Finite Group Databasebeta

← C2×#25297 #6448089

Subgroups — C2×#25297 #6448089

Order 512.

2292 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#464C211yesmaximal, characteristic
2C22×C8×Q8C211yesmaximal, characteristic, complemented
3C2×#2592C211yesmaximal, characteristic
4C2×#2581C211yesmaximal, characteristic, complemented
5#25297C2116yesmaximal, direct product factor
6C2×#2066C211yesmaximal, characteristic, complemented
7C2×#2067C211yesmaximal, characteristic
8C2×#2056C211yesmaximal, characteristic, complemented
9C2×#2055C211yesmaximal, characteristic
10C2×#1969C211yesmaximal, characteristic, complemented
11C2×#1968C211yesmaximal, characteristic
12C2×#1958C211yesmaximal, characteristic
13C2×#1949C211yesmaximal, characteristic, complemented
14C2×#1934C211yesmaximal, characteristic
15C2×#1933C211yesmaximal, characteristic, complemented
16C2×C4×#103C211yesmaximal, characteristic, complemented
17C23×Q16C2211yescharacteristic
18C23×SD16C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C2×#464—28no—
23C2×#464C2218yes—
24C2×C8×Q8—28no—
25C2×C8×Q8C2218yescomplemented
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31C22×#103C2211yescharacteristic
32C22×#103C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#102C2211yescharacteristic
37C22×#102C2211yescharacteristic
38C22×#102C2211yescharacteristic
39C22×C4×C8C2211yescharacteristic
40C22×C4×C8C2211yescharacteristic
41C22×C4×C8C2211yescharacteristic
42#2592—24no—
43#2592C2218yes—
44#2581—24no—
45#2581C2218yescomplemented
46C2×#400—21no—
47C2×#397—21no—
48C2×#395—21no—
49C2×#393—21no—
50C2×#391C2211yescharacteristic
51C2×#391C2211yescharacteristic
52C2×#391C2211yescharacteristic
53C2×#391C2211yescharacteristic
54C2×#390C2211yescharacteristic
55C2×#390C2211yescharacteristic
56C2×#389C2211yescharacteristic
57C2×#389C2211yescharacteristic
58C2×#388—21no—
59C2×#388—21no—
60C2×#387—21no—
61C2×#387—21no—
62C2×#382C2211yescharacteristic
63C2×C4×C4⋊C4C2211yescharacteristic
64#2067C2218yes—
65#2066C2218yescomplemented
66#2056C2218yescomplemented
67#2055C2218yes—
68#1969C2218yescomplemented
69#1968C2218yes—
70#1958C2218yes—
71#1949C2218yescomplemented
72#1934C2218yes—
73#1933C2218yescomplemented
74C4×#103C2218yescomplemented
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#340C2211yescharacteristic
78C2×#340C2211yescharacteristic
79C23×Q8D411yescharacteristic
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C22×Q16—24no—
84C22×Q16—24no—
85C22×Q16—42no—
86C22×Q16C2314yes—
87C22×SD16—24no—
88C22×SD16—24no—
89C22×SD16—42no—
90C22×SD16C2314yes—
91C23×C8C2311yescharacteristic
92C23×C8C2311yescharacteristic
93C23×C8C2311yescharacteristic
94C23×C8C2311yescharacteristic
95C2×#124—22no—
96C2×#124—24no—
97C2×#124—24no—
98C2×#124C2314yes—
99C2×C4×Q8—24no—
100C2×C4×Q8—22no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×Q8D414yes—
105C2×C4×Q8C2314yes—
106C2×C4×Q8C2314yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4D411yescharacteristic
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C42—21no—
125C22×C42—21no—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#464—232no—
130C8×Q8—232no—
131C2×#107—42no—
132C2×#107C2314yes—
133C2×#106—42no—
134C2×#106C2314yes—
135C2×C4⋊C8—24no—
136C2×C4⋊C8—24no—
137C2×C4⋊C8D418yes—
138C2×C4⋊C8C2314yes—
139C2×C4⋊C8C2314yes—
140C2×C4⋊C8C2314yes—
141C2×#103—24no—
142C2×#103—24no—
143C2×#103—24no—
144C2×#103—24no—
