Finite Group Databasebeta

← C2×#30167 #6433495

Subgroups — C2×#30167 #6433495

Order 512.

2270 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic, complemented
2C2×C4×#121C211yesmaximal, characteristic
3#30167C2116yesmaximal, direct product factor
4C2×#2085C211yesmaximal, characteristic, complemented
5C2×#2076C211yesmaximal, characteristic, complemented
6C2×#2077C211yesmaximal, characteristic, complemented
7C2×#2073C211yesmaximal, characteristic, complemented
8C2×#2059C211yesmaximal, characteristic, complemented
9C2×#2034C211yesmaximal, characteristic
10C2×#2033C211yesmaximal, characteristic
11C2×#2025C211yesmaximal, characteristic, complemented
12C2×#1920C211yesmaximal, characteristic, complemented
13C2×#1918C211yesmaximal, characteristic, complemented
14C2×#1864C211yesmaximal, characteristic, complemented
15C2×#1862C211yesmaximal, characteristic, complemented
16C2×#1817C211yesmaximal, characteristic, complemented
17C22×#125C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×C4×D4—21no—
20C22×C4×D4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#102C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32#2851—24no—
33#2851C2218yescomplemented
34C4×#121—24no—
35C4×#121C2218yes—
36C2×#390C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#388—21no—
41C2×#387—21no—
42C2×#387—21no—
43C2×#386—21no—
44C2×#385—21no—
45C2×#385—21no—
46C2×#382C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4—21no—
52#2085C2218yescomplemented
53#2077C2218yescomplemented
54#2076C2218yescomplemented
55#2073C2218yescomplemented
56#2059C2218yescomplemented
57#2034C2218yes—
58#2033C2218yes—
59#2025C2218yescomplemented
60#1920C2218yescomplemented
61#1918C2218yescomplemented
62#1864C2218yescomplemented
63#1862C2218yescomplemented
64#1817C2218yescomplemented
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4D411yescharacteristic
81C24×C4D411yescharacteristic
82C24×C4C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×#125—22no—
88C2×#125—42no—
89C2×#125—42no—
90C2×#125C2314yes—
91C2×#121—24no—
92C2×#121—22no—
93C2×#121—24no—
94C2×#121C2314yes—
95C2×C4×D4—28no—
96C2×C4×D4—22no—
97C2×C4×D4—42no—
98C2×C4×D4—42no—
99C2×C4×D4—48no—
100C2×C4×D4C2314yes—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—21no—
115C22×C22⋊C4—21no—
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C42—21no—
121C22×C42—42no—
122C22×C42—42no—
123C22×C42D414yes—
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2314yes—
129C2×#107—24no—
130C2×#107—42no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#106—24no—
134C2×#106—42no—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×#102—24no—
138C2×#102—24no—
139C2×#102—42no—
140C2×#102—42no—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8—42no—
146C2×C22⋊C8—42no—
147C2×C22⋊C8C2314yes—
148C2×C22⋊C8C2314yes—
149#390—22no—
150#390C2314yes—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#389C2314yes—
156#389C2314yes—
157#388—28no—
158#387—28no—
159#387—28no—
160#386—28no—
161#385—28no—
162#385—28no—
163#382—22no—
164#382C2314yes—
165#380—22no—
166#380—22no—
167#380—22no—
168#380C2314yes—
169#380C2314yes—
170#380C2314yes—
171C4×C4⋊C4—22no—
172C4×C4⋊C4C2314yes—
173C4×C22⋊C4—28no—
174C2×#95—21no—
175C2×#95—21no—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C25C2×D411yescharacteristic
197C22×D4—28no—
198C22×D4—24no—
199C22×D4—21no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4#4114yes—
208C22×D4C2412yes—
209C23×C4—24no—
210C23×C4—21no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—24no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4C2×D412yes—
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—22no—
245C22×C8—22no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—22no—
252C22×C8—22no—
253C22×C8—22no—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258#125—44no—
259#125—44no—
260#125—48no—
261#121—216no—
262#121—28no—
263#121—28no—
264C4×D4—44no—
265C4×D4—44no—
266C4×D4—48no—
267C4×D4—432no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4#4114yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—48no—
326C2×C42—44no—
327C2×C42—44no—
328C2×C42—42no—
329C2×C42—42no—
330C2×C42—44no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337#107—28no—
338#107—44no—
339#106—28no—
340#106—44no—
341#102—28no—
342#102—28no—
343#102—44no—
344#102—44no—
345C22⋊C8—44no—
346C22⋊C8—44no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—22no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24—42no—
