Finite Group Databasebeta

← C2×#30803 #7501878

Subgroups — C2×#30803 #7501878

Order 512.

2184 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C22×#484C211yesmaximal, characteristic
3C2×#2489C211yesmaximal, characteristic
4C2×#2440C211yesmaximal, characteristic
5#30803C2116yesmaximal, direct product factor
6C2×#2096C211yesmaximal, characteristic, complemented
7C2×#2077C211yesmaximal, characteristic
8C2×#2066C211yesmaximal, characteristic, complemented
9C2×#2059C211yesmaximal, characteristic, complemented
10C2×#2036C211yesmaximal, characteristic
11C2×#2024C211yesmaximal, characteristic, complemented
12C2×#1947C211yesmaximal, characteristic, complemented
13C2×#1935C211yesmaximal, characteristic, complemented
14C2×#1924C211yesmaximal, characteristic
15C2×#1918C211yesmaximal, characteristic, complemented
16C2×#1853C211yesmaximal, characteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125C2211yescharacteristic
19C22×C22⋊Q8—21no—
20C22×C22⋊Q8C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#501—28no—
23C2×#501C2218yescomplemented
24C2×#484—28no—
25C2×#484C2218yes—
26C22×#107C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C22×#107C2211yescharacteristic
29C22×#106C2211yescharacteristic
30C22×#106C2211yescharacteristic
31C22×#106C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C22×#103C2211yescharacteristic
36C22×#103C2211yescharacteristic
37C22×C22⋊C8C2211yescharacteristic
38C22×C22⋊C8C2211yescharacteristic
39C22×#97C2211yescharacteristic
40#2489—24no—
41#2489C2218yes—
42#2440—24no—
43#2440C2218yes—
44C2×#400—21no—
45C2×#398—21no—
46C2×#397—21no—
47C2×#395—21no—
48C2×#391C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#389C2211yescharacteristic
51C2×#389C2211yescharacteristic
52C2×#388—21no—
53C2×#385—21no—
54C2×#384C2211yescharacteristic
55C2×#382—21no—
56C2×#380C2211yescharacteristic
57C2×#380C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2211yescharacteristic
60#2096C2218yescomplemented
61#2077C2218yes—
62#2066C2218yescomplemented
63#2059C2218yescomplemented
64#2036C2218yes—
65#2024C2218yescomplemented
66#1947C2218yescomplemented
67#1935C2218yescomplemented
68#1924C2218yes—
69#1918C2218yescomplemented
70#1853C2218yes—
71C2×#340C2211yescharacteristic
72C2×#340C2211yescharacteristic
73C2×#340C2211yescharacteristic
74C2×#340C2211yescharacteristic
75C2×#340C2211yescharacteristic
76C2×#340C2211yescharacteristic
77C2×#336C2211yescharacteristic
78C2×#336C2211yescharacteristic
79C23×Q8—21no—
80C23×Q8C2311yescharacteristic
81C24×C4—21no—
82C24×C4C2311yescharacteristic
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86C23×C8C2311yescharacteristic
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—42no—
90C2×C4⋊Q8C2314yes—
91C2×#125—24no—
92C2×#125—22no—
93C2×#125—42no—
94C2×#125C2314yes—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—48no—
99C2×C22⋊Q8—42no—
100C2×C22⋊Q8C2314yes—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—21no—
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—22no—
111C22×C4⋊C4—21no—
112C22×C4⋊C4—42no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#501—232no—
130#484—232no—
131C2×#107—24no—
132C2×#107—24no—
133C2×#107D418yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#107C2314yes—
137C2×#106—24no—
138C2×#106—24no—
139C2×#106—42no—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8—24no—
144C2×C4⋊C8C2314yes—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103—24no—
148C2×#103—42no—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×C22⋊C8—24no—
154C2×C22⋊C8—24no—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2314yes—
157C2×#97—24no—
158C2×#97C2314yes—
159#400—28no—
160#398—28no—
161#397—28no—
162#395—28no—
163#391—22no—
164#391—22no—
165#391C2314yes—
166#391C2314yes—
167#389—22no—
168#389—22no—
169#389C2314yes—
170#389C2314yes—
171#388—28no—
172#385—28no—
173#384—22no—
174#384C2314yes—
175#382—28no—
176#380—22no—
177#380—22no—
178#380C2314yes—
179#380C2314yes—
180C4×C4⋊C4—22no—
181C4×C4⋊C4C2314yes—
182C4×C22⋊C4D414yes—
183C4×C22⋊C4C2314yes—
184C2×#95—21no—
185C2×#95—21no—
186C2×#95—21no—
187C2×#95—21no—
188C2×#95—21no—
189C2×#95—21no—
190C2×#95—21no—
191C2×#95—21no—
192C2×#95—21no—
193C2×#95D411yescharacteristic
194C2×#95D411yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#336C2314yes—
202#336C2314yes—
203C25C2×D411yescharacteristic
204C22×Q8—21no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—22no—
212C22×Q8—44no—
213C22×Q8—44no—
214C22×Q8—82no—
215C22×Q8C2412yes—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—21no—
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C23×C4—24no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—82no—
234C23×C4C2×D412yes—
235C23×C4C2×D411yescharacteristic
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
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×C8C2×D414yes—
253C22×C8#4114yes—
254C22×C8C2412yes—
255C22×C8C2412yes—
256C22×C8C2412yes—
257C22×C8C2412yes—
258C4⋊Q8—28no—
259C4⋊Q8—44no—
260C4⋊Q8—48no—
261#125—28no—
