Finite Group Databasebeta

← C2×#10900 #353929

Subgroups — C2×#10900 #353929

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2911C211yesmaximal, characteristic
2C2×#2873C211yesmaximal, characteristic
3C22×#344C211yesmaximal, characteristic
4C22×#341C211yesmaximal, characteristic
5C2×#1735C211yesmaximal, characteristic
6C2×#1733C211yesmaximal, characteristic
7C2×#1729C211yesmaximal, characteristic
8#10900C218yesmaximal, direct product factor
9C23×M16C2211yescharacteristic
10C23×M16C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C24×C8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16#2911—22no—
17#2911—22no—
18#2911C414yescomplemented
19#2911C2214yes—
20C2×#428—22no—
21C2×#428C414yescomplemented
22C2×#427—22no—
23#2873—22no—
24#2873—22no—
25#2873—22no—
26#2873C2214yes—
27C2×#405—22no—
28C2×#405—22no—
29C2×C4×M16—22no—
30C2×#344C418yescomplemented
31C2×#344C2214yes—
32C2×#341C418yescomplemented
33C2×#341C2214yes—
34#1735—22no—
35#1735C2214yes—
36C2×#340—22no—
37#1733—22no—
38#1733C2214yes—
39C2×#339—22no—
40#1729—22no—
41#1729C2214yes—
42C2×#336—22no—
43C24×C4C2311yescharacteristic
44C22×M16—21no—
45C22×M16—21no—
46C22×M16—21no—
47C22×M16—21no—
48C22×M16—21no—
49C22×M16—21no—
50C22×M16—21no—
51C22×M16—21no—
52C22×M16—21no—
53C22×M16—21no—
54C22×M16—22no—
55C22×M16—22no—
56C22×M16—22no—
57C22×M16—22no—
58C22×M16—22no—
59C22×M16—22no—
60C22×M16C2×C412yes—
61C22×M16C2×C412yes—
62C22×M16C2×C414yes—
63C22×M16C2×C414yes—
64C22×M16C2×C414yes—
65C22×M16C2×C414yes—
66C22×M16D412yes—
67C22×M16D412yes—
68C22×M16D412yes—
69C22×M16Q814yes—
70C22×M16Q814yes—
71C22×M16C2312yes—
72C22×M16C2312yes—
73C22×M16C2312yes—
74C23×C8—21no—
75C23×C8—21no—
76C23×C8—21no—
77C23×C8—21no—
78C23×C8—21no—
79C23×C8—21no—
80C23×C8—21no—
81C23×C8—21no—
82C23×C8—22no—
83C23×C8—22no—
84C23×C8—22no—
85C23×C8C2×C414yes—
86C23×C8D412yes—
87C23×C8C2312yes—
88C2×#117—21no—
89C2×#117—21no—
90C2×#117—21no—
91C2×#117—21no—
92C2×#117—21no—
93C2×#117—21no—
94C2×#117—21no—
95C2×#117—21no—
96C2×#117—21no—
97C2×#117—22no—
98C2×#117—21no—
99C2×#117—22no—
100C2×#117—21no—
101C2×#117—21no—
102C2×#117—21no—
103C2×#117—21no—
104C2×#117—21no—
105C2×#117—21no—
106C2×#117—21no—
107C2×#117—21no—
108C2×#117—21no—
109C2×#117—22no—
110C2×#117C2×C414yes—
111C2×#117C2×C412yes—
112C2×#117C2×C412yes—
113C2×#117C2×C412yes—
114C2×#117D412yes—
115C2×#117D412yes—
116C2×#117D412yes—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C2×#117C2312yes—
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2×C412yes—
122C22×C4⋊C4D412yes—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C42—21no—
127C22×C42—21no—
128C22×C42—21no—
129#428—28no—
130#428—28no—
131#428—28no—
132#428C2×C4116yescomplemented
133#427—28no—
134#427—28no—
135C2×#107—24no—
136C2×#106—24no—
137#405—28no—
138#405—28no—
139#405—28no—
140#405—28no—
141C4×M16—28no—
142C4×M16—28no—
143C2×#97—24no—
144C2×C4×C8—24no—
145#344C2×C4116yescomplemented
146#341C2×C4116yescomplemented
147#340—28no—
148#339—28no—
149#336—28no—
150C25#4111yescharacteristic
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—41no—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C412yes—
157C23×C4C22×C412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×Q812yes—
167C23×C4C2×Q812yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4#4111yescharacteristic
171C23×C4#4111yescharacteristic
172C23×C4#4111yescharacteristic
173C23×C4#4112yes—
174C23×C4#4112yes—
175C23×C4C2411yescharacteristic
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—24no—
179C2×M16—24no—
180C2×M16—24no—