370C24#13514yes—
371C24C22×D412yes—
372C24C22×D411yescharacteristic
373C2×D4—216no—
374C2×D4—24no—
375C2×D4—28no—
376C2×D4—28no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—22no—
382C2×D4—48no—
383C2×D4—432no—
384C2×D4—48no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4C2×#4114yes—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—416no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—416no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
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—48no—
456C22×C4—816no—
457C22×C4AGL(1,ℤ/8)14yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D411yescharacteristic
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C42−1+412yes—
469C22×C42−1+412yes—
470C22×C4C2511yescharacteristic
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—44no—
484C2×C8—44no—
485C2×C8—44no—
486C2×C8—44no—
487C4⋊C4—24no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—48no—
512C4⋊C4C2×#4114yes—
513C22⋊C4—216no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C42—24no—
521C42—24no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—22no—
526C42—44no—
527C42—44no—
528C42—48no—
529C23—28no—
530C23—22no—
531C23—22no—
532C23—28no—
533C23—22no—
534C23—28no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—24no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—48no—
544C23—48no—
545C23—48no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—84no—
553C23—84no—
554C23—84no—
555C23—84no—
556C23#52512yes—
557C23#53412yes—
558C23#53612yes—
559C23C2×#13514yes—
560C23C2×#13512yes—
561C23C2×AGL(1,ℤ/8)12yes—
562C23#57512yes—
563C23#57512yes—
564C23C23×D411yescharacteristic
565D4—24no—
566D4—24no—
567D4—432no—
568D4—48no—
569D4—42no—
570D4—42no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—416no—
588C2×C4—416no—
589C2×C4—48no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—48no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
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—816no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—84no—
623C2×C4—84no—
624C2×C4—84no—
625C2×C4—84no—
626C2×C4#52512yes—
627C2×C4#53412yes—
628C2×C4#53612yes—
629C2×C4#53914yes—
630C2×C4#53914yes—
631C2×C4#54214yes—
632C2×C4#54214yes—
633C2×C4C2×AGL(1,ℤ/8)14yes—
634C2×C4C22×#4111yescharacteristic
635C2×C4C22×#4111yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C2×C4C2×H(5,𝔽2)11yescharacteristic
638C2×C4C2×2−1+411yescharacteristic
639C2×C4C2×2−1+411yescharacteristic
640C8—42no—
641C8—42no—
642C8—42no—
643C8—42no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—416no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—816no—
660C22#263814yes—
661C22#310514yes—
662C22#311114yes—
663C22#319114yes—
664C22#319414yes—
665C22#323214yes—
666C22#323514yes—
667C22C2×#52511yescharacteristic
668C22C2×#53411yescharacteristic
669C22C2×#53611yescharacteristic
670C22C22×#13511yescharacteristic
671C22C22×AGL(1,ℤ/8)11yescharacteristic
672C22C2×#57511yescharacteristic
673C22C2×#57511yescharacteristic
674C4—24no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—44no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4#263814yes—
687C4C2×#53911yescharacteristic
688C4C2×#53911yescharacteristic
689C4C2×#54211yescharacteristic
690C4C2×#54211yescharacteristic
691C2—44no—
692C2—84no—
693C2#3016718yesdirect product factor
694C2C2×#263811yescharacteristic
695C2C2×#310511yescharacteristic
696C2C2×#311111yescharacteristic
697C2C2×#319111yescharacteristic
698C2C2×#319411yescharacteristic
699C2C2×#323211yescharacteristic
700C2C2×#323511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2851C2semidirect
3. #30167C2direct (N × H)
4. C2×#2085C2semidirect
5. C2×#2076C2semidirect
6. C2×#2077C2semidirect
7. C2×#2073C2semidirect
8. C2×#2059C2semidirect
11. C2×#2025C2semidirect
12. C2×#1920C2semidirect
13. C2×#1918C2semidirect
14. C2×#1864C2semidirect
15. C2×#1862C2semidirect
16. C2×#1817C2semidirect
33. #2851C22semidirect
52. #2085C22semidirect
53. #2077C22semidirect
54. #2076C22semidirect
55. #2073C22semidirect
56. #2059C22semidirect
59. #2025C22semidirect
60. #1920C22semidirect
61. #1918C22semidirect
62. #1864C22semidirect
63. #1862C22semidirect
64. #1817C22semidirect
693. C2#30167direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#28512
C2×C4×#1212
#301672
C2×#20852
C2×#20762
C2×#20772
C2×#20732
C2×#20592
C2×#20342
C2×#20332
C2×#20252
C2×#19202
C2×#19182
C2×#18642
C2×#18622
C2×#18172