Finite Group Databasebeta

← #6443492

Subgroups — #6443492

Order 512.

2030 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3163C211yesmaximal, characteristic, complemented
2C2×#2913C211yesmaximal, characteristic
3#35620C211yesmaximal, characteristic
4#35532C211yesmaximal, characteristic
5#26073C214yesmaximal, complemented
6#26072C214yesmaximal, complemented
7#26071C214yesmaximal, complemented
8#26070C214yesmaximal, complemented
9#23659C211yesmaximal, characteristic, complemented
10#23658C211yesmaximal, characteristic, complemented
11#23556C211yesmaximal, characteristic, complemented
12#23555C211yesmaximal, characteristic, complemented
13#21156C211yesmaximal, characteristic, complemented
14#21154C211yesmaximal, characteristic, complemented
15#20848C211yesmaximal, characteristic
16#20847C211yesmaximal, characteristic
17C2×#1847C211yesmaximal, characteristic, complemented
18#20662C211yesmaximal, characteristic
19#20384C211yesmaximal, characteristic
20C22×#131C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#544C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#520C2211yescharacteristic
25#3163—28no—
26C2×#501C2212yes—
27C2×#501C2212yes—
28C2×#499C2212yes—
29C2×#498C2212yes—
30#2913—28no—
31C2×#428C2211yescharacteristic
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#428C2212yes—
36C2×#428C2212yes—
37C2×#427C2211yescharacteristic
38C2×#427C2212yes—
39C2×#427C2211yescharacteristic
40C22×#107C2211yescharacteristic
41C22×#107C2212yes—
42C22×#107C2211yescharacteristic
43C22×#106C2211yescharacteristic
44C22×#106C2211yescharacteristic
45C2×#408C2211yescharacteristic
46C2×#405C2211yescharacteristic
47C22×#97C2211yescharacteristic
48#2568—21no—
49#2567—21no—
50#2566—21no—
51#2566—21no—
52#2565—21no—
53#2565—21no—
54#2563—21no—
55#2542—21no—
56#2450—24no—
57#2449—24no—
58#2442C2214yes—
59#2441C2214yes—
60#2440C2214yes—
61#2439C2214yes—
62#2317C2211yescharacteristic
63#2317C2211yescharacteristic
64#2311C2211yescharacteristic
65#2311C2211yescharacteristic
66#2297C2211yescharacteristic
67#2296C2211yescharacteristic
68#2296C2211yescharacteristic
69C2×#384C2211yescharacteristic
70C2×#382C2211yescharacteristic
71#2085C2214yes—
72#2085C2214yes—
73#2084C2214yes—
74#2083C2214yes—
75#2059C2214yes—
76#2059C2214yes—
77#2058C2214yes—
78#2057C2214yes—
79#1944C2212yes—
80#1944C2212yes—
81#1944C2214yes—
82#1943C2212yes—
83#1943C2212yes—
84#1942C2214yes—
85#1854C2214yes—
86#1854C2214yes—
87#1853C2214yes—
88#1852C2214yes—
89#1847C2214yes—
90#1847C2214yes—
91#1809C2214yes—
92#1807C2214yes—
93#1753C2214yes—
94#1752C2214yes—
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1736C2211yescharacteristic
98#1736C2211yescharacteristic
99#1735C2211yescharacteristic
100#1735C2211yescharacteristic
101#1729C2211yescharacteristic
102#1728C2211yescharacteristic
103C22×#41C2311yescharacteristic
104C2×#131—24no—
105C2×#131—24no—
106C2×#131Q818yescomplemented
107C22×M16C2312yes—
108C22×M16C2311yescharacteristic
109C22×M16C2311yescharacteristic
