Finite Group Databasebeta

← C22×#5712 #11702073

Subgroups — C22×#5712 #11702073

Order 768.

7238 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#126C211yesmaximal, characteristic, complemented
2C2×#5712C2148yesmaximal, direct product factor
3C22×#896C211yesmaximal, characteristic, complemented
4C22×#897C211yesmaximal, characteristic, complemented
5C22×#888C211yesmaximal, characteristic, complemented
6C22×#898C211yesmaximal, characteristic, complemented
7C22×#890C211yesmaximal, characteristic, complemented
8C22×#889C211yesmaximal, characteristic, complemented
9C22×#883C211yesmaximal, characteristic
10C22×#879C211yesmaximal, characteristic, complemented
11C22×#885C211yesmaximal, characteristic
12C22×#886C211yesmaximal, characteristic
13C22×#878C211yesmaximal, characteristic, complemented
14C22×#878C211yesmaximal, characteristic, complemented
15C22×#876C211yesmaximal, characteristic, complemented
16C22×#869C211yesmaximal, characteristic, complemented
17C22×#563—31nomaximal, Sylow
18C23×#67C2211yescharacteristic
19C24×Dic3C2211yescharacteristic
20C23×C4×D3C2211yescharacteristic
21C23×Dic6C2211yescharacteristic
22C23×Dic6C2211yescharacteristic
23C23×Dic6C2211yescharacteristic
24C2×C3×#126C22124yescomplemented
25C22×C3×C4⋊C4C2211yescharacteristic
26C22×C3×C4⋊C4C2211yescharacteristic
27C22×C3×C4⋊C4C2211yescharacteristic
28C22×C3×C22⋊C4C2211yescharacteristic
29C22×C3×C22⋊C4C2211yescharacteristic
30C22×C3×C22⋊C4C2211yescharacteristic
31C22×C3×C42C2211yescharacteristic
32C22×#217C2211yescharacteristic
33C22×#217C2211yescharacteristic
34C22×(C2×D3)⋊C4C2211yescharacteristic
35C22×(C2×D3)⋊C4C2211yescharacteristic
36C22×(C2×D3)⋊C4C2211yescharacteristic
37C22×(C2×D3)⋊C4C2211yescharacteristic
38C22×(C2×D3)⋊C4C2211yescharacteristic
39C22×C4⋊Dic3C2211yescharacteristic
40C22×C4⋊Dic3C2211yescharacteristic
41C22×C4⋊Dic3C2211yescharacteristic
42C22×C4⋊Dic3C2211yescharacteristic
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×Dic3⋊C4C2211yescharacteristic
52C22×C4×Dic3C2211yescharacteristic
53C22×C4×Dic3C2211yescharacteristic
54#5712C221256yesdirect product factor
55C2×#898C22124yescomplemented
56C2×#897C22124yescomplemented
57C2×#896C22124yescomplemented
58C2×#890C22124yescomplemented
59C2×#889C22124yescomplemented
60C2×#888C22124yescomplemented
61C2×#886C22124yes—
62C2×#885C22124yes—
63C2×#883C22124yes—
64C2×#879C22124yescomplemented
65C2×#878C22124yescomplemented
66C2×#878C22124yescomplemented
67C2×#876C22124yescomplemented
68C2×#869C22124yescomplemented
69C2×#563—348no—
70C22×C4⋊Q8—31no—
71C22×C4⋊Q8—31no—
72C22×#126—31no—
73C22×#126—31no—
74C22×#126—31no—
75C22×#126D311yescharacteristic, complemented
76C22×#125—31no—
77C22×#125—31no—
78C22×#124—31no—
79C22×#123—31no—
80C22×#123—31no—
81C22×C22⋊Q8—31no—
82C22×C22⋊Q8—31no—
83C22×C22⋊Q8—31no—
84C22×C22⋊Q8—31no—
