Finite Group Databasebeta

← C2×#32009 #7507403

Subgroups — C2×#32009 #7507403

Order 512.

2102 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2485C211yesmaximal, characteristic
3#32009C2116yesmaximal, direct product factor
4C2×#2099C211yesmaximal, characteristic, complemented
5C2×#2083C211yesmaximal, characteristic, complemented
6C2×#2077C211yesmaximal, characteristic, complemented
7C2×#2075C211yesmaximal, characteristic, complemented
8C2×#2036C211yesmaximal, characteristic, complemented
9C2×#2036C211yesmaximal, characteristic, complemented
10C2×#1892C211yesmaximal, characteristic, complemented
11C2×#1873C211yesmaximal, characteristic, complemented
12C2×#1869C211yesmaximal, characteristic
13C2×#1865C211yesmaximal, characteristic, complemented
14C2×#1863C211yesmaximal, characteristic, complemented
15C2×#1854C211yesmaximal, characteristic, complemented
16C2×#1808C211yesmaximal, characteristic, complemented
17C24×C8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×#103C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×C22⋊C8C2211yescharacteristic
34#2538—24no—
35#2538C2218yes—
36#2485—24no—
37#2485C2218yes—
38C2×#397—21no—
39C2×#396—21no—
40C2×#396—21no—
41C2×#394—21no—
42C2×#390—21no—
43C2×#389—21no—
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#388—21no—
47C2×#387—21no—
48C2×#384C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54#2099C2218yescomplemented
55#2083C2218yescomplemented
56#2077C2218yescomplemented
57#2075C2218yescomplemented
58#2036C2218yescomplemented
59#2036C2218yescomplemented
60#1892C2218yescomplemented
61#1873C2218yescomplemented
62#1869C2218yes—
63#1865C2218yescomplemented
64#1863C2218yescomplemented
65#1854C2218yescomplemented
66#1808C2218yescomplemented
67C2×#340C2211yescharacteristic
68C2×#340C2211yescharacteristic
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8C2311yescharacteristic
78C23×D4D411yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C23×C8—24no—
82C23×C8—42no—
83C23×C8—42no—
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2314yes—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—22no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—42no—
94C2×C22⋊Q8—42no—
95C2×C22⋊Q8C2314yes—
96C2×C22⋊Q8C2314yes—
97C2×C4×Q8—22no—
98C2×C4×Q8—42no—
99C2×C4×Q8—42no—
100C2×C4×Q8C2314yes—
101C2×C4×D4—42no—
102C2×C4×D4—42no—
103C2×C4×D4D414yes—
104C2×C4×D4C2314yes—
105C2×#117—22no—
106C2×#117—42no—
107C2×#117—42no—
108C2×#117C2314yes—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4D411yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42—21no—
135C22×C42—21no—
136C22×C42C2311yescharacteristic
137C2×#107—42no—
138C2×#107—42no—
139C2×#107—42no—
140C2×#107C2314yes—
141C2×#107C2314yes—
142C2×#107C2314yes—
143C2×#106—42no—
144C2×#106C2314yes—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103—42no—
148C2×#103—42no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×C22⋊C8—24no—
154C2×C22⋊C8—42no—
155C2×C22⋊C8—42no—
156C2×C22⋊C8C2314yes—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159#397—28no—
160#396—28no—
161#396—28no—
162#394—28no—
163#390—28no—
164#389—28no—
165#389—22no—
166#389—22no—
167#389C2314yes—
168#389C2314yes—
169#388—28no—
170#387—28no—
171#384—22no—
172#384C2314yes—
173#380—22no—
174#380—22no—
175#380—22no—
176#380—22no—
177#380—22no—
178#380C2314yes—
179#380C2314yes—
180#380C2314yes—
181#380C2314yes—
182#380C2314yes—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#336C2314yes—
199#336C2314yes—
200C25C2×D411yescharacteristic
201C25C2×D411yescharacteristic
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8#4114yes—
209C22×Q8C2412yes—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—42no—
213C22×D4C2×D414yes—
214C23×C4—21no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4C2×D412yes—
235C23×C4C2×D412yes—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
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×C8—28no—
255C22×C8—44no—
256C22×C8—44no—
257C22×C8—416no—
258C22×C8#4114yes—
259C22×C8C2412yes—
260C22×C8C2412yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22⋊Q8—28no—
264C22⋊Q8—28no—
265C22⋊Q8—44no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268C22⋊Q8—48no—
269C4×Q8—44no—
270C4×Q8—44no—
271C4×Q8—48no—
272C4×D4—48no—
273C4×D4—44no—
274C4×D4—44no—
275#117—44no—
276#117—44no—
277#117—48no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—82no—
313C2×C4⋊C4—82no—
314C2×C4⋊C4C2×D414yes—
