Finite Group Databasebeta

← C2×#19478 #6354546

Subgroups — C2×#19478 #6354546

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3231C211yesmaximal, characteristic
2C2×#3104C211yesmaximal, characteristic
3C2×#3078C211yesmaximal, characteristic
4C2×#2984C211yesmaximal, characteristic
5#19478C2116yesmaximal, direct product factor
6C2×#1704C211yesmaximal, characteristic, complemented
7C2×#1700C211yesmaximal, characteristic, complemented
8C2×#1696C211yesmaximal, characteristic, complemented
9C2×#1692C211yesmaximal, characteristic, complemented
10C2×#1684C211yesmaximal, characteristic, complemented
11C2×#1680C211yesmaximal, characteristic, complemented
12C2×#1676C211yesmaximal, characteristic, complemented
13C2×#1672C211yesmaximal, characteristic, complemented
14C2×#1574C211yesmaximal, characteristic
15C2×#1559C211yesmaximal, characteristic
16C2×#1554C211yesmaximal, characteristic
17C2×#533C2211yescharacteristic
18C2×#533C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20#3231—24no—
21#3231C2218yes—
22C2×#500C2211yescharacteristic
23C2×#498C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#487C2211yescharacteristic
26#3104—24no—
27#3104C2218yes—
28C2×#484—21no—
29C2×#485—21no—
30C2×#485—21no—
31#3078—24no—
32#3078C2218yes—
33C2×#479—21no—
34C2×#479—21no—
35C2×#479—21no—
36C2×#479—21no—
37C2×#477C2211yescharacteristic
38C2×#477C2211yescharacteristic
39C2×#477C2211yescharacteristic
40C2×#477C2211yescharacteristic
41C2×#475C2211yescharacteristic
42C2×#475C2211yescharacteristic
43C2×#462C2211yescharacteristic
44C2×#462C2211yescharacteristic
45C2×#451—21no—
46#2984—24no—
47#2984C2218yes—
48C2×#446—21no—
49C2×#446—21no—
50C2×#446C2211yescharacteristic
51C2×#446C2211yescharacteristic
52C2×C8×Q8C2211yescharacteristic
53C2×C8×Q8C2211yescharacteristic
54C2×C4×Q16—21no—
55C2×C4×Q16—21no—
56C2×#432—21no—
57C2×#432C2211yescharacteristic
58C2×#431—21no—
59C2×#431C2211yescharacteristic
60C2×#431C2211yescharacteristic
61C2×#428—21no—
62C2×#423C2211yescharacteristic
63C2×#419—21no—
64#1704C2218yescomplemented
65#1700C2218yescomplemented
66#1696C2218yescomplemented
67#1692C2218yescomplemented
68#1684C2218yescomplemented
69#1680C2218yescomplemented
70#1676C2218yescomplemented
71#1672C2218yescomplemented
72#1574C2218yes—
73C2×#335C2211yescharacteristic
74C2×#335C2211yescharacteristic
75C2×#334C2211yescharacteristic
76C2×#334C2211yescharacteristic
77#1559C2218yes—
78#1554C2218yes—
79C2×#333C2211yescharacteristic
80C2×#333C2211yescharacteristic
81C2×#333C2211yescharacteristic
82C2×#333C2211yescharacteristic
83C2×#332C2211yescharacteristic
84C2×#332C2211yescharacteristic
85C2×#332C2211yescharacteristic
86C2×#332C2211yescharacteristic
87C23×Q8D411yescharacteristic
88C22×Q16—21no—
89C22×Q16—21no—
90C22×M16—21no—
91#533—22no—
92#533—22no—
93#533—22no—
94#533—22no—
95#533—22no—
96#533—22no—
97#533C2314yes—
98#533C2314yes—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—21no—
104C2×#125—21no—
105C2×C22⋊Q8—21no—
106C2×C22⋊Q8—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8—22no—