85C25×C3C2311yescharacteristic
86C24×D3C2311yescharacteristic
87C23×C3×C4C2311yescharacteristic
88C23×C3×C4C2311yescharacteristic
89C23×C3×C4C2311yescharacteristic
90C23×C3×C4C2311yescharacteristic
91C23×C3×C4C2311yescharacteristic
92C23×C3×C4C2311yescharacteristic
93C22×#67—44no—
94C22×#67C23112yes—
95C23×Dic3—44no—
96C23×Dic3C2311yescharacteristic
97C23×Dic3C2311yescharacteristic
98C23×Dic3C2311yescharacteristic
99C23×Dic3C2311yescharacteristic
100C23×Dic3C2311yescharacteristic
101C23×Dic3C2311yescharacteristic
102C23×Dic3C2311yescharacteristic
103C23×Dic3C23112yes—
104C22×C4×D3—44no—
105C22×C4×D3C23112yes—
106C22×Dic6—44no—
107C22×Dic6—44no—
108C22×Dic6—44no—
109C22×Dic6C23112yes—
110C22×Dic6C23112yes—
111C22×Dic6C23112yes—
112C3×#126C23164yescomplemented
113C2×C3×C4⋊C4C23112yes—
114C2×C3×C4⋊C4C23112yes—
115C2×C3×C4⋊C4C23112yes—
116C2×C3×C22⋊C4C23112yes—
117C2×C3×C22⋊C4C23112yes—
118C2×C3×C22⋊C4C23112yes—
119C2×C3×C42C23112yes—
120C2×#217C23112yes—
121C2×#217C23112yes—
122C2×(C2×D3)⋊C4C23112yes—
123C2×(C2×D3)⋊C4C23112yes—
124C2×(C2×D3)⋊C4C23112yes—
125C2×(C2×D3)⋊C4C23112yes—
126C2×(C2×D3)⋊C4C23112yes—
127C2×C4⋊Dic3C23112yes—
128C2×C4⋊Dic3C23112yes—
129C2×C4⋊Dic3C23112yes—
130C2×C4⋊Dic3C23112yes—
131C2×C4⋊Dic3C23112yes—
132C2×C4⋊Dic3C23112yes—
133C2×Dic3⋊C4C23112yes—
134C2×Dic3⋊C4C23112yes—
135C2×Dic3⋊C4C23112yes—
136C2×Dic3⋊C4C23112yes—
137C2×Dic3⋊C4C23112yes—
138C2×Dic3⋊C4C23112yes—
139C2×Dic3⋊C4C23112yes—
140C2×C4×Dic3C23112yes—
141C2×C4×Dic3C23112yes—
142#898C23164yescomplemented
143#897C23164yescomplemented
144#896C23164yescomplemented
145#890C23164yescomplemented
146#889C23164yescomplemented
147#888C23164yescomplemented
148#886C23164yes—
149#885C23164yes—
150#883C23164yes—
151#879C23164yescomplemented
152#878C23164yescomplemented
153#878C23164yescomplemented
154#876C23164yescomplemented
155#869C23164yescomplemented
156C23×Q8—31no—
157C23×Q8—31no—
158C23×Q8—31no—
159C23×D4—31no—
160C24×C4—31no—
161C24×C4—31no—
162#563—3256no—
163C2×C4⋊Q8—324no—
164C2×C4⋊Q8—324no—
165C2×#126—324no—
166C2×#126—324no—
167C2×#126—324no—
168C2×#126C2×D3124yescomplemented
169C2×#125—324no—
170C2×#125—324no—
171C2×#124—324no—
172C2×#123—324no—
173C2×#123—324no—
174C2×C22⋊Q8—324no—
175C2×C22⋊Q8—324no—
176C2×C22⋊Q8—324no—
177C2×C22⋊Q8—324no—
178C22×C4⋊C4—31no—
179C22×C4⋊C4—31no—
180C22×C4⋊C4—31no—
181C22×C4⋊C4—31no—
182C22×C4⋊C4—31no—
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×C4⋊C4C2×D311yescharacteristic
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×C22⋊C4C2×D311yescharacteristic
204C22×C42—31no—
205C22×C42—31no—
206C22×C42C2×D311yescharacteristic
207C24×C3—24no—
208C24×C3—24no—
209C24×C3—24no—
210C24×C3C2416yes—
211C24×C3C2411yescharacteristic
212C23×D3—24no—
213C23×D3—24no—
214C23×D3—24no—
215C23×D3C2416yes—
216C22×C3×C4—24no—
217C22×C3×C4—24no—
218C22×C3×C4—24no—
