31no—
207C22×C42—324no—
208C22×C42C2×D311yescharacteristic
209C22×C42C2×D311yescharacteristic
210C24×C3—24no—
211C24×C3—24no—
212C24×C3—24no—
213C24×C3C2416yes—
214C24×C3C2411yescharacteristic
215C23×D3—24no—
216C23×D3—24no—
217C23×D3—24no—
218C23×D3—24no—
219C23×D3—24no—
220C23×D3#4118yes—
221C23×D3C2416yes—
222C23×D3C2416yes—
223C22×C3×C4—24no—
224C22×C3×C4—24no—
225C22×C3×C4—24no—
226C22×C3×C4—24no—
227C22×C3×C4—24no—
228C22×C3×C4—248no—
229C22×C3×C4#4118yes—
230C22×C3×C4C2416yes—
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#4118yes—
247C22×Dic3C2416yes—
248C22×Dic3C2416yes—
249C22×Dic3C2416yes—
250C22×Dic3C2416yes—
251C22×Dic3C2416yes—
252C22×Dic3C2416yes—
253C2×D12—248no—
254C2×D12C24116yes—
255C2×C4×D3—248no—
256C2×C4×D3—248no—
257C2×C4×D3—248no—
258C2×C4×D3—248no—
259C2×C4×D3C24116yes—
260C2×C4×D3C24116yes—
261C2×C4×D3C24116yes—
262C2×C4×D3C24116yes—
263C2×Dic6—248no—
264C2×Dic6C24116yes—
265C3×C4⋊C4C24116yes—
266C3×C4⋊C4C24116yes—
267C3×C22⋊C4C24116yes—
268C3×C22⋊C4C24116yes—
269C3×C42C24116yes—
270C3×C42C24116yes—
271#217C24116yes—
272#217C24116yes—
273(C2×D3)⋊C4C24116yes—
274(C2×D3)⋊C4C24116yes—
275(C2×D3)⋊C4C24116yes—
276(C2×D3)⋊C4C24116yes—
277(C2×D3)⋊C4C24116yes—
278(C2×D3)⋊C4C24116yes—
279C4⋊Dic3C24116yes—
280C4⋊Dic3C24116yes—
281Dic3⋊C4C24116yes—
282Dic3⋊C4C24116yes—
283Dic3⋊C4C24116yes—
284Dic3⋊C4C24116yes—
285Dic3⋊C4C24116yes—
286Dic3⋊C4C24116yes—
287C4×Dic3C24116yes—
288C4×Dic3C24116yes—
289C4×Dic3C24116yes—
290C4×Dic3C24116yes—
291C25—31no—
292C25—31no—
293C25C22×D311yescharacteristic
294C22×Q8—312no—
295C22×Q8—68no—
296C22×D4—312no—
297C22×D4—312no—
298C22×D4—312no—
299C22×D4—68no—
300C22×D4—68no—
301C22×D4—68no—
302C23×C4—28no—
303C23×C4—31no—
304C23×C4—31no—
305C23×C4—31no—
306C23×C4—31no—
307C23×C4—31no—
308C23×C4—31no—
309C23×C4—312no—
310C23×C4—312no—
311C23×C4—312no—
312C23×C4—312no—
313C23×C4—68no—
314C23×C4—68no—
315C23×C4—68no—
316C23×C4—68no—
317C23×C4C22×D311yescharacteristic
318C23×C4C22×D311yescharacteristic
319C23×C4C22×D311yescharacteristic
320C23×C4C22×D311yescharacteristic
321C23×C4C22×D311yescharacteristic
322C23×C4C22×D3112yes—
323C23×C4C22×D311yescharacteristic
324#126—364no—
325#126—364no—
326#125—364no—
327#124—364no—
328#123—364no—
329#123—364no—
330C22⋊Q8—364no—
331#121—364no—
332C4×Q8—364no—
333C4×D4—364no—
334C4×D4—364no—
335C4×D4—364no—
336#117—364no—
337#117C22×D3164yescomplemented
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⋊C4—312no—
356C2×C22⋊C4C22×D3112yes—
357C2×C22⋊C4C22×D3112yes—
358C2×C42—312no—
359C2×C42—312no—
360C2×C42—312no—
361C2×C42—312no—
362C2×C42—364no—
363C2×C42C22×D3112yes—
364C2×C42C22×D3112yes—
365C23×C3—212no—
366C23×C3—212no—
367C23×C3—212no—
368C23×C3—416no—
369C23×C3C2×#4114yes—
370C23×C3C2×#4114yes—
371C23×C3C2×#4114yes—
372C23×C3C2514yes—
373C23×C3C2513yes—
374C22×D3—232no—
375C22×D3—212no—
376C22×D3—212no—
377C22×D3—212no—
378C22×D3—212no—
379C22×D3—212no—