113C2×C4×Q8—21no—
114C2×C4×Q8—21no—
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8D411yescharacteristic
118C2×C4×Q8D411yescharacteristic
119C2×C4×Q8D414yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×Q8C2314yes—
123C2×#117—21no—
124C2×#117—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4D411yescharacteristic
127C22×C42—21no—
128C22×C42C2311yescharacteristic
129#500—22no—
130#500C2314yes—
131#498—22no—
132#498C2314yes—
133#487—22no—
134#487—22no—
135#487C2314yes—
136#487C2314yes—
137#485—28no—
138#485—28no—
139#484—28no—
140#479—28no—
141#479—28no—
142#479—28no—
143#479—28no—
144#477—22no—
145#477—22no—
146#477—22no—
147#477—22no—
148#477C2314yes—
149#477C2314yes—
150#477C2314yes—
151#477C2314yes—
152#475—22no—
153#475—22no—
154#475C2314yes—
155#475C2314yes—
156#462—22no—
157#462—22no—
158#462C2314yes—
159#462C2314yes—
160#451—28no—
161#446—28no—
162#446—28no—
163#446—22no—
164#446—22no—
165#446C2314yes—
166#446C2314yes—
167C8×Q8—22no—
168C8×Q8—22no—
169C8×Q8C2314yes—
170C8×Q8C2314yes—
171C4×Q16—28no—
172C4×Q16—28no—
173#432—22no—
174#432—28no—
175#432C2314yes—
176#431—22no—
177#431—28no—
178#431D414yes—
179#431C2314yes—
180#431C2314yes—
181#428—28no—
182C2×#107—21no—
183C2×#107—21no—
184C2×#107—21no—
185C2×#107—21no—
186C2×#107—21no—
187C2×#107—21no—
188C2×#106—21no—
189C2×#106—21no—
190C2×#106—21no—
191C2×#106—21no—
192#423—22no—
193#423C2314yes—
194C2×C4⋊C8—21no—
195C2×C4⋊C8—21no—
196C2×C4⋊C8—21no—
197C2×C4⋊C8—21no—
198C2×C4⋊C8—21no—
199C2×C4⋊C8D411yescharacteristic
200C2×C4⋊C8D411yescharacteristic
201C2×C4⋊C8C2311yescharacteristic
202C2×C4⋊C8C2311yescharacteristic
203C2×C4⋊C8C2311yescharacteristic
204C2×C4⋊C8C2311yescharacteristic
205C2×C4⋊C8C2311yescharacteristic
206C2×C4⋊C8C2311yescharacteristic
207#419—28no—
208C2×#103—21no—
209C2×#103—21no—
210C2×#103—21no—
211C2×#103—21no—
212C2×#103—21no—
213C2×#103—21no—
214C2×#103—21no—
215C2×#103—21no—
216C2×#103—21no—
217C2×#103—21no—
218C2×#103—21no—
219C2×#103—21no—
220C2×C22⋊C8—21no—
221C2×C22⋊C8—21no—
222C2×C22⋊C8D411yescharacteristic
223C2×C22⋊C8D411yescharacteristic
224C2×#97—21no—
225C2×#97—21no—
226C2×#97—21no—
227C2×C4×C8—21no—
228C2×C4×C8—21no—
229C2×C4×C8—21no—
230C2×C4×C8C2311yescharacteristic
231C2×C4×C8C2311yescharacteristic
232#335C2314yes—
233#335C2314yes—
234#334C2314yes—
235#334C2314yes—
236#333C2314yes—
237#333C2314yes—
238#333C2314yes—
239#333C2314yes—
240#332C2314yes—
241#332C2314yes—
242#332C2314yes—
243#332C2314yes—
244C22×Q8—21no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—21no—
248C22×Q8—21no—
249C22×Q8—21no—
250C22×Q8—21no—
251C22×Q8—42no—
252C22×Q8C2×D411yescharacteristic
253C22×Q8C2×D414yes—
254C22×Q8C2×D411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4#4111yescharacteristic
260C2×Q16—24no—
261C2×Q16—24no—
262C2×Q16—44no—
263C2×Q16—44no—
264C2×M16—24no—
265C2×M16—44no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8—21no—
269C22×C8—21no—
270C22×C8—21no—
271C22×C8—21no—
272C22×C8C2×D411yescharacteristic