219C22×C3×C4—24no—
220C22×C3×C4—24no—
221C22×C3×C4—24no—
222C22×C3×C4C2416yes—
223C22×C3×C4C2416yes—
224C22×C3×C4C2416yes—
225C22×C3×C4C2416yes—
226C22×C3×C4C2416yes—
227C22×C3×C4C2416yes—
228C2×#67—424no—
229C2×#67C24116yes—
230C22×Dic3—24no—
231C22×Dic3—24no—
232C22×Dic3—24no—
233C22×Dic3—24no—
234C22×Dic3—24no—
235C22×Dic3—24no—
236C22×Dic3—24no—
237C22×Dic3—424no—
238C22×Dic3C2416yes—
239C22×Dic3C2416yes—
240C22×Dic3C2416yes—
241C22×Dic3C2416yes—
242C22×Dic3C2416yes—
243C22×Dic3C2416yes—
244C22×Dic3C2416yes—
245C22×Dic3C24116yes—
246C2×C4×D3—424no—
247C2×C4×D3C24116yes—
248C2×Dic6—424no—
249C2×Dic6—424no—
250C2×Dic6—424no—
251C2×Dic6C24116yes—
252C2×Dic6C24116yes—
253C2×Dic6C24116yes—
254C3×C4⋊C4C24116yes—
255C3×C4⋊C4C24116yes—
256C3×C4⋊C4C24116yes—
257C3×C22⋊C4C24116yes—
258C3×C22⋊C4C24116yes—
259C3×C22⋊C4C24116yes—
260C3×C42C24116yes—
261#217C24116yes—
262#217C24116yes—
263(C2×D3)⋊C4C24116yes—
264(C2×D3)⋊C4C24116yes—
265(C2×D3)⋊C4C24116yes—
266(C2×D3)⋊C4C24116yes—
267(C2×D3)⋊C4C24116yes—
268C4⋊Dic3C24116yes—
269C4⋊Dic3C24116yes—
270C4⋊Dic3C24116yes—
271C4⋊Dic3C24116yes—
272C4⋊Dic3C24116yes—
273C4⋊Dic3C24116yes—
274Dic3⋊C4C24116yes—
275Dic3⋊C4C24116yes—
276Dic3⋊C4C24116yes—
277Dic3⋊C4C24116yes—
278Dic3⋊C4C24116yes—
279Dic3⋊C4C24116yes—
280Dic3⋊C4C24116yes—
281C4×Dic3C24116yes—
282C4×Dic3C24116yes—
283C25—31no—
284C25C22×D311yescharacteristic
285C22×Q8—312no—
286C22×Q8—312no—
287C22×Q8—312no—
288C22×Q8—124no—
289C22×Q8—124no—
290C22×Q8—124no—
291C22×D4—312no—
292C22×D4—124no—
293C23×C4—31no—
294C23×C4—31no—
295C23×C4—31no—
296C23×C4—31no—
297C23×C4—31no—
298C23×C4—31no—
299C23×C4—31no—
300C23×C4—312no—
301C23×C4—312no—
302C23×C4—124no—
303C23×C4—124no—
304C23×C4C22×D311yescharacteristic
305C23×C4C22×D311yescharacteristic
306C23×C4C22×D311yescharacteristic
307C23×C4C22×D311yescharacteristic
308C23×C4C22×D311yescharacteristic
309C23×C4C22×D311yescharacteristic
310C4⋊Q8—364no—
311C4⋊Q8—364no—
312#126—364no—
313#126—364no—
314#126—364no—
315#126C22×D3164yescomplemented
316#125—364no—
317#125—364no—
318#124—364no—
319#123—364no—
320#123—364no—
321C22⋊Q8—364no—
322C22⋊Q8—364no—
323C22⋊Q8—364no—
324C22⋊Q8—364no—
325C2×C4⋊C4—312no—
326C2×C4⋊C4—312no—
327C2×C4⋊C4—312no—
328C2×C4⋊C4—312no—
329C2×C4⋊C4—312no—
330C2×C4⋊C4—312no—
331C2×C4⋊C4—312no—
332C2×C4⋊C4—312no—
333C2×C4⋊C4—312no—
334C2×C4⋊C4—312no—
335C2×C4⋊C4—312no—
336C2×C4⋊C4—312no—
337C2×C4⋊C4—312no—
338C2×C4⋊C4C22×D3112yes—
339C2×C4⋊C4C22×D3112yes—
340C2×C4⋊C4C22×D3112yes—
341C2×C22⋊C4—312no—
342C2×C22⋊C4—312no—
343C2×C22⋊C4—312no—
344C2×C22⋊C4—312no—
345C2×C22⋊C4—312no—
346C2×C22⋊C4—312no—
347C2×C22⋊C4—312no—
348C2×C22⋊C4C22×D3112yes—
349C2×C22⋊C4C22×D3112yes—
350C2×C22⋊C4C22×D3112yes—
351C2×C42—312no—
352C2×C42—312no—
353C2×C42C22×D3112yes—
354C23×C3—212no—
355C23×C3—212no—
356C23×C3—212no—
357C23×C3—416no—
