Finite Group Databasebeta

← C2×#32298 #7508753

Subgroups — C2×#32298 #7508753

Order 512.

1988 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2504C211yesmaximal, characteristic
2C2×#2496C211yesmaximal, characteristic, complemented
3#32298C2116yesmaximal, direct product factor
4C2×#2101C211yesmaximal, characteristic, complemented
5C2×#2094C211yesmaximal, characteristic, complemented
6C2×#2085C211yesmaximal, characteristic, complemented
7C2×#2065C211yesmaximal, characteristic, complemented
8C2×#2033C211yesmaximal, characteristic
9C2×#2033C211yesmaximal, characteristic
10C2×#1949C211yesmaximal, characteristic, complemented
11C2×#1936C211yesmaximal, characteristic, complemented
12C2×#1926C211yesmaximal, characteristic, complemented
13C2×#1924C211yesmaximal, characteristic, complemented
14C2×#1853C211yesmaximal, characteristic, complemented
15C2×#1853C211yesmaximal, characteristic, complemented
16C2×#1847C211yesmaximal, characteristic, complemented
17C22×#126—21no—
18C22×#121C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C22×#107C2211yescharacteristic
21C22×#107C2211yescharacteristic
22C22×#106C2211yescharacteristic
23C22×#106C2211yescharacteristic
24C22×#106C2211yescharacteristic
25C22×#106C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×#102C2211yescharacteristic
28C22×#102C2211yescharacteristic
29C22×#102C2211yescharacteristic
30C22×#102C2211yescharacteristic
31C22×C22⋊C8C2211yescharacteristic
32C22×C22⋊C8C2211yescharacteristic
33C22×#97C2211yescharacteristic
34#2504—24no—
35#2504C2218yes—
36#2496—24no—
37#2496C2218yescomplemented
38C2×#400—21no—
39C2×#397—21no—
40C2×#397—21no—
41C2×#397—21no—
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#389C2211yescharacteristic
46C2×#387—21no—
47C2×#386—21no—
48C2×#386—21no—
49C2×#384C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56#2101C2218yescomplemented
57#2094C2218yescomplemented
58#2085C2218yescomplemented
59#2065C2218yescomplemented
60#2033C2218yes—
61#2033C2218yes—
62#1949C2218yescomplemented
63#1936C2218yescomplemented
64#1926C2218yescomplemented
65#1924C2218yescomplemented
66#1853C2218yescomplemented
67#1853C2218yescomplemented
68#1847C2218yescomplemented
69C2×#340C2211yescharacteristic
70C2×#340C2211yescharacteristic
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#336C2211yescharacteristic
76C2×#336C2211yescharacteristic
77C23×D4—21no—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C24×C4C2311yescharacteristic
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C2×#126—28no—
86C2×#126—48no—
87C2×#121—22no—
88C2×#121—42no—
89C2×#121—42no—
90C2×#121C2314yes—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—42no—
98C22×C4⋊C4—42no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C22⋊C4—21no—
111C22×C22⋊C4—21no—
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4D411yescharacteristic
115C22×C22⋊C4D411yescharacteristic
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C2×#107—42no—
120C2×#107—42no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106—42no—
124C2×#106—42no—
125C2×#106—42no—
126C2×#106—42no—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×C4⋊C8—24no—
132C2×C4⋊C8C2314yes—
133C2×#102—24no—
134C2×#102—24no—
135C2×#102—42no—
136C2×#102—42no—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8—24no—
142C2×C22⋊C8—24no—
143C2×C22⋊C8C2314yes—
144C2×C22⋊C8C2314yes—
145C2×#97—24no—
146C2×#97C2314yes—
147#400—28no—
148#397—28no—
149#397—28no—
150#397—28no—
151#390—22no—
152#390—22no—
153#390C2314yes—
154#390C2314yes—
155#389—22no—
156#389—22no—
157#389C2314yes—
158#389C2314yes—
159#387—28no—
160#386—28no—
161#386—28no—
162#384—22no—
163#384C2314yes—
164#382—22no—
165#382—22no—
166#382C2314yes—
167#382C2314yes—
168#380—22no—
169#380—22no—
170#380C2314yes—
171#380C2314yes—
172C4×C4⋊C4—22no—
173C4×C4⋊C4C2314yes—
174C4×C22⋊C4D414yes—
175C4×C22⋊C4C2314yes—
176C2×#95—21no—
177C2×#95—21no—
178C2×#95—21no—
179C2×#95—21no—
180C2×#95—21no—
181C2×#95—21no—
182C2×#95—21no—
183C2×#95—21no—
184C2×#95—21no—
185C2×#95D411yescharacteristic
186C2×#95D411yescharacteristic
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195C25—21no—
196C25C2×D411yescharacteristic
197C22×D4—24no—
198C22×D4—21no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4C2412yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—24no—
223C23×C4—44no—
224C23×C4—82no—
225C23×C4—82no—
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
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#4114yes—
245C22×C8#4114yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C22×C8C2412yes—
249C22×C8C2412yes—
250#126—432no—
251#121—44no—
252#121—44no—
253#121—48no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
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—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—82no—
308C2×C22⋊C4—82no—
309C2×C22⋊C4—82no—
310C2×C22⋊C4—82no—
