Finite Group Databasebeta

← C2×#32665 #7511132

Subgroups — C2×#32665 #7511132

Order 512.

2074 conjugacy classes of subgroups.

The subgroup lattice has 679 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic, complemented
2C2×#2538C211yesmaximal, characteristic
3#32665C2116yesmaximal, direct product factor
4C2×#2095C211yesmaximal, characteristic, complemented
5C2×#2093C211yesmaximal, characteristic
6C2×#2083C211yesmaximal, characteristic, complemented
7C2×#2083C211yesmaximal, characteristic, complemented
8C2×#2068C211yesmaximal, characteristic, complemented
9C2×#2067C211yesmaximal, characteristic
10C2×#1985C211yesmaximal, characteristic, complemented
11C2×#1973C211yesmaximal, characteristic
12C2×#1943C211yesmaximal, characteristic, complemented
13C2×#1880C211yesmaximal, characteristic
14C2×#1877C211yesmaximal, characteristic
15C2×#1865C211yesmaximal, characteristic, complemented
16C2×#1864C211yesmaximal, characteristic
17C23×SD16C2211yescharacteristic
18C22×C22⋊Q8—21no—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#107C2211yescharacteristic
23C22×#107C2211yescharacteristic
24C22×#107C2211yescharacteristic
25C22×#107C2211yescharacteristic
26C22×#106C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28C22×#103C2211yescharacteristic
29C22×#103C2211yescharacteristic
30C22×#103C2211yescharacteristic
31C22×#102C2211yescharacteristic
32C22×#102C2211yescharacteristic
33C22×#102C2211yescharacteristic
34C22×#97C2211yescharacteristic
35#2582—24no—
36#2582C2218yescomplemented
37#2538—24no—
38#2538C2218yes—
39C2×#398—21no—
40C2×#398—21no—
41C2×#396—21no—
42C2×#396—21no—
43C2×#391—21no—
44C2×#391C2211yescharacteristic
45C2×#391C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#389—21no—
49C2×#389C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#389C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#384C2211yescharacteristic
55C2×#380—21no—
56#2095C2218yescomplemented
57#2093C2218yes—
58#2083C2218yescomplemented
59#2083C2218yescomplemented
60#2068C2218yescomplemented
61#2067C2218yes—
62#1985C2218yescomplemented
63#1973C2218yes—
64#1943C2218yescomplemented
65#1880C2218yes—
66#1877C2218yes—
67#1865C2218yescomplemented
68#1864C2218yes—
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#336C2211yescharacteristic
74C2×#336C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×Q8D411yescharacteristic
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4—21no—
81C22×SD16—42no—
82C22×SD16—42no—
83C22×SD16Q818yescomplemented
84C22×SD16C2314yes—
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C23×C8C2311yescharacteristic
88C23×C8C2311yescharacteristic
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—48no—
91C2×C4×Q8—22no—
92C2×C4×Q8—42no—
93C2×C4×Q8—42no—
94C2×C4×Q8—42no—
95C2×C4×Q8—42no—
96C2×C4×Q8D414yes—
97C2×C4×Q8C2314yes—
98C2×C4×Q8C2314yes—
99C2×C4×D4—22no—
100C2×C4×D4—42no—
101C2×C4×D4—42no—
102C2×C4×D4C2314yes—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4D411yescharacteristic
113C22×C4⋊C4D411yescharacteristic
114C22×C4⋊C4D411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C2×#107—24no—
131C2×#107—42no—
132C2×#107—42no—
133C2×#107—42no—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106—42no—
139C2×#106C2314yes—
140C2×C4⋊C8—42no—
141C2×C4⋊C8C2314yes—
142C2×#103—24no—
143C2×#103—42no—
144C2×#103—42no—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#102—24no—
149C2×#102—42no—
150C2×#102—42no—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#97—42no—
155C2×#97C2314yes—
156#398—28no—
157#398—28no—
158#396—28no—
159#396—28no—
160#391—28no—
161#391—22no—
162#391—22no—
163#391C2314yes—
164#391C2314yes—
165#390—22no—
166#390—22no—
167#390C2314yes—
168#390C2314yes—
169#389—28no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389C2314yes—
175#389C2314yes—
176#389C2314yes—
177#389C2314yes—
178#384—22no—
179#384—22no—
180#384C2314yes—
181#384C2314yes—
182#380—28no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#336C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2314yes—
199C25—21no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—42no—
209C22×Q8C2×D414yes—
210C22×Q8C2×Q814yes—
211C22×Q8C2412yes—
212C22×D4—21no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4C2×Q814yes—
219C22×D4C2412yes—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—24no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2411yescharacteristic
238C2×SD16—416no—
239C2×SD16—44no—
240C2×SD16—44no—
241C2×SD16C2×Q8116yescomplemented
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×C8C2×Q814yes—
253C22×C8#4114yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C22⋊Q8—432no—
259C4×Q8—48no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—48no—
265C4×D4—44no—
266C4×D4—44no—
267C4×D4—48no—
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—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4—82no—
305C2×C4⋊C4—82no—
306C2×C4⋊C4—82no—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×D414yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2412yes—