181C2×M16—24no—
182C2×M16—24no—
183C2×M16—24no—
184C2×M16—24no—
185C2×M16—24no—
186C2×M16—24no—
187C2×M16—24no—
188C2×M16—24no—
189C2×M16—24no—
190C2×M16—24no—
191C2×M16—24no—
192C2×M16—24no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—24no—
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—24no—
201C2×M16—24no—
202C2×M16—24no—
203C2×M16—24no—
204C2×M16—24no—
205C2×M16—24no—
206C2×M16—24no—
207C2×M16—22no—
208C2×M16—22no—
209C2×M16—22no—
210C2×M16—22no—
211C2×M16—22no—
212C2×M16—22no—
213C2×M16—22no—
214C2×M16—22no—
215C2×M16—22no—
216C2×M16—22no—
217C2×M16C4218yes—
218C2×M16C4218yes—
219C2×M16C4⋊C418yes—
220C2×M16C4⋊C418yes—
221C2×M16C4⋊C418yes—
222C2×M16C22×C414yes—
223C2×M16C22×C414yes—
224C2×M16C22×C414yes—
225C2×M16C22×C414yes—
226C2×M16C22×C414yes—
227C2×M16C2×D414yes—
228C2×M16C2×Q814yes—
229C2×M16C2×Q814yes—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—24no—
233C22×C8—24no—
234C22×C8—24no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—24no—
240C22×C8—24no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8—24no—
246C22×C8—24no—
247C22×C8—24no—
248C22×C8—24no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8—22no—
255C22×C8—41no—
256C22×C8—41no—
257C22×C8—41no—
258C22×C8—41no—
259C22×C8—42no—
260C22×C8—42no—
261C22×C8—42no—
262C22×C8—42no—
263C22×C8—42no—
264C22×C8—42no—
265C22×C8C4⋊C418yes—
266C22×C8C22×C414yes—
267C22×C8C2×D414yes—
268#117—22no—
269#117—22no—
270#117—22no—
271#117—22no—
272#117—22no—
273#117—22no—
274#117—22no—
275#117—22no—
276#117—22no—
277#117—22no—
278#117—22no—
279#117—22no—
280#117—24no—
281#117—22no—
282#117—24no—
283#117—22no—
284#117—22no—
285#117—22no—
286#117—22no—
287#117—22no—
288#117—24no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—22no—
294#117—41no—
295#117—41no—
296#117—41no—
297#117—41no—
298#117—41no—
299#117—41no—
300#117—42no—
301#117—41no—
302#117—41no—
303#117—42no—
304#117—42no—
305#117C22⋊C414yes—
306#117C22⋊C414yes—
307#117C22×C414yes—
308#117C22×C414yes—
309#117C2×D414yes—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—41no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4C22⋊C414yes—
323C2×C4⋊C4C22⋊C414yes—
324C2×C4⋊C4C22×C414yes—
325C2×C4⋊C4C2×D414yes—
326C2×C22⋊C4—22no—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—41no—
337C2×C22⋊C4—41no—
338C2×C22⋊C4—41no—
339C2×C22⋊C4—41no—
340C2×C22⋊C4—41no—
341C2×C22⋊C4—42no—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—41no—
353C2×C42—41no—
354C2×C42—41no—
355C2×C42—41no—
356C2×C42—41no—
357C2×C42—42no—
358#107—216no—
359#106—216no—
360#97—216no—
361C4×C8—216no—
362C24—44no—
363C24#11712yes—
364C24C4×D411yescharacteristic
365C24C4×D411yescharacteristic
366C24C4×Q812yes—
367C24#12111yescharacteristic
368C24#12111yescharacteristic
369C24#12311yescharacteristic
370C24#12311yescharacteristic
371C24#12512yes—
372C24#12612yes—
373C24C2×#4111yescharacteristic
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—48no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4C2×C4212yes—
386C22×C4C2×C4212yes—
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C4⋊C412yes—
392C22×C4C2×C4⋊C412yes—
393C22×C4C2×C4⋊C412yes—
394C22×C4C2×C4⋊C412yes—
395C22×C4#11712yes—
396C22×C4#11712yes—
397C22×C4#11712yes—
398C22×C4#11712yes—
399C22×C4#11712yes—
400C22×C4C4×D412yes—
401C22×C4C4×D412yes—
402C22×C4C4×D412yes—
403C22×C4C4×D412yes—
404C22×C4C4×D412yes—
405C22×C4C4×D412yes—
406C22×C4C4×D412yes—
407C22×C4C4×D412yes—
408C22×C4C4×D412yes—
409C22×C4C4×D412yes—
410C22×C4C4×D411yescharacteristic
411C22×C4C4×D411yescharacteristic