110C23×C8C2312yes—
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#544—24no—
114#544—24no—
115#544—42no—
116#544—42no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8C2312yes—
120C2×C4⋊Q8C2312yes—
121C2×#125—21no—
122C2×#125—21no—
123C2×#125C2312yes—
124C2×#125C2312yes—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8—24no—
128#520—24no—
129#520—24no—
130#520—42no—
131#520—42no—
132C2×C4×Q8—21no—
133C2×C4×Q8—21no—
134C2×C4×D4—21no—
135C2×C4×D4—21no—
136C2×C4×D4—22no—
137C2×C4×D4—22no—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2312yes—
141C2×#117C2312yes—
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C2×#117C2311yescharacteristic
145C22×C4⋊C4—22no—
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2311yescharacteristic
150C22×C4⋊C4C2312yes—
151C22×C4⋊C4C2312yes—
152C22×C42C2311yescharacteristic
153#501—28no—
154#501—28no—
155#499—28no—
156#498—28no—
157#428—24no—
158#428—24no—
159#428—24no—
160#428—24no—
161#428—28no—
162#428—28no—
163#427—24no—
164#427—28no—
165#427D418yes—
166C2×#107—24no—
167C2×#107—28no—
168C2×#107—42no—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#107C2312yes—
173C2×#107C2312yes—
174C2×#107C2312yes—
175C2×#107C2312yes—
176C2×#107C2314yes—
177C2×#107C2311yescharacteristic
178C2×#107C2311yescharacteristic
179C2×#106—24no—
180C2×#106—42no—
181C2×#106C2312yes—
182C2×#106C2312yes—
183C2×#106C2312yes—
184C2×#106C2312yes—
185C2×#106C2312yes—
186C2×#106C2312yes—
187C2×#106C2312yes—
188C2×#106C2314yes—
189C2×#106C2311yescharacteristic
190C2×#106C2311yescharacteristic
191#408Q818yescomplemented
192C2×C22⋊C8C2314yes—
193#405—24no—
194C2×#97—24no—
195C2×#97C2312yes—
196C2×#97C2312yes—
197C2×#97C2312yes—
198C2×C4×C8C2312yes—
199#400—24no—
200#399—24no—
201#398—24no—
202#398—24no—
203#396—24no—
204#396—24no—
205#395—24no—
206#393—24no—
207#391—21no—
208#391—21no—
209#391—24no—
210#391D411yescharacteristic
211#391D411yescharacteristic
212#390—21no—
213#390—21no—
214#390D411yescharacteristic
215#390D411yescharacteristic
216#389—24no—
217#389C2314yes—
218#389C2314yes—
219#389C2312yes—
220#389C2312yes—
221#389C2312yes—
222#389C2312yes—
223#387—24no—
224#387—24no—
225#386—22no—
226#386—22no—
227#386D414yes—
228#385—22no—
229#385—22no—
230#385—24no—
231#385—24no—
232#385—22no—
233#385—22no—
234#384—21no—
235#384—21no—
236#384—21no—
237#384—21no—
238#384C2314yes—
239#382—21no—
240#382—21no—
241#382—21no—
242#382—21no—
243#382C2314yes—
244#380—24no—
245#380C2314yes—
246#380C2314yes—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4—21no—
251C4×C22⋊C4C2314yes—
252C2×#95—22no—
253#340C2312yes—
254#340C2314yes—
255#340C2312yes—
256#340C2314yes—
257#340C2312yes—
258#340C2312yes—
259#340C2314yes—
260#340C2314yes—
261#336C2314yes—
262#336C2314yes—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—22no—
267C2×#41C2×Q814yes—
268C2×#41C2×Q814yes—
269C22×Q8—22no—
270C22×Q8—22no—
271C22×Q8C2×D411yescharacteristic
272C22×D4C2×D412yes—
273C22×D4C2×D411yescharacteristic