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2412yes—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—42no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—82no—
337C2×C22⋊C4—82no—
338C2×C22⋊C4—82no—
339C2×C22⋊C4—82no—
340C2×C22⋊C4C2×D414yes—
341C2×C22⋊C4C2×D414yes—
342C2×C42—22no—
343C2×C42—22no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—44no—
347C2×C42—44no—
348C2×C42C2×D412yes—
349C2×C42C2412yes—
350#107—44no—
351#107—44no—
352#107—44no—
353#106—44no—
354#103—28no—
355#103—28no—
356#103—44no—
357#103—44no—
358C22⋊C8—28no—
359C22⋊C8—44no—
360C22⋊C8—44no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24#13714yes—
382C24C22×D412yes—
383C24C22×D412yes—
384C24C22×D411yescharacteristic
385C2×Q8—24no—
386C2×Q8—22no—
387C2×Q8—28no—
388C2×Q8—28no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—42no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8C2×#4114yes—
398C2×D4—28no—
399C2×D4—28no—
400C2×D4—44no—
401C2×D4—416no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4#13514yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D411yescharacteristic
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)12yes—
480C22×C4H(5,𝔽2)12yes—
481C22×C4H(5,𝔽2)12yes—
482C22×C42−1+412yes—
483C22×C4C2511yescharacteristic
484C2×C8—28no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
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—416no—
497C2×C8—44no—
498C2×C8—44no—
499C2×C8—44no—
500C2×C8C2×#4114yes—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—44no—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—48no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C42—24no—
541C42—22no—
542C42—22no—
543C42—48no—
544C42—48no—
545C23—28no—
546C23—22no—
547C23—28no—
548C23—22no—
549C23—22no—
550C23—28no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23#53412yes—
573C23#54612yes—
574C23#54712yes—
575C23C2×#13512yes—
576C23C2×#13714yes—
577C23C2×#13712yes—
578C23#57612yes—
579C23#57812yes—
580C23C23×D411yescharacteristic
581Q8—24no—
582Q8—24no—
583Q8—42no—
584Q8—48no—
585Q8—42no—
586D4—416no—
587C2×C4—22no—
588C2×C4—28no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—84no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4#53412yes—
645C2×C4#53814yes—
646C2×C4#53914yes—
647C2×C4#54612yes—
648C2×C4#54712yes—
649C2×C4#55314yes—
650C2×C4#56214yes—
651C2×C4C2×#13514yes—
652C2×C4C22×#4111yescharacteristic
653C2×C4C22×#4111yescharacteristic
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4C2×H(5,𝔽2)11yescharacteristic
657C2×C4C2×2−1+411yescharacteristic
658C8—24no—
659C8—42no—
660C8—42no—
661C8—42no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—416no—
666C22—42no—
667C22—42no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—88no—
678C22#270814yes—
679C22#319014yes—
680C22#320314yes—
681C22#330514yes—
682C22#331214yes—
683C22#334214yes—
684C22#334614yes—
685C22C2×#53411yescharacteristic
686C22C2×#54611yescharacteristic
687C22C2×#54711yescharacteristic
688C22C22×#13511yescharacteristic
689C22C22×#13711yescharacteristic
690C22C2×#57611yescharacteristic
691C22C2×#57811yescharacteristic
692C4—24no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4#270814yes—
705C4C2×#53811yescharacteristic
706C4C2×#53911yescharacteristic
707C4C2×#55311yescharacteristic
708C4C2×#56211yescharacteristic
709C2—44no—
710C2—82no—
711C2#3200918yesdirect product factor
712C2C2×#270811yescharacteristic
713C2C2×#319011yescharacteristic
714C2C2×#320311yescharacteristic
715C2C2×#330511yescharacteristic
716C2C2×#331211yescharacteristic
717C2C2×#334211yescharacteristic
718C2C2×#334611yescharacteristic
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. #32009C2direct (N × H)
4. C2×#2099C2semidirect
5. C2×#2083C2semidirect
6. C2×#2077C2semidirect
7. C2×#2075C2semidirect
8. C2×#2036C2semidirect
9. C2×#2036C2semidirect
10. C2×#1892C2semidirect
11. C2×#1873C2semidirect
13. C2×#1865C2semidirect
14. C2×#1863C2semidirect
15. C2×#1854C2semidirect
16. C2×#1808C2semidirect
54. #2099C22semidirect
55. #2083C22semidirect
56. #2077C22semidirect
57. #2075C22semidirect
58. #2036C22semidirect
59. #2036C22semidirect
60. #1892C22semidirect
61. #1873C22semidirect
63. #1865C22semidirect
64. #1863C22semidirect
65. #1854C22semidirect
66. #1808C22semidirect
711. C2#32009direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#24852
#320092
C2×#20992
C2×#20832
C2×#20772
C2×#20752
C2×#20362
C2×#18922
C2×#18732
C2×#18692
C2×#18652
C2×#18632
C2×#18542
C2×#18082