412C22×C4C4×D411yescharacteristic
413C22×C4C4×D411yescharacteristic
414C22×C4C4×D412yes—
415C22×C4C4×D412yes—
416C22×C4C4×D411yescharacteristic
417C22×C4C4×D411yescharacteristic
418C22×C4C4×Q812yes—
419C22×C4C4×Q812yes—
420C22×C4C4×Q812yes—
421C22×C4C4×Q812yes—
422C22×C4C4×Q812yes—
423C22×C4C4×Q812yes—
424C22×C4C4×Q812yes—
425C22×C4#12011yescharacteristic
426C22×C4#12011yescharacteristic
427C22×C4#12011yescharacteristic
428C22×C4#12011yescharacteristic
429C22×C4#12111yescharacteristic
430C22×C4#12111yescharacteristic
431C22×C4#12111yescharacteristic
432C22×C4#12111yescharacteristic
433C22×C4#12111yescharacteristic
434C22×C4#12111yescharacteristic
435C22×C4C22⋊Q812yes—
436C22×C4C22⋊Q812yes—
437C22×C4C22⋊Q812yes—
438C22×C4C22⋊Q812yes—
439C22×C4C22⋊Q812yes—
440C22×C4C22⋊Q812yes—
441C22×C4C22⋊Q812yes—
442C22×C4C22⋊Q812yes—
443C22×C4#12312yes—
444C22×C4#12312yes—
445C22×C4#12312yes—
446C22×C4#12312yes—
447C22×C4#12311yescharacteristic
448C22×C4#12311yescharacteristic
449C22×C4#12412yes—
450C22×C4#12412yes—
451C22×C4#12512yes—
452C22×C4#12512yes—
453C22×C4#12512yes—
454C22×C4#12612yes—
455C22×C4#12612yes—
456C22×C4#12612yes—
457C22×C4C4⋊Q812yes—
458C22×C4C4⋊Q812yes—
459C22×C4C23×C411yescharacteristic
460C22×C4C23×C411yescharacteristic
461C22×C4C23×C411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×Q811yescharacteristic
467C22×C4C22×Q811yescharacteristic
468C22×C4C2×#4111yescharacteristic
469C22×C4C2×#4111yescharacteristic
470C22×C4C2×#4111yescharacteristic
471C22×C4C2×#4111yescharacteristic
472C22×C4C2×#4111yescharacteristic
473M16—24no—
474M16—24no—
475M16—24no—
476M16—24no—
477M16—24no—
478M16—24no—
479M16—24no—
480M16—24no—
481M16—24no—
482M16—24no—
483M16—28no—
484M16—28no—
485M16—28no—
486M16—28no—
487M16—28no—
488M16—28no—
489M16C2×C4218yes—
490M16C2×C4⋊C418yes—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—28no—
500C2×C8—28no—
501C2×C8—28no—
502C2×C8—28no—
503C2×C8—28no—
504C2×C8—28no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—24no—
508C2×C8—42no—
509C2×C8—42no—
510C2×C8—42no—
511C2×C8—42no—
512C2×C8—44no—
513C2×C8—44no—
514C2×C8—44no—
515C2×C8—44no—
516C2×C8C2×C4⋊C418yes—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4C2×C22⋊C418yes—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—24no—
531C22⋊C4—24no—
532C22⋊C4—24no—
533C22⋊C4—42no—
534C22⋊C4—42no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—42no—
538C22⋊C4—44no—
539C42—24no—
540C42—24no—
541C42—24no—
542C42—24no—
543C42—24no—
544C42—42no—
545C42—42no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—44no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—44no—
554C23—42no—
555C23—44no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—42no—
561C23C4×C4⋊C412yes—
562C23C4×C4⋊C412yes—
563C23#38611yescharacteristic
564C23#38611yescharacteristic
565C23#38611yescharacteristic
566C23#38611yescharacteristic
567C23#38712yes—
568C23#38712yes—
569C23#38712yes—
570C23#38712yes—
571C23#38712yes—
572C23#38712yes—
573C23#38812yes—
574C23#38812yes—
575C23#38912yes—
576C23#38912yes—
577C23C2×#11711yescharacteristic
578C23C2×C4×D411yescharacteristic
579C23C2×C4×Q811yescharacteristic
580C23C2×#12111yescharacteristic
581C23C2×#12311yescharacteristic
582C23C2×#12511yescharacteristic
583C23C2×#12611yescharacteristic
584C2×C4—28no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—84no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—88no—
593C2×C4—84no—
594C2×C4—84no—
595C2×C4C4×C22⋊C412yes—
596C2×C4C4×C22⋊C412yes—
597C2×C4C4×C22⋊C412yes—
598C2×C4C4×C22⋊C412yes—
599C2×C4C4×C4⋊C412yes—