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—22no—
277C23×C4—21no—
278C23×C4—22no—
279C23×C4—21no—
280C23×C4C2×D412yes—
281C23×C4C2412yes—
282C23×C4C2411yescharacteristic
283#131—232no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—24no—
289C2×M16—24no—
290C2×M16C2×Q814yes—
291C2×M16C2×Q818yes—
292C2×M16C2×Q814yes—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—24no—
296C22×C8—24no—
297C22×C8C2×D414yes—
298C22×C8C2×Q818yes—
299C22×C8C2×Q814yes—
300C22×C8C2×Q814yes—
301C22×C8#4114yes—
302C22×C8C2412yes—
303C22×C8C2412yes—
304C22×C8C2412yes—
305C22×C8C2411yescharacteristic
306C22×C8C2411yescharacteristic
307C4⋊Q8—24no—
308C4⋊Q8—24no—
309C4⋊Q8—42no—
310C4⋊Q8—42no—
311C4⋊Q8—44no—
312C4⋊Q8—44no—
313#125—24no—
314#125—24no—
315#125—42no—
316#125—42no—
317#125—44no—
318#125—44no—
319C22⋊Q8—48no—
320C22⋊Q8—48no—
321C22⋊Q8—88no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×D4—44no—
325C4×D4—44no—
326C4×D4—48no—
327C4×D4—48no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—24no—
337#117—24no—
338#117—42no—
339#117—42no—
340#117C2×D414yes—
341#117C2×D414yes—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—42no—
373C2×C4⋊C4—42no—
374C2×C4⋊C4—84no—
375C2×C4⋊C4C2×D411yescharacteristic
376C2×C4⋊C4C2×D411yescharacteristic
377C2×C4⋊C4C2412yes—
378C2×C4⋊C4C2412yes—
379C2×C4⋊C4C2412yes—
380C2×C4⋊C4C2412yes—
381C2×C4⋊C4C2412yes—
382C2×C4⋊C4C2412yes—
383C2×C4⋊C4C2411yescharacteristic
384C2×C4⋊C4C2411yescharacteristic
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2411yescharacteristic
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C22⋊C4—24no—
394C2×C22⋊C4—24no—
395C2×C22⋊C4C2×D412yes—
396C2×C22⋊C4C2×D414yes—
397C2×C22⋊C4C2×D414yes—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2412yes—
409C2×C42C2412yes—
410#107—24no—
411#107—24no—
412#107—24no—
413#107—24no—
414#107—24no—
415#107—24no—
416#107—24no—
417#107—44no—
418#107C2×D414yes—
419#107C2×D414yes—
420#106—24no—
421#106—24no—
422#106—24no—
423#106—24no—
424#106—24no—
425#106—24no—
426#106—24no—
427#106—44no—
428#106C2×D414yes—
429#106C2×D414yes—
430C22⋊C8C2×Q8116yes—
431#97—24no—
432#97—24no—
433#97—24no—
434C4×C8—24no—
435#95—24no—
436#95—24no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—24no—
442#95—24no—
443#95—24no—
444#95—24no—
445#95C2×D414yes—
446#95C2×D414yes—
447C24C22×D412yes—
448C24C22×D411yescharacteristic
449#41—28no—
450#41—28no—
451#41—28no—
452#41—28no—
453C2×Q8—22no—
454C2×Q8—22no—
455C2×Q8—44no—
456C2×Q8—44no—
457C2×Q8—84no—
458C2×Q8—84no—
459C2×Q8C4⋊Q814yes—
460C2×D4—24no—
461C2×D4—24no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4C4⋊Q818yes—
465C2×D4C4⋊Q814yes—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—82no—
490C22×C4—82no—
491C22×C4—84no—
492C22×C4—84no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×Q814yes—
503C22×C4C22×Q814yes—
504C22×C4C22×Q812yes—
505C22×C4C22×Q812yes—
506C22×C4C2×#4112yes—
507C22×C4H(5,𝔽2)12yes—
508C22×C42−1+412yes—
509C22×C4C2511yescharacteristic
510M16—28no—
511M16—28no—
512M16—216no—