311C2×C22⋊C4C2×D414yes—
312C2×C22⋊C4C2×D414yes—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—44no—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2412yes—
322C2×C42C2412yes—
323#107—44no—
324#107—44no—
325#106—44no—
326#106—44no—
327#106—44no—
328#106—44no—
329C4⋊C8—28no—
330#102—28no—
331#102—28no—
332#102—44no—
333#102—44no—
334C22⋊C8—28no—
335C22⋊C8—28no—
336#97—28no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95C2×D414yes—
347#95C2×D414yes—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—42no—
356C24—42no—
357C24—42no—
358C24—42no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24AGL(1,ℤ/8)14yes—
363C24C22×D412yes—
364C24C22×D411yescharacteristic
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—42no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—816no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
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—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—44no—
414C22×C4—44no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—48no—
418C22×C4—42no—
419C22×C4—42no—
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—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—84no—
440C22×C4—84no—
441C22×C4—816no—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D411yescharacteristic
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C42−1+412yes—
453C22×C42−1+412yes—
454C22×C4C2511yescharacteristic
455C2×C8—28no—
456C2×C8—28no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—44no—
468C2×C8—44no—
469C2×C8C2×#4114yes—
470C2×C8C2×#4114yes—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
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—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C22⋊C4—28no—
498C22⋊C4—44no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C42—24no—
507C42—24no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—48no—
513C23—28no—
514C23—28no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—28no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—84no—
534C23—84no—
535C23—84no—
536C23—84no—
537C23—84no—
538C23—84no—
539C23—84no—
540C23#53512yes—
541C23#54712yes—
542C23#54712yes—
543C23C2×AGL(1,ℤ/8)14yes—
544C23C2×AGL(1,ℤ/8)12yes—
545C23#57512yes—
546C23#57712yes—
547C23#57712yes—
548C23C23×D411yescharacteristic
549D4—42no—
550D4—42no—
551D4—42no—
552D4—42no—
553D4—816no—
554D4—84no—
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—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
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—816no—
602C2×C4—84no—
603C2×C4—84no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4—84no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4#53512yes—
610C2×C4#53914yes—
611C2×C4#54214yes—
612C2×C4#54712yes—
613C2×C4#54712yes—
614C2×C4#55314yes—
615C2×C4#56314yes—
616C2×C4C22×#4111yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4C2×H(5,𝔽2)11yescharacteristic
620C2×C4C2×2−1+411yescharacteristic
621C2×C4C2×2−1+411yescharacteristic
622C8—24no—
623C8—24no—
624C8—42no—
625C8—42no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—416no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—168no—
642C22#271814yes—
643C22#321314yes—
644C22#322214yes—
645C22#334514yes—
646C22#334514yes—
647C22#335014yes—
648C22#335114yes—
649C22C2×#53511yescharacteristic
650C22C2×#54711yescharacteristic
651C22C2×#54711yescharacteristic
652C22C22×AGL(1,ℤ/8)11yescharacteristic
653C22C2×#57511yescharacteristic
654C22C2×#57711yescharacteristic
655C22C2×#57711yescharacteristic
656C4—42no—
657C4—42no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4#271814yes—
669C4C2×#53911yescharacteristic
670C4C2×#54211yescharacteristic
671C4C2×#55311yescharacteristic
672C4C2×#56311yescharacteristic
673C2—44no—
674C2—162no—
675C2#3229818yesdirect product factor
676C2C2×#271811yescharacteristic
677C2C2×#321311yescharacteristic
678C2C2×#322211yescharacteristic
679C2C2×#334511yescharacteristic
680C2C2×#334511yescharacteristic
681C2C2×#335011yescharacteristic
682C2C2×#335111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2496C2semidirect
3. #32298C2direct (N × H)
4. C2×#2101C2semidirect
5. C2×#2094C2semidirect
6. C2×#2085C2semidirect
7. C2×#2065C2semidirect
10. C2×#1949C2semidirect
11. C2×#1936C2semidirect
12. C2×#1926C2semidirect
13. C2×#1924C2semidirect
14. C2×#1853C2semidirect
15. C2×#1853C2semidirect
16. C2×#1847C2semidirect
37. #2496C22semidirect
56. #2101C22semidirect
57. #2094C22semidirect
58. #2085C22semidirect
59. #2065C22semidirect
62. #1949C22semidirect
63. #1936C22semidirect
64. #1926C22semidirect
65. #1924C22semidirect
66. #1853C22semidirect
67. #1853C22semidirect
68. #1847C22semidirect
675. C2#32298direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#25042
C2×#24962
#322982
C2×#21012
C2×#20942
C2×#20852
C2×#20652
C2×#20332
C2×#19492
C2×#19362
C2×#19262
C2×#19242
C2×#18532
C2×#18472