358C23×C3H(5,𝔽2)14yes—
359C23×C32−1+414yes—
360C23×C32−1+414yes—
361C23×C3C2514yes—
362C23×C3C2513yes—
363C22×D3—212no—
364C22×D3—212no—
365C22×D3—212no—
366C22×D3—416no—
367C22×D3C2514yes—
368C2×C3×C4—212no—
369C2×C3×C4—212no—
370C2×C3×C4—212no—
371C2×C3×C4—212no—
372C2×C3×C4—212no—
373C2×C3×C4—212no—
374C2×C3×C4C2514yes—
375C2×C3×C4C2514yes—
376C2×C3×C4C2514yes—
377C2×C3×C4C2514yes—
378C2×C3×C4C2514yes—
379C2×C3×C4C2514yes—
380#67—416no—
381C2×Dic3—212no—
382C2×Dic3—212no—
383C2×Dic3—212no—
384C2×Dic3—212no—
385C2×Dic3—212no—
386C2×Dic3—212no—
387C2×Dic3—212no—
388C2×Dic3—416no—
389C2×Dic3C2514yes—
390C2×Dic3C2514yes—
391C2×Dic3C2514yes—
392C2×Dic3C2514yes—
393C2×Dic3C2514yes—
394C2×Dic3C2514yes—
395C2×Dic3C2514yes—
396C4×D3—416no—
397Dic6—416no—
398Dic6—416no—
399Dic6—416no—
400C24—24no—
401C24—24no—
402C24—24no—
403C24—36no—
404C24—64no—
405C24—64no—
406C24—64no—
407C24C23×D316yes—
408C24C23×D311yescharacteristic
409C2×Q8—316no—
410C2×Q8—316no—
411C2×Q8—316no—
412C2×Q8—1224no—
413C2×Q8—1224no—
414C2×Q8—1224no—
415C2×D4—316no—
416C2×D4—1224no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—36no—
424C22×C4—36no—
425C22×C4—36no—
426C22×C4—36no—
427C22×C4—36no—
428C22×C4—36no—
429C22×C4—36no—
430C22×C4—316no—
431C22×C4—316no—
432C22×C4—64no—
433C22×C4—64no—
434C22×C4—64no—
435C22×C4—64no—
436C22×C4—64no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—1224no—
440C22×C4—1224no—
441C22×C4C23×D316yes—
442C22×C4C23×D316yes—
443C22×C4C23×D316yes—
444C22×C4C23×D316yes—
445C22×C4C23×D316yes—
446C22×C4C23×D316yes—
447C4⋊C4—316no—
448C4⋊C4—316no—
449C4⋊C4—316no—
450C4⋊C4—316no—
451C4⋊C4—316no—
452C4⋊C4—316no—
453C4⋊C4—316no—
454C4⋊C4—316no—
455C4⋊C4—316no—
456C4⋊C4—316no—
457C4⋊C4—316no—
458C4⋊C4—316no—
459C4⋊C4—316no—
460C4⋊C4C23×D3116yes—
461C4⋊C4C23×D3116yes—
462C4⋊C4C23×D3116yes—
463C22⋊C4—316no—
464C22⋊C4—316no—
465C22⋊C4—316no—
466C22⋊C4—316no—
467C22⋊C4—316no—
468C22⋊C4—316no—
469C22⋊C4—316no—
470C22⋊C4C23×D3116yes—
471C22⋊C4C23×D3116yes—
472C22⋊C4C23×D3116yes—
473C42—316no—
474C42—316no—
475C42C23×D3116yes—
476C22×C3—24no—
477C22×C3—24no—
478C22×C3—24no—
479C22×C3—424no—
480C22×C3#563116yescomplemented
481C22×C3C2×H(5,𝔽2)16yes—
482C22×C3C2×2−1+416yes—
483C22×C3C2×2−1+416yes—
484C22×C3C2611yescharacteristic
485C2×D3—24no—
486C2×D3—24no—
487C2×D3—24no—
488C2×D3—424no—
489C3×C4—24no—
490C3×C4—24no—
491C3×C4—24no—
492C3×C4—24no—
493C3×C4—24no—
494C3×C4—24no—
495Dic3—24no—
496Dic3—24no—
497Dic3—24no—
498Dic3—24no—
499Dic3—24no—
500Dic3—24no—
501Dic3—24no—
502C23—212no—
503C23—212no—
504C23—212no—
505C23—34no—
506C23—416no—
507C23—612no—
508C23—612no—
509C23—612no—
510C23—1216no—
511C23#100914yes—
512C23#101014yes—
513C23#101014yes—
514C23C24×D314yes—
515C23C24×D313yes—
516Q8—1216no—
517Q8—1216no—
518Q8—1216no—
519D4—1216no—
520C2×C4—212no—
521C2×C4—212no—