262#125—44no—
263#125—48no—
264C22⋊Q8—28no—
265C22⋊Q8—432no—
266C22⋊Q8—44no—
267C22⋊Q8—48no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
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—28no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—48no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4—44no—
305C2×C4⋊C4—44no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—82no—
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×C4⋊C4C2412yes—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4C2×D414yes—
326C2×C22⋊C4C2×D414yes—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—44no—
333C2×C42C2×D412yes—
334C2×C42C2×D412yes—
335C2×C42C2412yes—
336C2×C42C2412yes—
337#107—28no—
338#107—28no—
339#107C2×D4116yes—
340#106—28no—
341#106—28no—
342#106—44no—
343C4⋊C8—28no—
344#103—28no—
345#103—28no—
346#103—28no—
347#103—44no—
348C22⋊C8—28no—
349C22⋊C8—28no—
350#97—28no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—24no—
360#95C2×D414yes—
361#95C2×D414yes—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24#13714yes—
369C24C22×D412yes—
370C24C22×D411yescharacteristic
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—84no—
387C2×Q8—816no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
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—24no—
407C22×C4—24no—
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—48no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
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—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—84no—
450C22×C4—816no—
451C22×C4AGL(1,ℤ/8)14yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D412yes—
459C22×C4C2×#4112yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C42−1+412yes—
462C22×C42−1+412yes—
463C22×C42−1+412yes—
464C22×C4C2511yescharacteristic
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×D414yes—
480C2×C8C2×#4114yes—
481C4⋊C4—24no—
482C4⋊C4—24no—
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—22no—
494C4⋊C4—22no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—84no—
509C4⋊C4C22×D414yes—
510C4⋊C4C22×D414yes—
511C22⋊C4—28no—
512C22⋊C4—28no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C42—24no—
517C42—24no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—22no—
522C42—48no—
523C23—28no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—28no—
528C23—28no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23#53612yes—
537C23D4212yes—
538C23#54712yes—
539C23C2×AGL(1,ℤ/8)12yes—
540C23C2×#13714yes—
541C23C2×#13712yes—
542C23#57712yes—
543C23#57812yes—
544C23C23×D411yescharacteristic
545Q8—42no—
546Q8—42no—
547Q8—42no—
548Q8—42no—
549Q8—816no—
550Q8—84no—
551C2×C4—28no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
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—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—42no—
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#53612yes—
609C2×C4#53614yes—
610C2×C4#54214yes—
611C2×C4D4212yes—
612C2×C4#54712yes—
613C2×C4#54814yes—
614C2×C4#56314yes—
615C2×C4C2×AGL(1,ℤ/8)14yes—
616C2×C4C23×D411yescharacteristic
617C2×C4C22×#4111yescharacteristic
618C2×C4C2×H(5,𝔽2)11yescharacteristic
619C2×C4C2×2−1+411yescharacteristic
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—42no—
630C22—416no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22#267414yes—
635C22#323914yes—
636C22#324214yes—
637C22#327014yes—
638C22#327214yes—
639C22#334314yes—
640C22#334414yes—
641C22C2×#53611yescharacteristic
642C22C2×D4211yescharacteristic
643C22C2×#54711yescharacteristic
644C22C22×AGL(1,ℤ/8)11yescharacteristic
645C22C22×#13711yescharacteristic
646C22C2×#57711yescharacteristic
647C22C2×#57811yescharacteristic
648C4—24no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—44no—
653C4—44no—
654C4—42no—
655C4—82no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4#267414yes—
662C4C2×#53611yescharacteristic
663C4C2×#54211yescharacteristic
664C4C2×#54811yescharacteristic
665C4C2×#56311yescharacteristic
666C2—44no—
667C2#3080318yesdirect product factor
668C2C2×#267411yescharacteristic
669C2C2×#323911yescharacteristic
670C2C2×#324211yescharacteristic
671C2C2×#327011yescharacteristic
672C2C2×#327211yescharacteristic
673C2C2×#334311yescharacteristic
674C2C2×#334411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
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
1. C22×#501C2semidirect
5. #30803C2direct (N × H)
6. C2×#2096C2semidirect
8. C2×#2066C2semidirect
9. C2×#2059C2semidirect
11. C2×#2024C2semidirect
12. C2×#1947C2semidirect
13. C2×#1935C2semidirect
15. C2×#1918C2semidirect
23. C2×#501C22semidirect
60. #2096C22semidirect
62. #2066C22semidirect
63. #2059C22semidirect
65. #2024C22semidirect
66. #1947C22semidirect
67. #1935C22semidirect
69. #1918C22semidirect
667. C2#30803direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5012
C22×#4842
C2×#24892
C2×#24402
#308032
C2×#20962
C2×#20772
C2×#20662
C2×#20592
C2×#20362
C2×#20242
C2×#19472
C2×#19352
C2×#19242
C2×#19182
C2×#18532