Finite Group Databasebeta

← C2×#32264 #7508683

Subgroups — C2×#32264 #7508683

Order 512.

2080 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic
3#32264C2116yesmaximal, direct product factor
4C2×#2076C211yesmaximal, characteristic, complemented
5C2×#2077C211yesmaximal, characteristic, complemented
6C2×#2077C211yesmaximal, characteristic, complemented
7C2×#2075C211yesmaximal, characteristic, complemented
8C2×#2059C211yesmaximal, characteristic, complemented
9C2×#2059C211yesmaximal, characteristic, complemented
10C2×#1921C211yesmaximal, characteristic, complemented
11C2×#1920C211yesmaximal, characteristic, complemented
12C2×#1869C211yesmaximal, characteristic
13C2×#1853C211yesmaximal, characteristic, complemented
14C2×#1852C211yesmaximal, characteristic, complemented
15C2×#1808C211yesmaximal, characteristic, complemented
16C2×#1807C211yesmaximal, characteristic, complemented
17C24×C8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×#123—21no—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C22×C22⋊C8C2211yescharacteristic
31#2562—24no—
32#2562C2218yes—
33#2486—24no—
34#2486C2218yes—
35C2×#400—21no—
36C2×#398—21no—
37C2×#397—21no—
38C2×#394—21no—
39C2×#389C2211yescharacteristic
40C2×#389C2211yescharacteristic
41C2×#388—21no—
42C2×#388—21no—
43C2×#381—21no—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52#2077C2218yescomplemented
53#2077C2218yescomplemented
54#2076C2218yescomplemented
55#2075C2218yescomplemented
56#2059C2218yescomplemented
57#2059C2218yescomplemented
58#1921C2218yescomplemented
59#1920C2218yescomplemented
60#1869C2218yes—
61#1853C2218yescomplemented
62#1852C2218yescomplemented
63#1808C2218yescomplemented
64#1807C2218yescomplemented
65C2×#340C2211yescharacteristic
66C2×#340C2211yescharacteristic
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4D411yescharacteristic
78C24×C4—21no—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C23×C8—24no—
82C23×C8—24no—
83C23×C8—42no—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2314yes—
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×#125—22no—
90C2×#125—42no—
91C2×#125—42no—
92C2×#125C2314yes—
93C2×#123—28no—
94C2×#123—48no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C2×C4×D4D414yes—
98C2×C4×D4C2314yes—
99C22×C4⋊C4—21no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—42no—
106C22×C4⋊C4—42no—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4—21no—
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×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C2×#107—24no—
129C2×#107—42no—
130C2×#107—42no—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106—24no—
135C2×#106—42no—
136C2×#106—42no—
137C2×#106C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140C2×C22⋊C8—24no—
141C2×C22⋊C8—24no—
142C2×C22⋊C8—42no—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×C22⋊C8C2314yes—
146#400—28no—
147#398—28no—
148#397—28no—
149#394—28no—
150#389—22no—
151#389—22no—
152#389C2314yes—
153#389C2314yes—
154#388—28no—
155#388—28no—
156#381—28no—
157#380—22no—
158#380—22no—
159#380—22no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380—22no—
164#380C2314yes—
165#380C2314yes—
166#380C2314yes—
167#380C2314yes—
168#380C2314yes—
169#380C2314yes—
170#380C2314yes—
171C4×C4⋊C4—22no—
172C4×C4⋊C4C2314yes—
173C2×#95—21no—
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#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25C2×D411yescharacteristic
196C25C2×D411yescharacteristic
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—42no—
200C22×D4C2×D414yes—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D412yes—
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C22×C8—22no—
235C22×C8—22no—
236C22×C8—22no—
237C22×C8—22no—
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—28no—
245C22×C8—28no—
246C22×C8—44no—
247C22×C8—416no—
248C22×C8#4114yes—
249C22×C8#4114yes—
250C22×C8C2412yes—
251C22×C8C2412yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254#125—44no—
255#125—44no—
256#125—48no—
257#123—432no—
258C4×D4—48no—
259C4×D4—44no—
260C4×D4—44no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
