Finite Group Databasebeta

← #6444704

Subgroups — #6444704

Order 512.

2094 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic, complemented
2C2×#2951C211yesmaximal, characteristic
3#35639C211yesmaximal, characteristic, complemented
4#35574C211yesmaximal, characteristic, complemented
5#26499C214yesmaximal, complemented
6#26498C214yesmaximal, complemented
7#26489C214yesmaximal, complemented
8#26488C214yesmaximal, complemented
9#23611C211yesmaximal, characteristic, complemented
10#23610C211yesmaximal, characteristic, complemented
11#23453C211yesmaximal, characteristic, complemented
12#23452C211yesmaximal, characteristic, complemented
13#21229C211yesmaximal, characteristic
14#21228C211yesmaximal, characteristic
15#21158C211yesmaximal, characteristic, complemented
16#21155C211yesmaximal, characteristic, complemented
17#20882C211yesmaximal, characteristic, complemented
18#20881C211yesmaximal, characteristic, complemented
19#20436C211yesmaximal, characteristic, complemented
20C22×#135C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#125C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#520C2211yescharacteristic
25C2×#501—28no—
26C2×#501C2214yes—
27C2×#501C2214yes—
28#2951—28no—
29C2×#440C2212yes—
30C2×#440C2212yes—
31C2×C4×Q16C2212yes—
32C2×C4×D8C2212yes—
33C2×#427C2211yescharacteristic
34C22×#107C2211yescharacteristic
35C22×#107C2211yescharacteristic
36C22×#106C2211yescharacteristic
37C22×#106C2211yescharacteristic
38C2×#424C2211yescharacteristic
39C2×#417C2211yescharacteristic
40C2×#417C2211yescharacteristic
41C2×#417C2211yescharacteristic
42C2×#417C2211yescharacteristic
43C2×#416C2211yescharacteristic
44C2×#416C2211yescharacteristic
45C2×#405C2211yescharacteristic
46C2×#405C2211yescharacteristic
47C22×#97C2211yescharacteristic
48#2579—22no—
49#2579—22no—
50#2487—22no—
51#2454C2214yes—
52#2453C2214yes—
53#2452C2214yes—
54#2451C2214yes—
55#2442—22no—
56#2441—22no—
57#2440—22no—
58#2439—22no—
59#2409—22no—
60#2317C2211yescharacteristic
61#2317C2211yescharacteristic
62#2316C2211yescharacteristic
63#2316C2211yescharacteristic
64#2311C2211yescharacteristic
65#2311C2211yescharacteristic
66#2297C2211yescharacteristic
67#2297C2211yescharacteristic
68#2296C2211yescharacteristic
69#2068C2214yes—
70#2067C2214yes—
71#2066C2214yes—
72#2065C2214yes—
73#2026C2214yes—
74#2025C2214yes—
75#2024C2214yes—
76#2023C2214yes—
77#1973C2212yes—
78#1973C2212yes—
79#1972C2212yes—
80#1972C2212yes—
81#1971C2212yes—
82#1971C2212yes—
83#1968C2212yes—
84#1968C2212yes—
85#1943C2214yes—
86#1943C2214yes—
87#1942C2214yes—
88#1941C2214yes—
89#1865C2214yes—
90#1864C2214yes—
91#1863C2214yes—
92#1862C2214yes—
93#1771C2214yes—
94#1770C2214yes—
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1736C2211yescharacteristic
98#1736C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C22×#41C2311yescharacteristic
103C23×Q8—21no—
104C2×#135—24no—
105C2×#135—24no—
106C2×#135Q818yescomplemented
107C22×Q16C2311yescharacteristic
108C22×SD16C2311yescharacteristic
109C22×SD16C2311yescharacteristic
110C22×D8C2311yescharacteristic
111C22×M16C2311yescharacteristic
112C22×M16C2311yescharacteristic
113C23×C8C2311yescharacteristic
114C23×C8C2311yescharacteristic
115C2×C4⋊Q8—24no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8—42no—
118C2×C4⋊Q8C2314yes—
119C2×#125—24no—
120C2×#125—22no—
121C2×#125—42no—
122C2×#125C2314yes—
123C2×C22⋊Q8—24no—
124#520—24no—
125#520—24no—
126#520—42no—
127#520—42no—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2312yes—
130C2×C4×D4—22no—
131C2×C4×D4—22no—
132C2×C4×D4C2312yes—
133C2×C4×D4C2312yes—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4C2311yescharacteristic
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2311yescharacteristic
148C22×C4⋊C4C2311yescharacteristic
149C22×C42C2311yescharacteristic
150#501—216no—
151#501—216no—
152#440—28no—
153#440—28no—