145C2×#103—42no—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102—24no—
152C2×#102—24no—
153C2×#102—42no—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×C4×C8—24no—
158C2×C4×C8—24no—
159C2×C4×C8—24no—
160C2×C4×C8C2314yes—
161C2×C4×C8C2314yes—
162C2×C4×C8C2314yes—
163#400—28no—
164#397—28no—
165#395—28no—
166#393—28no—
167#391—22no—
168#391—22no—
169#391—22no—
170#391—22no—
171#391C2314yes—
172#391C2314yes—
173#391C2314yes—
174#391C2314yes—
175#390—22no—
176#390—22no—
177#390C2314yes—
178#390C2314yes—
179#389—22no—
180#389—22no—
181#389C2314yes—
182#389C2314yes—
183#388—28no—
184#388—28no—
185#387—28no—
186#387—28no—
187#382—22no—
188#382C2314yes—
189C4×C4⋊C4—22no—
190C4×C4⋊C4C2314yes—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95—21no—
194C2×#95—21no—
195C2×#95—21no—
196C2×#95—21no—
197C2×#95—21no—
198C2×#95—21no—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208C25—21no—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—21no—
213C22×Q8—21no—
214C22×Q8—22no—
215C22×Q8—22no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—22no—
220C22×Q8—22no—
221C22×Q8—22no—
222C22×Q8—22no—
223C22×Q8—22no—
224C22×Q8—22no—
225C22×Q8C2×D414yes—
226C22×Q8#4114yes—
227C22×Q8C2412yes—
228C22×Q8C2412yes—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4C2412yes—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2411yescharacteristic
251C2×Q16—28no—
252C2×Q16—28no—
253C2×Q16—44no—
254C2×Q16—416no—
255C2×SD16—28no—
256C2×SD16—28no—
257C2×SD16—44no—
258C2×SD16—416no—
259C22×C8—22no—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—22no—
265C22×C8—22no—
266C22×C8—22no—
267C22×C8—22no—
268C22×C8C2×D414yes—
269C22×C8C2×D414yes—
270C22×C8#4114yes—
271C22×C8C2412yes—
272C22×C8C2412yes—
273C22×C8C2412yes—
274C22×C8C2412yes—
275#124—28no—
276#124—28no—
277#124—48no—
278C4×Q8—216no—
279C4×Q8—28no—
280C4×Q8—28no—
281C4×Q8—28no—
282C4×Q8—28no—
283C4×Q8—48no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D414yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4—82no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—44no—
335C2×C42—82no—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D414yes—
340C2×C42#4114yes—
341C2×C42C2412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344#107—44no—
345#106—44no—
346C4⋊C8—28no—
347C4⋊C8—28no—
348C4⋊C8C2×D4116yes—
349#103—28no—
350#103—28no—
351#103—28no—
352#103—28no—
353#103—44no—
354#102—28no—
355#102—28no—
356#102—44no—
357C4×C8—28no—
358C4×C8—28no—
359C4×C8—28no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373C24—22no—
374C24—42no—
375C24—42no—
376C24—42no—
377C24—42no—
378C24—42no—
379C24—42no—
380C24—42no—
381C24C22×D411yescharacteristic
382C2×Q8—232no—
383C2×Q8—28no—
384C2×Q8—24no—
385C2×Q8—28no—
386C2×Q8—28no—
387C2×Q8—22no—
388C2×Q8—28no—
389C2×Q8—28no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8—42no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—44no—
411C2×Q8C2×#4114yes—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—42no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—42no—
424C2×D4—42no—
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—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
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—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4#12114yes—
468C22×C4#12114yes—
469C22×C4#13514yes—
470C22×C4#13514yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D412yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C42−1+412yes—
