Finite Group Databasebeta

← C2×#11216 #358075

Subgroups — C2×#11216 #358075

Order 512.

1970 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#428C211yesmaximal, characteristic
2C22×#424C211yesmaximal, characteristic
3C22×#343C211yesmaximal, characteristic
4C22×#343C211yesmaximal, characteristic
5C22×#341C211yesmaximal, characteristic
6C2×#1736C211yesmaximal, characteristic
7C22×#340C211yesmaximal, characteristic
8#11216C218yesmaximal, direct product factor
9C23×M16C2211yescharacteristic
10C23×M16C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C24×C8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C2×#428—24no—
17C2×#428—22no—
18C2×#428C418yescomplemented
19C2×#428C2214yes—
20C22×#107—21no—
21C22×#106—21no—
22C2×#424—24no—
23C2×#424—22no—
24C2×#424—24no—
25C2×#424C2214yes—
26C22×C4⋊C8—22no—
27C2×#343—24no—
28C2×#343—24no—
29C2×#343C2214yes—
30C2×#343C2214yes—
31C2×#341—24no—
32C2×#341C2214yes—
33#1736—22no—
34#1736C2214yes—
35C2×#340—22no—
36C2×#340—22no—
37C2×#340—21no—
38C2×#340—21no—
39C2×#340C2214yes—
40C24×C4—21no—
41C24×C4—21no—
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C22×M16—21no—
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—21no—
49C22×M16—21no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16C2×C414yes—
60C22×M16D414yes—
61C22×M16D414yes—
62C22×M16Q814yes—
63C22×M16Q814yes—
64C22×M16C2312yes—
65C22×M16C2312yes—
66C22×M16C2312yes—
67C23×C8—21no—
68C23×C8—21no—
69C23×C8—22no—
70C23×C8—22no—
71C23×C8—21no—
72C23×C8—22no—
73C23×C8—22no—
74C23×C8—22no—
75C23×C8—22no—
76C23×C8D412yes—
77C23×C8D412yes—
78C23×C8C2312yes—
79C2×#117—22no—
80C2×#117—21no—
81C2×#117—22no—
82C2×#117—21no—
83C2×#117C2×C414yes—
84C2×#117D414yes—
85C2×#117D412yes—
86C2×#117C2312yes—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2×C414yes—
104C22×C4⋊C4D411yescharacteristic
105C22×C4⋊C4D411yescharacteristic
106C22×C4⋊C4Q814yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4—21no—
110C22×C42—21no—
111#428—216no—
112#428—28no—
113#428C2×C4116yescomplemented
114C2×#107—28no—
115C2×#107—24no—
116C2×#106—28no—
117C2×#106—24no—
118#424—216no—
119#424—28no—
120#424—28no—
121C2×C4⋊C8—216no—
122C2×C4⋊C8—28no—
123#343—28no—
124#343—28no—
125#341—28no—
126#340—28no—
127#340—24no—
128#340—24no—
129C25#4111yescharacteristic
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—41no—
137C23×C4—41no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—82no—
143C23×C4—82no—
144C23×C4—82no—
145C23×C4—82no—
146C23×C4C22×C412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×Q812yes—
155C23×C4C2×Q812yes—
156C23×C4C2×Q812yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4#4111yescharacteristic
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4C2411yescharacteristic
164C2×M16—24no—
165C2×M16—28no—
166C2×M16—24no—
167C2×M16—28no—
168C2×M16—24no—
169C2×M16—28no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—24no—
174C2×M16—24no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—28no—
178C2×M16—28no—
179C2×M16—28no—
180C2×M16—28no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—42no—
190C2×M16—42no—
191C2×M16—42no—
192C2×M16—44no—
193C2×M16—44no—
194C2×M16—42no—
195C2×M16—44no—
196C2×M16—44no—
197C2×M16C4⋊C4116yes—
198C2×M16C22×C414yes—
199C2×M16C2×D414yes—
200C2×M16C2×D414yes—
201C2×M16C2×Q814yes—
202C2×M16C2×Q814yes—
203C22×C8—24no—
204C22×C8—24no—
205C22×C8—24no—
206C22×C8—22no—
207C22×C8—24no—
208C22×C8—24no—
209C22×C8—24no—