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—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—48no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4C2×D414yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
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—24no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—82no—
323C2×C22⋊C4—82no—
324C2×C22⋊C4C2×D414yes—
325C2×C22⋊C4C2×D414yes—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—82no—
330C2×C42C2×D412yes—
331C2×C42C2412yes—
332#107—28no—
333#107—44no—
334#107—44no—
335#106—28no—
336#106—44no—
337#106—44no—
338C22⋊C8—28no—
339C22⋊C8—28no—
340C22⋊C8—44no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24#13514yes—
364C24AGL(1,ℤ/8)14yes—
365C24C22×D412yes—
366C24C22×D412yes—
367C24C22×D411yescharacteristic
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—44no—
371C2×D4—416no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—48no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
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—816no—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C42−1+412yes—
456C22×C42−1+412yes—
457C22×C4C2511yescharacteristic
458C2×C8—28no—
459C2×C8—28no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—416no—
471C2×C8—44no—
472C2×C8—44no—
473C2×C8C2×#4114yes—
474C2×C8C2×#4114yes—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C42—24no—
515C42—22no—
516C42—22no—
517C42—84no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23#53412yes—
545C23#54712yes—
546C23#54712yes—
547C23C2×#13514yes—
548C23C2×#13512yes—
549C23C2×AGL(1,ℤ/8)14yes—
550C23C2×AGL(1,ℤ/8)12yes—
551C23#57712yes—
552C23#57812yes—
553C23C23×D411yescharacteristic
554D4—416no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
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—816no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4—84no—
611C2×C4—84no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4#53412yes—
615C2×C4#54214yes—
616C2×C4#54214yes—
617C2×C4#54712yes—
618C2×C4#54712yes—
619C2×C4#55114yes—
620C2×C4#56314yes—
621C2×C4C22×#4111yescharacteristic
622C2×C4C22×#4111yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×H(5,𝔽2)11yescharacteristic
625C2×C4C2×2−1+411yescharacteristic
626C2×C4C2×2−1+411yescharacteristic
627C8—24no—
628C8—24no—
629C8—42no—
630C8—42no—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—416no—
638C22—416no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22#271714yes—
648C22#318814yes—
649C22#320114yes—
650C22#333614yes—
651C22#333714yes—
652C22#334314yes—
653C22#334514yes—
654C22C2×#53411yescharacteristic
655C22C2×#54711yescharacteristic
656C22C2×#54711yescharacteristic
657C22C22×#13511yescharacteristic
658C22C22×AGL(1,ℤ/8)11yescharacteristic
659C22C2×#57711yescharacteristic
660C22C2×#57811yescharacteristic
661C4—42no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4#271714yes—
674C4C2×#54211yescharacteristic
675C4C2×#54211yescharacteristic
676C4C2×#55111yescharacteristic
677C4C2×#56311yescharacteristic
678C2—44no—
679C2—44no—
680C2#3226418yesdirect product factor
681C2C2×#271711yescharacteristic
682C2C2×#318811yescharacteristic
683C2C2×#320111yescharacteristic
684C2C2×#333611yescharacteristic
685C2C2×#333711yescharacteristic
686C2C2×#334311yescharacteristic
687C2C2×#334511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32264C2direct (N × H)
4. C2×#2076C2semidirect
5. C2×#2077C2semidirect
6. C2×#2077C2semidirect
7. C2×#2075C2semidirect
8. C2×#2059C2semidirect
9. C2×#2059C2semidirect
10. C2×#1921C2semidirect
11. C2×#1920C2semidirect
13. C2×#1853C2semidirect
14. C2×#1852C2semidirect
15. C2×#1808C2semidirect
16. C2×#1807C2semidirect
52. #2077C22semidirect
53. #2077C22semidirect
54. #2076C22semidirect
55. #2075C22semidirect
56. #2059C22semidirect
57. #2059C22semidirect
58. #1921C22semidirect
59. #1920C22semidirect
61. #1853C22semidirect
62. #1852C22semidirect
63. #1808C22semidirect
64. #1807C22semidirect
680. C2#32264direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#24862
#322642
C2×#20762
C2×#20772
C2×#20752
C2×#20592
C2×#19212
C2×#19202
C2×#18692
C2×#18532
C2×#18522
C2×#18082
C2×#18072