Finite Group Databasebeta

← C2×#28197 #6463769

Subgroups — C2×#28197 #6463769

Order 512.

2488 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#446C211yesmaximal, characteristic, complemented
2C22×#442C211yesmaximal, characteristic
3C2×#2538C211yesmaximal, characteristic, complemented
4C2×#2580C211yesmaximal, characteristic, complemented
5#28197C2116yesmaximal, direct product factor
6C2×#2088C211yesmaximal, characteristic, complemented
7C2×#2088C211yesmaximal, characteristic, complemented
8C2×#2054C211yesmaximal, characteristic
9C2×#2054C211yesmaximal, characteristic
10C2×#1936C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic, complemented
12C2×#1934C211yesmaximal, characteristic, complemented
13C2×#1934C211yesmaximal, characteristic, complemented
14C2×#1836C211yesmaximal, characteristic, complemented
15C2×#1804C211yesmaximal, characteristic, complemented
16C2×#1802C211yesmaximal, characteristic, complemented
17C23×D8C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#446—28no—
23C2×#446C2218yescomplemented
24C2×#442—28no—
25C2×#442C2218yes—
26C22×#106C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#102C2211yescharacteristic
35C22×#102C2211yescharacteristic
36C22×#97C2211yescharacteristic
37C22×#97C2211yescharacteristic
38C22×C4×C8C2211yescharacteristic
39#2580—24no—
40#2580C2218yescomplemented
41#2538—24no—
42#2538C2218yescomplemented
43C2×#399—21no—
44C2×#396—21no—
45C2×#396—21no—
46C2×#393—21no—
47C2×#390C2211yescharacteristic
48C2×#390C2211yescharacteristic
49C2×#390C2211yescharacteristic
50C2×#390C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#389C2211yescharacteristic
54C2×#389C2211yescharacteristic
55C2×#385—21no—
56C2×#385—21no—
57C2×#384C2211yescharacteristic
58C2×#382C2211yescharacteristic
59C2×#380—21no—
60#2088C2218yescomplemented
61#2088C2218yescomplemented
62#2054C2218yes—
63#2054C2218yes—
64#1936C2218yescomplemented
65#1936C2218yescomplemented
66#1934C2218yescomplemented
67#1934C2218yescomplemented
68#1836C2218yescomplemented
69#1804C2218yescomplemented
70#1802C2218yescomplemented
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8D411yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C24×C4—21no—
83C24×C4—21no—
84C22×D8—24no—
85C22×D8—24no—
86C22×D8—24no—
87C22×D8C2314yes—
88C23×C8C2311yescharacteristic
89C23×C8C2311yescharacteristic
90C23×C8C2311yescharacteristic
91C23×C8C2311yescharacteristic
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—48no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8D414yes—
97C2×C4×Q8C2314yes—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4C2314yes—
105C2×C4×D4C2314yes—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#446—232no—
135#442—232no—
136C2×#106—24no—
137C2×#106C2314yes—
138C2×C4⋊C8—24no—
139C2×C4⋊C8—24no—
140C2×C4⋊C8—24no—
141C2×C4⋊C8C2314yes—
142C2×C4⋊C8C2314yes—
143C2×C4⋊C8C2314yes—
144C2×#102—24no—
145C2×#102—24no—
146C2×#102—24no—
147C2×#102—24no—
148C2×#102—24no—
149C2×#102—24no—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#97—24no—
157C2×#97D418yes—
158C2×#97C2314yes—
159C2×#97C2314yes—
160C2×C4×C8—24no—
161C2×C4×C8C2314yes—
162#399—28no—
163#396—28no—
164#396—28no—
165#393—28no—
166#390—22no—
167#390—22no—
168#390—22no—
169#390—22no—
170#390C2314yes—
171#390C2314yes—
172#390C2314yes—
173#390C2314yes—
174#389—22no—
175#389—22no—
176#389—22no—
177#389—22no—
178#389C2314yes—
179#389C2314yes—
180#389C2314yes—
181#389C2314yes—
182#385—28no—
183#385—28no—
184#384—22no—
185#384C2314yes—
186#382—22no—
187#382C2314yes—
188#380—28no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203C25—21no—
204C25—21no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—42no—
208C22×Q8C2×D414yes—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—21no—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4#4114yes—
222C22×D4#4114yes—
223C22×D4C2412yes—
224C22×D4C2412yes—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4—82no—
238C23×C4—82no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2411yescharacteristic
246C2×D8—232no—
247C2×D8—28no—
248C2×D8—28no—
249C2×D8—28no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—22no—
257C22×C8—22no—
258C22×C8—22no—
259C22×C8—22no—
260C22×C8C2×D414yes—
261C22×C8C2×D414yes—
262C22×C8C2412yes—
263C22×C8C2412yes—
264C22×C8C2412yes—
265C22×C8C2412yes—
266C22⋊Q8—432no—
267C4×Q8—216no—
268C4×Q8—28no—
269C4×Q8—28no—
270C4×D4—28no—
271C4×D4—28no—
272C4×D4—28no—
273C4×D4—28no—
274C4×D4—48no—
275C4×D4—48no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4—82no—
308C2×C4⋊C4—82no—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2×D414yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—82no—
335C2×C22⋊C4—82no—
336C2×C22⋊C4—82no—
337C2×C22⋊C4—82no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—22no—