484C22×C4C2511yescharacteristic
485Q16—416no—
486SD16—416no—
487C2×C8—28no—
488C2×C8—28no—
489C2×C8—28no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—44no—
500C2×C8C22×D414yes—
501C2×C8C22×D414yes—
502C2×C8C2×#4114yes—
503C4⋊C4—28no—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—48no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—84no—
529C22⋊C4—84no—
530C42—24no—
531C42—24no—
532C42—22no—
533C42—22no—
534C42—22no—
535C42—22no—
536C42—48no—
537C42—84no—
538C42#12118yes—
539C42C22×D414yes—
540C42C2×#4114yes—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—84no—
549C23—84no—
550C23—84no—
551C23—84no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23C2×#12112yes—
556C23#54812yes—
557C23#55012yes—
558C23C2×#13512yes—
559C23C2×#13512yes—
560C23#57712yes—
561C23#57812yes—
562C23C23×D411yescharacteristic
563Q8—232no—
564Q8—24no—
565Q8—24no—
566Q8—48no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—84no—
574D4—42no—
575D4—42no—
576D4—42no—
577D4—42no—
578D4—84no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—42no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
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—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4C2×#12114yes—
628C2×C4C2×#12114yes—
629C2×C4C2×#12112yes—
630C2×C4#52914yes—
631C2×C4#54214yes—
632C2×C4#54614yes—
633C2×C4#54812yes—
634C2×C4#54814yes—
635C2×C4#55012yes—
636C2×C4C2×#13514yes—
637C2×C4C2×#13514yes—
638C2×C4C23×D411yescharacteristic
639C2×C4C22×#4111yescharacteristic
640C2×C4C22×#4111yescharacteristic
641C2×C4C22×#4111yescharacteristic
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×2−1+411yescharacteristic
644C8—24no—
645C8—24no—
646C8—24no—
647C8—42no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—168no—
656C22#242614yes—
657C22#305014yes—
658C22#306314yes—
659C22#336414yes—
660C22#336514yes—
661C22#339314yes—
662C22#339414yes—
663C22C22×#12111yescharacteristic
664C22C2×#54811yescharacteristic
665C22C2×#55011yescharacteristic
666C22C22×#13511yescharacteristic
667C22C22×#13511yescharacteristic
668C22C2×#57711yescharacteristic
669C22C2×#57811yescharacteristic
670C4—24no—
671C4—24no—
672C4—24no—
673C4—24no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#242614yes—
684C4C2×#52911yescharacteristic
685C4C2×#54211yescharacteristic
686C4C2×#54611yescharacteristic
687C4C2×#54811yescharacteristic
688C2—162no—
689C2#2529718yesdirect product factor
690C2C2×#242611yescharacteristic
691C2C2×#305011yescharacteristic
692C2C2×#306311yescharacteristic
693C2C2×#336411yescharacteristic
694C2C2×#336511yescharacteristic
695C2C2×#339311yescharacteristic
696C2C2×#339411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
2. C22×C8×Q8C2semidirect
4. C2×#2581C2semidirect
5. #25297C2direct (N × H)
6. C2×#2066C2semidirect
8. C2×#2056C2semidirect
10. C2×#1969C2semidirect
13. C2×#1949C2semidirect
15. C2×#1933C2semidirect
16. C2×C4×#103C2semidirect
25. C2×C8×Q8C22semidirect
45. #2581C22semidirect
65. #2066C22semidirect
66. #2056C22semidirect
68. #1969C22semidirect
71. #1949C22semidirect
73. #1933C22semidirect
74. C4×#103C22semidirect
689. C2#25297direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4642
C22×C8×Q82
C2×#25922
C2×#25812
#252972
C2×#20662
C2×#20672
C2×#20562
C2×#20552
C2×#19692
C2×#19682
C2×#19582
C2×#19492
C2×#19342
C2×#19332
C2×C4×#1032