154C4×Q16—28no—
155C4×D8—28no—
156#427—24no—
157C2×#107—24no—
158C2×#107D418yes—
159C2×#107C2314yes—
160C2×#107C2312yes—
161C2×#107C2312yes—
162C2×#107C2312yes—
163C2×#106—24no—
164C2×#106—24no—
165C2×#106C2314yes—
166C2×#106C2312yes—
167C2×#106C2312yes—
168C2×#106C2312yes—
169#424—24no—
170C2×C4⋊C8C2314yes—
171#417—24no—
172#417—24no—
173#417—42no—
174#417—42no—
175#416—24no—
176#416—24no—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#102C2312yes—
184C2×#102C2312yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189#405—24no—
190#405—24no—
191C2×#97—24no—
192C2×#97C2312yes—
193C2×#97C2312yes—
194C2×#97C2312yes—
195C2×#97C2312yes—
196C2×C4×C8C2312yes—
197C2×C4×C8C2312yes—
198#400—24no—
199#399—24no—
200#398—24no—
201#398—24no—
202#396—24no—
203#396—24no—
204#395—24no—
205#393—24no—
206#391—24no—
207#391C2312yes—
208#391C2312yes—
209#391C2312yes—
210#391C2312yes—
211#390C2312yes—
212#390C2312yes—
213#390C2312yes—
214#390C2312yes—
215#389—22no—
216#389—22no—
217#389—24no—
218#389D412yes—
219#389D412yes—
220#389C2314yes—
221#389C2314yes—
222#389C2314yes—
223#389C2314yes—
224#387—24no—
225#387—24no—
226#386—21no—
227#386—21no—
228#386—21no—
229#386—21no—
230#386—22no—
231#386—22no—
232#385—24no—
233#385—22no—
234#385—22no—
235#385—24no—
236#385D412yes—
237#385D412yes—
238#384—21no—
239#384—21no—
240#382—21no—
241#382—21no—
242#380—22no—
243#380—22no—
244#380—22no—
245#380—22no—
246#380—24no—
247C4×C4⋊C4C2314yes—
248C4×C22⋊C4—22no—
249C4×C22⋊C4—22no—
250#340C2314yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#336C2314yes—
255#336C2314yes—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41C2×Q814yes—
267C2×#41C2×Q814yes—
268C22×Q8—24no—
269C22×Q8—44no—
270C22×Q8—84no—
271C22×Q8C2411yescharacteristic
272C22×Q8C2411yescharacteristic
273C22×D4—21no—
274C22×D4—21no—
275C22×D4C2411yescharacteristic
276C22×D4C2411yescharacteristic
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—21no—
281C23×C4—21no—
282C23×C4—21no—
283C23×C4—22no—
284C23×C4—22no—
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2411yescharacteristic
288#135—232no—
289C2×Q16—22no—
290C2×Q16—22no—
291C2×Q16C2×Q814yescomplemented
292C2×SD16—22no—
293C2×SD16—22no—
294C2×SD16—22no—
295C2×SD16—22no—
296C2×SD16C2×Q814yescomplemented
297C2×SD16C2×Q814yescomplemented
298C2×D8—22no—
299C2×D8—22no—
300C2×D8C2×Q814yescomplemented
301C2×M16—22no—
302C2×M16—22no—
303C2×M16—22no—
304C2×M16—22no—
305C2×M16—22no—
306C2×M16—22no—
307C22×C8—22no—
308C22×C8—22no—
309C22×C8C2×D414yes—
310C22×C8C2×Q814yes—
311C22×C8#4114yes—
312C22×C8#4114yes—
313C22×C8C2412yes—
314C22×C8C2412yes—
315C22×C8C2412yes—
316C22×C8C2411yescharacteristic
317C22×C8C2411yescharacteristic
318C4⋊Q8—28no—
319C4⋊Q8—44no—
320C4⋊Q8—48no—
321#125—28no—
322#125—44no—
323#125—48no—
324C22⋊Q8—88no—
325C4×Q8—24no—
326C4×Q8—24no—
327C4×Q8—42no—
328C4×Q8—42no—
329C4×D4—24no—
330C4×D4—24no—
331C4×D4—42no—
332C4×D4—42no—
333C4×D4—48no—
334C4×D4—48no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—22no—
339#117—22no—
340#117—22no—
341#117—22no—
342#117—22no—
343#117—44no—
344#117—44no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—24no—
361C2×C4⋊C4—24no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—44no—
365C2×C4⋊C4—44no—
366C2×C4⋊C4—44no—
367C2×C4⋊C4—44no—
368C2×C4⋊C4—44no—
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2×D412yes—
372C2×C4⋊C4C2×D414yes—
373C2×C4⋊C4C2×D414yes—
374C2×C4⋊C4C2×D414yes—
375C2×C4⋊C4C2×D414yes—
376C2×C4⋊C4C2412yes—
377C2×C4⋊C4C2412yes—
378C2×C4⋊C4C2412yes—
379C2×C4⋊C4C2412yes—
380C2×C4⋊C4C2412yes—
381C2×C4⋊C4C2412yes—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—22no—
387C2×C22⋊C4—22no—
388C2×C22⋊C4—24no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4C2×D412yes—
393C2×C22⋊C4C2×D412yes—
394C2×C42—22no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—22no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42C2×D412yes—