522C2×C4—212no—
523C2×C4—212no—
524C2×C4—212no—
525C2×C4—212no—
526C2×C4—34no—
527C2×C4—34no—
528C2×C4—34no—
529C2×C4—34no—
530C2×C4—34no—
531C2×C4—34no—
532C2×C4—34no—
533C2×C4—612no—
534C2×C4—612no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—612no—
539C2×C4—612no—
540C2×C4—1216no—
541C2×C4—1216no—
542C2×C4C24×D314yes—
543C2×C4C24×D314yes—
544C2×C4C24×D314yes—
545C2×C4C24×D314yes—
546C2×C4C24×D314yes—
547C2×C4C24×D314yes—
548C2×C3—44no—
549C2×C3C2×#563112yescomplemented
550C2×C3C22×H(5,𝔽2)11yescharacteristic
551C2×C3C22×2−1+411yescharacteristic
552C2×C3C22×2−1+411yescharacteristic
553D3—44no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—424no—
558C22—64no—
559C22—64no—
560C22—64no—
561C22—1224no—
562C22#5712116yesdirect product factor
563C22C2×#100916yes—
564C22C2×#101016yes—
565C22C2×#101016yes—
566C22C25×D311yescharacteristic
567C4—24no—
568C4—24no—
569C4—24no—
570C4—24no—
571C4—24no—
572C4—24no—
573C4—64no—
574C4—64no—
575C4—64no—
576C4—64no—
577C4—64no—
578C4—64no—
579C4—64no—
580C3C22×#56311yescharacteristic, complemented, Sylow
581C2—44no—
582C2—124no—
583C2C2×#5712112yesdirect product factor
584C2C22×#101011yescharacteristic
585C2C22×#101011yescharacteristic
586C2C22×#100911yescharacteristic
subgroupG/H
centerC24C23×D3
commutator, cosocleC22×C3C26
FrattiniC22C25×D3
FittingC22×C3×#126C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC22×#126D3
hypocenterC3C22×#563
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
1. C22×C3×#126C2semidirect
2. C2×#5712C2direct (N × H)
3. C22×#896C2semidirect
4. C22×#897C2semidirect
5. C22×#888C2semidirect
6. C22×#898C2semidirect
7. C22×#890C2semidirect
8. C22×#889C2semidirect
10. C22×#879C2semidirect
13. C22×#878C2semidirect
14. C22×#878C2semidirect
15. C22×#876C2semidirect
16. C22×#869C2semidirect
24. C2×C3×#126C22semidirect
54. #5712C22direct (N × H)
55. C2×#898C22semidirect
56. C2×#897C22semidirect
57. C2×#896C22semidirect
58. C2×#890C22semidirect
59. C2×#889C22semidirect
60. C2×#888C22semidirect
64. C2×#879C22semidirect
65. C2×#878C22semidirect
66. C2×#878C22semidirect
67. C2×#876C22semidirect
68. C2×#869C22semidirect
75. C22×#126D3semidirect
112. C3×#126C23semidirect
142. #898C23semidirect
143. #897C23semidirect
144. #896C23semidirect
145. #890C23semidirect
146. #889C23semidirect
147. #888C23semidirect
151. #879C23semidirect
152. #878C23semidirect
153. #878C23semidirect
154. #876C23semidirect
155. #869C23semidirect
168. C2×#126C2×D3semidirect
315. #126C22×D3semidirect
480. C22×C3#563semidirect
549. C2×C3C2×#563semidirect
562. C22#5712direct (N × H)
580. C3C22×#563semidirect
583. C2C2×#5712direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1262
C2×#57122
C22×#8962
C22×#8972
C22×#8882
C22×#8982
C22×#8902
C22×#8892
C22×#8832
C22×#8792
C22×#8852
C22×#8862
C22×#8782
C22×#8762
C22×#8692
C22×#5633