380C22×D3—416no—
381C22×D3C2×#41124yes—
382C22×D3C2514yes—
383C22×D3C2514yes—
384C2×C3×C4—212no—
385C2×C3×C4—212no—
386C2×C3×C4—212no—
387C2×C3×C4—212no—
388C2×C3×C4—212no—
389C2×C3×C4—232no—
390C2×C3×C4C2×#41124yes—
391C2×C3×C4C2514yes—
392C2×C3×C4C2514yes—
393C2×C3×C4C2514yes—
394C2×C3×C4C2514yes—
395C2×C3×C4C2514yes—
396C2×C3×C4C2514yes—
397#67—232no—
398#67—232no—
399C2×Dic3—212no—
400C2×Dic3—212no—
401C2×Dic3—212no—
402C2×Dic3—212no—
403C2×Dic3—212no—
404C2×Dic3C2×#41124yes—
405C2×Dic3C2514yes—
406C2×Dic3C2514yes—
407C2×Dic3C2514yes—
408C2×Dic3C2514yes—
409C2×Dic3C2514yes—
410C2×Dic3C2514yes—
411D12—232no—
412C4×D3—232no—
413C4×D3—232no—
414C4×D3—232no—
415C4×D3—232no—
416Dic6—232no—
417C24—24no—
418C24—24no—
419C24—24no—
420C24—38no—
421C24—36no—
422C24—36no—
423C24—64no—
424C24—64no—
425C24—64no—
426C24—64no—
427C24—64no—
428C24C23×D316yes—
429C24C23×D311yescharacteristic
430C2×Q8—316no—
431C2×Q8—648no—
432C2×D4—316no—
433C2×D4—316no—
434C2×D4—316no—
435C2×D4—648no—
436C2×D4—648no—
437C2×D4—648no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—248no—
444C22×C4—38no—
445C22×C4—36no—
446C22×C4—36no—
447C22×C4—36no—
448C22×C4—36no—
449C22×C4—36no—
450C22×C4—36no—
451C22×C4—316no—
452C22×C4—316no—
453C22×C4—316no—
454C22×C4—316no—
455C22×C4—648no—
456C22×C4—648no—
457C22×C4—64no—
458C22×C4—64no—
459C22×C4—64no—
460C22×C4—64no—
461C22×C4—64no—
462C22×C4—648no—
463C22×C4—648no—
464C22×C4#23518yes—
465C22×C4C23×D316yes—
466C22×C4C23×D316yes—
467C22×C4C23×D316yes—
468C22×C4C23×D316yes—
469C22×C4C23×D316yes—
470C22×C4C23×D3116yes—
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⋊C4—316no—
490C22⋊C4C23×D3116yes—
491C22⋊C4C23×D3116yes—
492C42—316no—
493C42—316no—
494C42—316no—
495C42—316no—
496C42C23×D3116yes—
497C42C23×D3116yes—
498C22×C3—24no—
499C22×C3—24no—
500C22×C3—24no—
501C22×C3—424no—
502C22×C3#529116yescomplemented
503C22×C3C22×#4116yes—
504C22×C3C22×#4116yes—
505C22×C3C22×#4116yes—
506C22×C3C2611yescharacteristic
507C2×D3—24no—
508C2×D3—24no—
509C2×D3—248no—
510C2×D3—24no—
511C2×D3—24no—
512C2×D3—24no—
513C2×D3—424no—
514C2×D3C22×#4118yes—
515C3×C4—24no—
516C3×C4—24no—
517C3×C4—24no—
518C3×C4—24no—
519C3×C4—24no—
520C3×C4C22×#4118yes—
521Dic3—24no—
522Dic3—24no—
523Dic3—24no—
524Dic3—24no—
525Dic3—24no—
526Dic3C22×#4118yes—
527C23—212no—
528C23—212no—
529C23—212no—
530C23—324no—
531C23—34no—
532C23—34no—
533C23—416no—
534C23—612no—
535C23—612no—
536C23—632no—
537C23—612no—
538C23—612no—
539C23—612no—
540C23—1216no—
541C23C2×#23514yes—
542C23D3×#4114yes—
543C23D3×#4114yes—
544C23C24×D314yes—
545C23C24×D313yes—
546Q8—632no—
547D4—632no—
548D4—632no—
549D4—632no—
550C2×C4—212no—
551C2×C4—212no—
552C2×C4—212no—
553C2×C4—212no—
554C2×C4—212no—
555C2×C4—232no—
556C2×C4—324no—
557C2×C4—34no—
558C2×C4—34no—
559C2×C4—34no—
560C2×C4—34no—
561C2×C4—34no—
562C2×C4—34no—
563C2×C4—632no—
564C2×C4—632no—
565C2×C4—612no—
566C2×C4—612no—