210C22×C8—24no—
211C22×C8—24no—
212C22×C8—24no—
213C22×C8—24no—
214C22×C8—24no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8—22no—
221C22×C8—22no—
222C22×C8—22no—
223C22×C8—44no—
224C22×C8—44no—
225C22×C8—42no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8—44no—
229C22×C8—42no—
230C22×C8—44no—
231C22×C8C2×D414yes—
232#117—24no—
233#117—24no—
234#117—24no—
235#117—24no—
236#117—24no—
237#117—22no—
238#117—22no—
239#117C22⋊C418yes—
240#117C22×C414yes—
241#117C2×D414yes—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4C22⋊C414yes—
281C2×C4⋊C4C22⋊C414yes—
282C2×C4⋊C4C22×C414yes—
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×Q814yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294#107—216no—
295#106—216no—
296C4⋊C8—232no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24C4×D412yes—
302C24#12111yescharacteristic
303C24#12111yescharacteristic
304C24C22⋊Q812yes—
305C24#12312yes—
306C24#12512yes—
307C24#12512yes—
308C24#12612yes—
309C24C2×#4111yescharacteristic
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—88no—
329C22×C4—88no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—88no—
339C22×C4—88no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C4⋊C414yes—
347C22×C4#11714yes—
348C22×C4C4×D412yes—
349C22×C4C4×D412yes—
350C22×C4C4×D412yes—
351C22×C4C4×Q814yes—
352C22×C4C4×Q814yes—
353C22×C4#12012yes—
354C22×C4#12012yes—
355C22×C4#12114yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4#12112yes—
360C22×C4#12112yes—
361C22×C4#12112yes—
362C22×C4#12114yes—
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4C22⋊Q814yes—
366C22×C4C22⋊Q812yes—
367C22×C4C22⋊Q812yes—
368C22×C4C22⋊Q812yes—
369C22×C4C22⋊Q812yes—
370C22×C4C22⋊Q814yes—
371C22×C4C22⋊Q814yes—
372C22×C4C22⋊Q812yes—
373C22×C4C22⋊Q812yes—
374C22×C4C22⋊Q812yes—
375C22×C4C22⋊Q812yes—
376C22×C4C22⋊Q812yes—
377C22×C4#12312yes—
378C22×C4#12312yes—
379C22×C4#12314yes—
380C22×C4#12312yes—
381C22×C4#12414yes—
382C22×C4#12514yes—
383C22×C4#12514yes—
384C22×C4#12512yes—
385C22×C4#12512yes—
386C22×C4#12614yes—
387C22×C4#12612yes—
388C22×C4C4⋊Q812yes—
389C22×C4C4⋊Q812yes—
390C22×C4C4⋊Q814yes—
391C22×C4C4⋊Q814yes—
392C22×C4C23×C411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×Q811yescharacteristic
399C22×C4C22×Q811yescharacteristic
400C22×C4C22×Q811yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406M16—24no—
407M16—24no—
408M16—24no—
409M16—216no—
410M16—24no—
411M16—24no—
412M16—24no—
413M16—24no—
414M16—42no—
415M16—42no—
416M16—42no—
417M16—42no—
418M16—48no—
419M16—48no—
420M16C2×C4⋊C418yes—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—216no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—48no—
439C2×C8—48no—
440C2×C8—48no—
441C2×C8—48no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4C2×C22⋊C414yes—
459C4⋊C4C2×C22⋊C414yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—28no—
463C42—24no—
464C42—24no—
465C42—28no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23#38612yes—
487C23#38612yes—
488C23#38714yes—
489C23#38914yes—
490C23#39612yes—
491C23#39612yes—
492C23#39814yes—
493C23#39812yes—
494C23#39812yes—
495C23#40014yes—
496C23#47112yes—
497C23#47112yes—
498C23#47212yes—
499C23#47312yes—
500C23C2×C4×D411yescharacteristic
501C23C2×#12111yescharacteristic
502C23C2×C22⋊Q811yescharacteristic
503C23C2×#12311yescharacteristic
504C23C2×#12511yescharacteristic
505C23C2×#12511yescharacteristic
506C23C2×#12611yescharacteristic