341C2×C42—22no—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D414yes—
346C2×C42#4114yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349C2×C42C2412yes—
350#106—28no—
351C4⋊C8—28no—
352C4⋊C8—28no—
353C4⋊C8—28no—
354#102—28no—
355#102—28no—
356#102—28no—
357#102—28no—
358#102—28no—
359#102—28no—
360#97—28no—
361#97C2×D4116yes—
362C4×C8—28no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369C24—24no—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—42no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24—42no—
382C24—42no—
383C24—42no—
384C24—42no—
385C24C22×D411yescharacteristic
386C2×Q8—28no—
387C2×Q8—28no—
388C2×Q8—44no—
389C2×Q8—416no—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4—24no—
393C2×D4—24no—
394C2×D4—28no—
395C2×D4—28no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4C2×#4114yes—
415C2×D4C2×#4114yes—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—816no—
462C22×C4—816no—
463C22×C4#12114yes—
464C22×C4#12114yes—
465C22×C4AGL(1,ℤ/8)14yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D412yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C42−1+412yes—
478C22×C42−1+412yes—
479C22×C4C2511yescharacteristic
480D8—232no—
481C2×C8—28no—
482C2×C8—28no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—44no—
494C2×C8—44no—
495C2×C8C22×D414yes—
496C2×C8C22×D414yes—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—48no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—48no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C42—24no—
535C42—24no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—22no—
540C42#12118yes—
541C42C22×D414yes—
542C42C2×#4114yes—
543C23—24no—
544C23—24no—
545C23—48no—
546C23—48no—
547C23—48no—
548C23—48no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—48no—
554C23—48no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—84no—
564C23—84no—
565C23—84no—
566C23—84no—
567C23—84no—
568C23—84no—
569C23—84no—
570C23—84no—
571C23C2×#12112yes—
572C23#54812yes—
573C23#54812yes—
574C23C2×AGL(1,ℤ/8)12yes—
575C23#57512yes—
576C23#57612yes—
577C23#57712yes—
578C23C23×D411yescharacteristic
579Q8—416no—
580D4—24no—
581D4—24no—
582D4—24no—
583D4—24no—
584D4—42no—
585D4—42no—
586D4—42no—
587D4—42no—
588D4—48no—
589D4—48no—
590C2×C4—22no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—816no—
630C2×C4—816no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—84no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4C2×#12114yes—
638C2×C4C2×#12114yes—
639C2×C4C2×#12112yes—
640C2×C4#52514yes—
641C2×C4#53614yes—
642C2×C4#54812yes—
643C2×C4#54812yes—
644C2×C4#55214yes—
645C2×C4#55214yes—
646C2×C4C2×AGL(1,ℤ/8)14yes—
647C2×C4C23×D411yescharacteristic
648C2×C4C22×#4111yescharacteristic
649C2×C4C22×#4111yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C2×C4C2×2−1+411yescharacteristic
652C2×C4C2×2−1+411yescharacteristic
653C8—24no—
654C8—24no—
655C8—42no—
656C8—42no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22—816no—
668C22—816no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22#253614yes—
674C22#305514yes—
675C22#306214yes—
676C22#336614yes—
677C22#337014yes—
678C22#337314yes—
679C22#337614yes—
680C22C22×#12111yescharacteristic
681C22C2×#54811yescharacteristic
682C22C2×#54811yescharacteristic
683C22C22×AGL(1,ℤ/8)11yescharacteristic
684C22C2×#57511yescharacteristic
685C22C2×#57611yescharacteristic
686C22C2×#57711yescharacteristic
687C4—24no—
688C4—24no—
689C4—24no—
690C4—42no—
691C4—42no—
692C4—42no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4#253614yes—
700C4C2×#52511yescharacteristic
701C4C2×#53611yescharacteristic
702C4C2×#55211yescharacteristic
703C4C2×#55211yescharacteristic
704C2—84no—
705C2—84no—
706C2#2819718yesdirect product factor
707C2C2×#253611yescharacteristic
708C2C2×#305511yescharacteristic
709C2C2×#306211yescharacteristic
710C2C2×#336611yescharacteristic
711C2C2×#337311yescharacteristic
712C2C2×#337011yescharacteristic
713C2C2×#337611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#446C2semidirect
3. C2×#2538C2semidirect
4. C2×#2580C2semidirect
5. #28197C2direct (N × H)
6. C2×#2088C2semidirect
7. C2×#2088C2semidirect
10. C2×#1936C2semidirect
11. C2×#1936C2semidirect
12. C2×#1934C2semidirect
13. C2×#1934C2semidirect
14. C2×#1836C2semidirect
15. C2×#1804C2semidirect
16. C2×#1802C2semidirect
23. C2×#446C22semidirect
40. #2580C22semidirect
42. #2538C22semidirect
60. #2088C22semidirect
61. #2088C22semidirect
64. #1936C22semidirect
65. #1936C22semidirect
66. #1934C22semidirect
67. #1934C22semidirect
68. #1836C22semidirect
69. #1804C22semidirect
70. #1802C22semidirect
706. C2#28197direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4462
C22×#4422
C2×#25382
C2×#25802
#281972
C2×#20882
C2×#20542
C2×#19362
C2×#19342
C2×#18362
C2×#18042
C2×#18022