315C2×C4⋊C4C2412yes—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—82no—
325C2×C22⋊C4—82no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—24no—
333C2×C42—44no—
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2412yes—
338C2×C42C2412yes—
339C2×C42C2412yes—
340#107—28no—
341#107—44no—
342#107—44no—
343#107—44no—
344#106—44no—
345C4⋊C8—44no—
346#103—28no—
347#103—44no—
348#103—44no—
349#102—28no—
350#102—44no—
351#102—44no—
352#97—44no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95—24no—
361C24—24no—
362C24—22no—
363C24—42no—
364C24—42no—
365C24—42no—
366C24—42no—
367C24—42no—
368C24—42no—
369C24C22×D411yescharacteristic
370C2×Q8—24no—
371C2×Q8—28no—
372C2×Q8—28no—
373C2×Q8—22no—
374C2×Q8—28no—
375C2×Q8—28no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—44no—
381C2×Q8—416no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8C22×Q814yes—
387C2×D4—28no—
388C2×D4—24no—
389C2×D4—28no—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—42no—
396C2×D4—44no—
397C2×D4—42no—
398C2×D4—44no—
399C2×D4C22×Q814yes—
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—24no—
409C22×C4—22no—
410C22×C4—22no—
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—22no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
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—42no—
446C22×C4—48no—
447C22×C4—816no—
448C22×C4AGL(1,ℤ/8)14yes—
449C22×C4#13714yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D411yescharacteristic
457C22×C4C22×Q812yes—
458C22×C4C2×#4112yes—
459C22×C4H(5,𝔽2)12yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C42−1+412yes—
463C22×C4C2511yescharacteristic
464SD16—416no—
465C2×C8—28no—
466C2×C8—28no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—44no—
478C2×C8—44no—
479C2×C8C22×Q814yes—
480C2×C8C2×#4114yes—
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—22no—
492C4⋊C4—22no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—48no—
506C4⋊C4—48no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—84no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—84no—
515C22⋊C4—84no—
516C42—24no—
517C42—24no—
518C42—24no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—22no—
524C42—22no—
525C42—48no—
526C23—24no—
527C23—48no—
528C23—48no—
529C23—42no—
530C23—42no—
531C23—48no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23#53412yes—
541C23D4×Q812yes—
542C23#55012yes—
543C23C2×AGL(1,ℤ/8)12yes—
544C23C2×#13712yes—
545C23#57612yes—
546C23#57812yes—
547C23C23×D411yescharacteristic
548Q8—24no—
549Q8—24no—
550Q8—416no—
551Q8—42no—
552Q8—48no—
553Q8—42no—
554D4—24no—
555D4—24no—
556D4—42no—
557D4—48no—
558D4—42no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—48no—
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—816no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4#53412yes—
611C2×C4#53914yes—
612C2×C4#54314yes—
613C2×C4D4×Q812yes—
614C2×C4#55012yes—
615C2×C4#55414yes—
616C2×C4#56214yes—
617C2×C4C2×AGL(1,ℤ/8)14yes—
618C2×C4C2×#13714yes—
619C2×C4C23×Q811yescharacteristic
620C2×C4C22×#4111yescharacteristic
621C2×C4C2×H(5,𝔽2)11yescharacteristic
622C2×C4C2×H(5,𝔽2)11yescharacteristic
623C2×C4C2×H(5,𝔽2)11yescharacteristic
624C2×C4C2×2−1+411yescharacteristic
625C8—24no—
626C8—24no—
627C8—42no—
628C8—42no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—82no—
633C22—82no—
634C22—816no—
635C22—82no—
636C22—82no—
637C22#273214yes—
638C22#319314yes—
639C22#320314yes—
640C22#338514yes—
641C22#338614yes—
642C22#339814yes—
643C22#339914yes—
644C22C2×#53411yescharacteristic
645C22C2×D4×Q811yescharacteristic
646C22C2×#55011yescharacteristic
647C22C22×AGL(1,ℤ/8)11yescharacteristic
648C22C22×#13711yescharacteristic
649C22C2×#57611yescharacteristic
650C22C2×#57811yescharacteristic
651C4—24no—
652C4—24no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—42no—
657C4—44no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4#273214yes—
665C4C2×#53911yescharacteristic
666C4C2×#54311yescharacteristic
667C4C2×#55411yescharacteristic
668C4C2×#56211yescharacteristic
669C2—84no—
670C2#3266518yesdirect product factor
671C2C2×#273211yescharacteristic
672C2C2×#319311yescharacteristic
673C2C2×#320311yescharacteristic
674C2C2×#338511yescharacteristic
675C2C2×#338611yescharacteristic
676C2C2×#339811yescharacteristic
677C2C2×#339911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2582C2semidirect
3. #32665C2direct (N × H)
4. C2×#2095C2semidirect
6. C2×#2083C2semidirect
7. C2×#2083C2semidirect
8. C2×#2068C2semidirect
10. C2×#1985C2semidirect
12. C2×#1943C2semidirect
15. C2×#1865C2semidirect
36. #2582C22semidirect
56. #2095C22semidirect
58. #2083C22semidirect
59. #2083C22semidirect
60. #2068C22semidirect
62. #1985C22semidirect
64. #1943C22semidirect
67. #1865C22semidirect
83. C22×SD16Q8semidirect
241. C2×SD16C2×Q8semidirect
670. C2#32665direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#25382
#326652
C2×#20952
C2×#20932
C2×#20832
C2×#20682
C2×#20672
C2×#19852
C2×#19732
C2×#19432
C2×#18802
C2×#18772
C2×#18652
C2×#18642