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—84no—
518C2×C4—88no—
519C2×C4—84no—
520C2×C4—88no—
521C2×C4—88no—
522C2×C4—88no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4#38014yes—
530C2×C4#38014yes—
531C2×C4#38612yes—
532C2×C4#38612yes—
533C2×C4#38714yes—
534C2×C4#38714yes—
535C2×C4#38714yes—
536C2×C4#38914yes—
537C2×C4#38914yes—
538C2×C4#38914yes—
539C2×C4#39114yes—
540C2×C4#39114yes—
541C2×C4#39414yes—
542C2×C4#39414yes—
543C2×C4#39514yes—
544C2×C4#39514yes—
545C2×C4#39614yes—
546C2×C4#39614yes—
547C2×C4#39612yes—
548C2×C4#39612yes—
549C2×C4#39814yes—
550C2×C4#39814yes—
551C2×C4#39814yes—
552C2×C4#39812yes—
553C2×C4#39812yes—
554C2×C4#40014yes—
555C2×C4C22×C22⋊C411yescharacteristic
556C2×C4C22×C4⋊C411yescharacteristic
557C2×C4C2×#11711yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C2×C4×Q811yescharacteristic
560C2×C4C2×C4×Q811yescharacteristic
561C2×C4C2×#12011yescharacteristic
562C2×C4C2×#12111yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×#12111yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×C22⋊Q811yescharacteristic
568C2×C4C2×C22⋊Q811yescharacteristic
569C2×C4C2×C22⋊Q811yescharacteristic
570C2×C4C2×C22⋊Q811yescharacteristic
571C2×C4C2×C22⋊Q811yescharacteristic
572C2×C4C2×C22⋊Q811yescharacteristic
573C2×C4C2×C22⋊Q811yescharacteristic
574C2×C4C2×#12311yescharacteristic
575C2×C4C2×#12311yescharacteristic
576C2×C4C2×#12411yescharacteristic
577C2×C4C2×#12511yescharacteristic
578C2×C4C2×#12511yescharacteristic
579C2×C4C2×#12611yescharacteristic
580C2×C4C2×C4⋊Q811yescharacteristic
581C2×C4C2×C4⋊Q811yescharacteristic
582C2×C4C2×C4⋊Q811yescharacteristic
583C8—28no—
584C8—44no—
585C8—44no—
586C8—44no—
587C22—21no—
588C22—21no—
589C22—22no—
590C22—21no—
591C22—21no—
592C22—22no—
593C22—21no—
594C22—21no—
595C22—22no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—21no—
600C22—21no—
601C22—22no—
602C22—21no—
603C22—21no—
604C22—22no—
605C22—28no—
606C22—28no—
607C22—28no—
608C22—28no—
609C22#144114yes—
610C22#144114yes—
611C22#190714yes—
612C22#190814yes—
613C22#206314yes—
614C22#206414yes—
615C22#208914yes—
616C22#209014yes—
617C22C2×#38611yescharacteristic
618C22C2×#38711yescharacteristic
619C22C2×#38911yescharacteristic
620C22C2×#39611yescharacteristic
621C22C2×#39811yescharacteristic
622C22C2×#39811yescharacteristic
623C22C2×#40011yescharacteristic
624C22C2×#47111yescharacteristic
625C22C2×#47111yescharacteristic
626C22C2×#47211yescharacteristic
627C22C2×#47311yescharacteristic
628C4—44no—
629C4—44no—
630C4—84no—
631C4—84no—
632C4#144114yes—
633C4#144114yes—
634C4C2×#38011yescharacteristic
635C4C2×#38711yescharacteristic
636C4C2×#38911yescharacteristic
637C4C2×#39111yescharacteristic
638C4C2×#39411yescharacteristic
639C4C2×#39511yescharacteristic
640C4C2×#39611yescharacteristic
641C4C2×#39811yescharacteristic
642C2—21no—
643C2—21no—
644C2—24no—
645C2—21no—
646C2—21no—
647C2#1121618yesdirect product factor
648C2C2×#144111yescharacteristic
649C2C2×#190711yescharacteristic
650C2C2×#190811yescharacteristic
651C2C2×#206311yescharacteristic
652C2C2×#206411yescharacteristic
653C2C2×#208911yescharacteristic
654C2C2×#209011yescharacteristic
subgroupG/H
center, socleC24#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11216C2direct (N × H)
18. C2×#428C4semidirect
113. #428C2×C4semidirect
647. C2#11216direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#4282
C22×#4242
C22×#3432
C22×#3412
C2×#17362
C22×#3402
#112162