567C2×C4—612no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—632no—
571C2×C4—632no—
572C2×C4C2×#235124yes—
573C2×C4C24×D314yes—
574C2×C4C24×D314yes—
575C2×C4C24×D314yes—
576C2×C4C24×D314yes—
577C2×C4C24×D314yes—
578C2×C4C24×D314yes—
579C2×C3—44no—
580C2×C3C2×#529112yescomplemented
581C2×C3C23×#4111yescharacteristic
582C2×C3C23×#4111yescharacteristic
583C2×C3C23×#4111yescharacteristic
584D3—28no—
585D3—44no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—38no—
590C22—424no—
591C22—64no—
592C22—64no—
593C22—648no—
594C22—64no—
595C22—64no—
596C22—64no—
597C22—1224no—
598C22#5528116yesdirect product factor
599C22C22×#23516yes—
600C22C2×D3×#4116yes—
601C22C2×D3×#4116yes—
602C22C25×D311yescharacteristic
603C4—24no—
604C4—24no—
605C4—24no—
606C4—24no—
607C4—24no—
608C4—38no—
609C4—64no—
610C4—64no—
611C4—64no—
612C4—64no—
613C4—64no—
614C4C22×#23518yes—
615C3C22×#52911yescharacteristic, complemented, Sylow
616C2—44no—
617C2—68no—
618C2—124no—
619C2C2×#5528112yesdirect product factor
620C2C23×#23511yescharacteristic
621C2C22×D3×#4111yescharacteristic
622C2C22×D3×#4111yescharacteristic
subgroupG/H
centerC23×C4C22×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×#117C2
socleC24×C3C24
prime radicalC22×#884C2
hypercenterC22×#117D3
hypocenterC3C22×#529
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C4 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
1. C22×C3×#117C2semidirect
2. C22×C4×#67C2semidirect
3. C22×C4×#67C2semidirect
4. C2×#5528C2direct (N × H)
5. C22×#894C2semidirect
6. C22×#896C2semidirect
7. C22×#898C2semidirect
8. C22×#889C2semidirect
9. C22×#883C2semidirect
11. C22×#879C2semidirect
12. C22×#885C2semidirect
13. C22×C4×D12C2semidirect
14. C22×#872C2semidirect
15. C22×C42×D3C2semidirect
16. C22×C4×Dic6C2semidirect
27. C2×C3×#117C22semidirect
36. C2×C4×#67C22semidirect
37. C2×C4×#67C22semidirect
56. #5528C22direct (N × H)
57. C2×#898C22semidirect
58. C2×#896C22semidirect
59. C2×#894C22semidirect
60. C2×#889C22semidirect
61. C2×#885C22semidirect
63. C2×#883C22semidirect
64. C2×#879C22semidirect
65. C2×C4×D12C22semidirect
66. C2×#872C22semidirect
67. C2×C42×D3C22semidirect
68. C2×C4×Dic6C22semidirect
83. C22×#117D3semidirect
118. C3×#117C23semidirect
127. C4×#67C23semidirect
128. C4×#67C23semidirect
147. #898C23semidirect
148. #896C23semidirect
149. #894C23semidirect
150. #889C23semidirect
151. #885C23semidirect
153. #883C23semidirect
154. #879C23semidirect
155. C4×D12C23semidirect
156. #872C23semidirect
157. C42×D3C23semidirect
158. C4×Dic6C23semidirect
182. C2×#117C2×D3semidirect
337. #117C22×D3semidirect
502. C22×C3#529semidirect
580. C2×C3C2×#529semidirect
598. C22#5528direct (N × H)
615. C3C22×#529semidirect
619. C2C2×#5528direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1172
C22×C4×#672
C2×#55282
C22×#8942
C22×#8962
C22×#8982
C22×#8892
C22×#8832
C22×#8842
C22×#8792
C22×#8852
C22×C4×D122
C22×#8722
C22×C42×D32
C22×C4×Dic62
C22×#5293