513C2×C8—28no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—216no—
521C2×C8C22⋊Q8116yescomplemented
522C2×C8C22×D412yes—
523C2×C8C22×D412yes—
524C2×C8C22×Q814yes—
525C2×C8C22×Q814yes—
526C2×C8C22×Q814yes—
527C2×C8C22×Q812yes—
528C2×C8C22×Q812yes—
529C2×C8C2×#4112yes—
530C2×C8C2×#4112yes—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—48no—
559C4⋊C4—44no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—42no—
564C4⋊C4—84no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—84no—
568C4⋊C4—84no—
569C4⋊C4C22×D412yes—
570C4⋊C4C22×D412yes—
571C4⋊C4C22×D412yes—
572C4⋊C4C22×D412yes—
573C22⋊C4—24no—
574C22⋊C4—24no—
575C22⋊C4—24no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—44no—
580C22⋊C4—42no—
581C22⋊C4—42no—
582C22⋊C4—84no—
583C22⋊C4—84no—
584C42—24no—
585C42—24no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—44no—
593C42—44no—
594C42—42no—
595C42—42no—
596C23—22no—
597C23—22no—
598C23—24no—
599C23—24no—
600C23—22no—
601C23—22no—
602C23C2×C4⋊Q814yes—
603C23C2×C4⋊Q812yes—
604C23D4212yes—
605C23#54712yes—
606C23C23×D411yescharacteristic
607Q8—28no—
608Q8—164no—
609Q8—164no—
610D4—216no—
611D4—28no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—48no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—164no—
652C2×C4—164no—
653C2×C4—162no—
654C2×C4—162no—
655C2×C4—162no—
656C2×C4—162no—
657C2×C4C2×C22⋊Q814yes—
658C2×C4C2×C4⋊Q814yes—
659C2×C4C2×C4⋊Q812yes—
660C2×C4#54414yes—
661C2×C4D4212yes—
662C2×C4#54712yes—
663C2×C4D4×Q814yes—
664C2×C4#55414yes—
665C2×C4C23×D411yescharacteristic
666C2×C4C23×Q811yescharacteristic
667C2×C4C23×Q811yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C2×H(5,𝔽2)11yescharacteristic
670C2×C4C2×2−1+411yescharacteristic
671C8—24no—
672C8—24no—
673C8—24no—
674C8C2×C22⋊Q814yes—
675C8C2×C22⋊Q814yes—
676C22—24no—
677C22—22no—
678C22—22no—
679C22—24no—
680C22—24no—
681C22—28no—
682C22—44no—
683C22—42no—
684C22—42no—
685C22#244014yes—
686C22C22×C4⋊Q811yescharacteristic
687C22C2×D4211yescharacteristic
688C22C2×#54711yescharacteristic
689C22#358512yes—
690C22#358612yes—
691C4—24no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4#244014yes—
707C4C22×C22⋊Q811yescharacteristic
708C4C2×#54411yescharacteristic
709C4C2×D4×Q811yescharacteristic
710C4C2×#55411yescharacteristic
711C2—24no—
712C2—44no—
713C2C2×#244011yescharacteristic
714C2#6066812yes—
715C2#6108211yescharacteristic
716C2#6108311yescharacteristic
717C2#6126612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3163C2semidirect
5. #26073C2semidirect
6. #26072C2semidirect
7. #26071C2semidirect
8. #26070C2semidirect
9. #23659C2semidirect
10. #23658C2semidirect
11. #23556C2semidirect
12. #23555C2semidirect
13. #21156C2semidirect
14. #21154C2semidirect
17. C2×#1847C2semidirect
106. C2×#131Q8semidirect
191. #408Q8semidirect
521. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31632
C2×#29132
#356202
#355322
#260732
#260722
#260712
#260702
#236592
#236582
#235562
#235552
#211562
#211542
#208482
#208472
C2×#18472
#206622
#203842