273C22×C8C2×D411yescharacteristic
274C22×C8C2×D411yescharacteristic
275C22×C8C2×D411yescharacteristic
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8—21no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8C2412yes—
289C4⋊Q8C2412yes—
290C4⋊Q8C2412yes—
291C4⋊Q8C2412yes—
292#125—24no—
293#125—24no—
294#125—42no—
295#125—42no—
296C22⋊Q8—24no—
297C22⋊Q8—24no—
298C22⋊Q8—24no—
299C22⋊Q8—24no—
300C22⋊Q8—42no—
301C22⋊Q8—42no—
302C22⋊Q8—42no—
303C22⋊Q8—42no—
304C4×Q8—24no—
305C4×Q8—22no—
306C4×Q8—24no—
307C4×Q8—22no—
308C4×Q8—21no—
309C4×Q8—24no—
310C4×Q8—21no—
311C4×Q8—24no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—42no—
319C4×Q8—42no—
320C4×Q8—42no—
321C4×Q8—42no—
322C4×Q8C2×D414yes—
323C4×Q8C2×D414yes—
324C4×Q8C2×D412yes—
325C4×Q8C2×D412yes—
326C4×Q8C2412yes—
327C4×Q8C2412yes—
328#117—24no—
329#117—24no—
330#117—42no—
331#117—42no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—41no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4—41no—
351C2×C4⋊C4—41no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D414yes—
355C2×C22⋊C4—41no—
356C2×C22⋊C4—41no—
357C2×C42—21no—
358C2×C42—24no—
359C2×C42—21no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—24no—
363C2×C42—41no—
364C2×C42—41no—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42#4111yescharacteristic
370C2×C42#4112yes—
371C2×C42C2411yescharacteristic
372C2×C42C2412yes—
373#107—24no—
374#107—24no—
375#107—24no—
376#107—24no—
377#107—24no—
378#107—24no—
379#106—24no—
380#106—24no—
381#106—24no—
382#106—24no—
383C4⋊C8—24no—
384C4⋊C8—24no—
385C4⋊C8—24no—
386C4⋊C8—24no—
387C4⋊C8—24no—
388C4⋊C8—21no—
389C4⋊C8—21no—
390C4⋊C8—21no—
391C4⋊C8—21no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8C2×D414yes—
395C4⋊C8C2×D414yes—
396C4⋊C8C2412yes—
397C4⋊C8C2412yes—
398C4⋊C8C2412yes—
399C4⋊C8C2412yes—
400C4⋊C8C2412yes—
401C4⋊C8C2412yes—
402#103—24no—
403#103—24no—
404#103—24no—
405#103—24no—
406#103—24no—
407#103—24no—
408#103—24no—
409#103—24no—
410#103—24no—
411#103—24no—
412#103—24no—
413#103—24no—
414C22⋊C8—24no—
415C22⋊C8—24no—
416C22⋊C8C2×D414yes—
417C22⋊C8C2×D414yes—
418#97—24no—
419#97—24no—
420#97—24no—
421C4×C8—24no—
422C4×C8—24no—
423C4×C8—24no—
424C4×C8C2×D412yes—
425C4×C8C2×D412yes—
426C4×C8C2412yes—
427C4×C8C2412yes—
428C24#12111yescharacteristic
429C2×Q8—21no—
430C2×Q8—24no—
431C2×Q8—21no—
432C2×Q8—22no—
433C2×Q8—28no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—48no—
443C2×Q8—41no—
444C2×Q8—41no—
445C2×Q8—41no—
446C2×Q8—41no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—82no—
452C2×Q8—82no—
453C2×Q8#13714yes—
454C2×Q8C22×D412yes—
455C2×Q8C22×D412yes—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4#12112yes—
474C22×C4#12112yes—
475C22×C4#12111yescharacteristic
476C22×C4#12111yescharacteristic
477C22×C4#12111yescharacteristic
478C22×C4#12112yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4111yescharacteristic
484C22×C4C2×#4112yes—
485Q16—48no—
486Q16—48no—
487M16—48no—
488C2×C8—22no—
489C2×C8—22no—
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—42no—