600C2×C4C4×C4⋊C412yes—
601C2×C4C4×C4⋊C412yes—
602C2×C4C4×C4⋊C412yes—
603C2×C4C4×C4⋊C412yes—
604C2×C4C4×C4⋊C412yes—
605C2×C4#38512yes—
606C2×C4#38512yes—
607C2×C4#38512yes—
608C2×C4#38512yes—
609C2×C4#38512yes—
610C2×C4#38512yes—
611C2×C4#38512yes—
612C2×C4#38512yes—
613C2×C4#38611yescharacteristic
614C2×C4#38611yescharacteristic
615C2×C4#38611yescharacteristic
616C2×C4#38611yescharacteristic
617C2×C4#38712yes—
618C2×C4#38712yes—
619C2×C4#38712yes—
620C2×C4#38712yes—
621C2×C4#38712yes—
622C2×C4#38712yes—
623C2×C4#38712yes—
624C2×C4#38712yes—
625C2×C4#38712yes—
626C2×C4#38712yes—
627C2×C4#38812yes—
628C2×C4#38812yes—
629C2×C4#38812yes—
630C2×C4#38812yes—
631C2×C4#38812yes—
632C2×C4#38812yes—
633C2×C4#38912yes—
634C2×C4#38912yes—
635C2×C4#38912yes—
636C2×C4#38912yes—
637C2×C4#38912yes—
638C2×C4#38912yes—
639C2×C4#39112yes—
640C2×C4#39112yes—
641C2×C4#39112yes—
642C2×C4#39112yes—
643C2×C4C22×C4211yescharacteristic
644C2×C4C22×C22⋊C411yescharacteristic
645C2×C4C22×C4⋊C411yescharacteristic
646C2×C4C22×C4⋊C411yescharacteristic
647C2×C4C2×#11711yescharacteristic
648C2×C4C2×#11711yescharacteristic
649C2×C4C2×C4×D411yescharacteristic
650C2×C4C2×C4×D411yescharacteristic
651C2×C4C2×C4×D411yescharacteristic
652C2×C4C2×C4×D411yescharacteristic
653C2×C4C2×C4×D411yescharacteristic
654C2×C4C2×C4×D411yescharacteristic
655C2×C4C2×C4×D411yescharacteristic
656C2×C4C2×C4×Q811yescharacteristic
657C2×C4C2×C4×Q811yescharacteristic
658C2×C4C2×C4×Q811yescharacteristic
659C2×C4C2×#12011yescharacteristic
660C2×C4C2×#12111yescharacteristic
661C2×C4C2×C22⋊Q811yescharacteristic
662C2×C4C2×C22⋊Q811yescharacteristic
663C2×C4C2×C22⋊Q811yescharacteristic
664C2×C4C2×C22⋊Q811yescharacteristic
665C2×C4C2×#12311yescharacteristic
666C2×C4C2×#12311yescharacteristic
667C2×C4C2×#12411yescharacteristic
668C2×C4C2×#12511yescharacteristic
669C2×C4C2×#12611yescharacteristic
670C2×C4C2×C4⋊Q811yescharacteristic
671C8—28no—
672C8—28no—
673C8—28no—
674C8—44no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—24no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—42no—
692C22—42no—
693C22—42no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22#143612yes—
698C22#143612yes—
699C22#143612yes—
700C22#143612yes—
701C22#188114yes—
702C22C2×C4×C4⋊C411yescharacteristic
703C22C2×#38611yescharacteristic
704C22C2×#38711yescharacteristic
705C22C2×#38711yescharacteristic
706C22C2×#38711yescharacteristic
707C22C2×#38811yescharacteristic
708C22C2×#38911yescharacteristic
709C4—48no—
710C4—84no—
711C4—84no—
712C4#143612yes—
713C4#143612yes—
714C4#143612yes—
715C4#143612yes—
716C4C2×C4×C22⋊C411yescharacteristic
717C4C2×C4×C4⋊C411yescharacteristic
718C4C2×#38511yescharacteristic
719C4C2×#38511yescharacteristic
720C4C2×#38711yescharacteristic
721C4C2×#38811yescharacteristic
722C4C2×#38911yescharacteristic
723C4C2×#39111yescharacteristic
724C2—21no—
725C2—22no—
726C2—22no—
727C2—22no—
728C2—21no—
729C2—21no—
730C2—21no—
731C2—21no—
732C2—21no—
733C2#1090014yesdirect product factor
734C2C2×#143611yescharacteristic
735C2C2×#188112yes—
subgroupG/H
centerC22×C4#120
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23#386
prime radicalC25#41
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #10900C2direct (N × H)
18. #2911C4semidirect
21. C2×#428C4semidirect
30. C2×#344C4semidirect
32. C2×#341C4semidirect
132. #428C2×C4semidirect
145. #344C2×C4semidirect
146. #341C2×C4semidirect
733. C2#10900direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#29112
C2×#28732
C22×#3442
C22×#3412
C2×#17352
C2×#17332
C2×#17292
#109002