499C2×C8—42no—
500C2×C8—42no—
501C2×C8—42no—
502C2×C8—42no—
503C2×C8—42no—
504C2×C8C22×D412yes—
505C2×C8C22×D412yes—
506C2×C8C22×D412yes—
507C2×C8C22×D412yes—
508C4⋊C4—24no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—41no—
526C4⋊C4—41no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—41no—
531C4⋊C4—41no—
532C4⋊C4—41no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4C22×D412yes—
538C4⋊C4C22×D412yes—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C42—24no—
542C42—22no—
543C42—24no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—44no—
548C42—44no—
549C42#12112yes—
550C42#12112yes—
551C42#12112yes—
552C42C22×D412yes—
553C42C22×D411yescharacteristic
554C42C22×D412yes—
555C42C22×D411yescharacteristic
556C42C2×#4112yes—
557C42C2×#4111yescharacteristic
558C42C2511yescharacteristic
559C23—22no—
560C23—22no—
561C23—22no—
562C23C2×#12112yes—
563C23C2×#12111yescharacteristic
564Q8—24no—
565Q8—22no—
566Q8—22no—
567Q8—22no—
568Q8—48no—
569Q8—42no—
570Q8—42no—
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—82no—
575Q8—82no—
576Q8—82no—
577Q8—82no—
578Q8—82no—
579Q8—82no—
580Q8C2×#13714yes—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4C2×#12112yes—
612C2×C4C2×#12112yes—
613C2×C4C2×#12112yes—
614C2×C4C2×#12111yescharacteristic
615C2×C4C2×#12111yescharacteristic
616C2×C4C2×#12111yescharacteristic
617C2×C4C2×#13712yes—
618C2×C4#57512yes—
619C2×C4#57812yes—
620C2×C4#57812yes—
621C2×C4C23×D411yescharacteristic
622C2×C4C23×D411yescharacteristic
623C2×C4C22×#4111yescharacteristic
624C8—22no—
625C8—22no—
626C8—22no—
627C8—22no—
628C8—42no—
629C8—42no—
630C8—42no—
631C8—42no—
632C8—42no—
633C8—42no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—44no—
638C22#304412yes—
639C22#305612yes—
640C22#306712yes—
641C22C22×#12111yescharacteristic
642C4—22no—
643C4—22no—
644C4—24no—
645C4—22no—
646C4—21no—
647C4—21no—
648C4—42no—
649C4—42no—
650C4—42no—
651C4—82no—
652C4—82no—
653C4—82no—
654C4—82no—
655C4#304412yes—
656C4#305612yes—
657C4C22×#13711yescharacteristic
658C4C2×#57511yescharacteristic
659C4C2×#57811yescharacteristic
660C4C2×#57811yescharacteristic
661C2—42no—
662C2#1947814yesdirect product factor
663C2C2×#304411yescharacteristic
664C2C2×#305611yescharacteristic
665C2C2×#306711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC42C25
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #19478C2direct (N × H)
6. C2×#1704C2semidirect
7. C2×#1700C2semidirect
8. C2×#1696C2semidirect
9. C2×#1692C2semidirect
10. C2×#1684C2semidirect
11. C2×#1680C2semidirect
12. C2×#1676C2semidirect
13. C2×#1672C2semidirect
64. #1704C22semidirect
65. #1700C22semidirect
66. #1696C22semidirect
67. #1692C22semidirect
68. #1684C22semidirect
69. #1680C22semidirect
70. #1676C22semidirect
71. #1672C22semidirect
662. C2#19478direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#32312
C2×#31042
C2×#30782
C2×#29842
#194782
C2×#17042
C2×#17002
C2×#16962
C2×#16922
C2×#16842
C2×#16802
C2×#16762
C2×#16722
C2×#15742
C2×#15592
C2×#15542