401C2×C42C2412yes—
402C2×C42C2412yes—
403C2×C42C2412yes—
404#107—24no—
405#107—28no—
406#107C2×D418yes—
407#107C2×D418yes—
408#106—24no—
409#106—24no—
410#106—24no—
411#106—28no—
412C4⋊C8—28no—
413#103—24no—
414#103—24no—
415#103—24no—
416#103—24no—
417#103—42no—
418#103—42no—
419#102—24no—
420#102—24no—
421#102—24no—
422#102—24no—
423#102—42no—
424#102—42no—
425#97—24no—
426#97—24no—
427#97—24no—
428#97—24no—
429C4×C8—24no—
430C4×C8—24no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—24no—
436#95—24no—
437#95—24no—
438#95—24no—
439#95—24no—
440#95—24no—
441#95C2×D414yes—
442#95C2×D414yes—
443C24—21no—
444C24—21no—
445C24C22×D411yescharacteristic
446#41—28no—
447#41—28no—
448#41—28no—
449#41—28no—
450#41—44no—
451#41—44no—
452#41—44no—
453#41—44no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—48no—
465C2×Q8—48no—
466C2×Q8—84no—
467C2×Q8—816no—
468C2×Q8C22×Q812yes—
469C2×Q8C22×Q812yes—
470C2×D4—24no—
471C2×D4—24no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C2×D4—21no—
479C2×D4—21no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—42no—
483C2×D4—42no—
484C2×D4—42no—
485C2×D4—42no—
486C2×D4C22×Q812yes—
487C2×D4C22×Q812yes—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—42no—
509C22×C4—42no—
510C22×C4—42no—
511C22×C4—42no—
512C22×C4—84no—
513C22×C4—84no—
514C22×C4—82no—
515C22×C4—82no—
516C22×C4—82no—
517C22×C4—82no—
518C22×C4C22×D414yes—
519C22×C4C22×D412yes—
520C22×C4C22×D412yes—
521C22×C4C22×D412yes—
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C22×D411yescharacteristic
525C22×C4C22×Q812yes—
526C22×C4C2×#4112yes—
527C22×C4C2×#4112yes—
528C22×C4H(5,𝔽2)12yes—
529C22×C42−1+412yes—
530C22×C4C2511yescharacteristic
531Q16—28no—
532SD16—28no—
533SD16—28no—
534D8—28no—
535M16—44no—
536M16—44no—
537C2×C8—28no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—22no—
546C2×C8C22⋊Q8116yescomplemented
547C2×C8C22×D412yes—
548C2×C8C22×D412yes—
549C2×C8C22×Q812yes—
550C2×C8C22×Q812yes—
551C2×C8C2×#4114yes—
552C2×C8C2×#4112yes—
553C2×C8C2×#4112yes—
554C4⋊C4—24no—
555C4⋊C4—24no—
556C4⋊C4—24no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—22no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—22no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—44no—
572C4⋊C4—44no—
573C4⋊C4—48no—
574C4⋊C4—48no—
575C4⋊C4—48no—
576C4⋊C4—42no—
577C4⋊C4—42no—
578C4⋊C4—84no—
579C4⋊C4—84no—
580C4⋊C4—84no—
581C4⋊C4—84no—
582C4⋊C4C22×D414yes—
583C4⋊C4C22×D414yes—
584C22⋊C4—24no—
585C22⋊C4—24no—
586C22⋊C4—44no—
587C22⋊C4—44no—
588C22⋊C4—44no—
589C22⋊C4—44no—
590C22⋊C4—42no—
591C22⋊C4—42no—
592C22⋊C4—82no—
593C22⋊C4—84no—
594C22⋊C4—82no—
595C22⋊C4—84no—
596C42—24no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—44no—
604C42—44no—
605C42—42no—
606C42—42no—
607C23—22no—
608C23—22no—
609C23—22no—
610C23—22no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—41no—
616C23—41no—
617C23—41no—
618C23—41no—
619C23#53212yes—
620C23D4212yes—
621C23D4×Q812yes—
622C23C23×D411yescharacteristic
623Q8—22no—
624Q8—22no—
625Q8—22no—
626Q8—22no—
627Q8—41no—
628Q8—41no—
629Q8—41no—
630Q8—41no—
631Q8—816no—
632D4—22no—
633D4—22no—
634D4—22no—
635D4—22no—
636D4—48no—
637D4—48no—
638D4—41no—
639D4—41no—
640D4—41no—
641D4—41no—
642C2×C4—22no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—24no—
646C2×C4—22no—
647C2×C4—24no—
648C2×C4—22no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—22no—
658C2×C4—22no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—48no—
668C2×C4—48no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4—84no—
683C2×C4—164no—
684C2×C4—164no—
685C2×C4C2×C22⋊Q814yes—
686C2×C4#53212yes—
687C2×C4#54214yes—
688C2×C4D4212yes—
689C2×C4#54814yes—
690C2×C4D4×Q812yes—
691C2×C4#55414yes—
692C2×C4C23×D411yescharacteristic
693C2×C4C23×Q811yescharacteristic
694C2×C4C22×#4111yescharacteristic
695C2×C4C22×#4111yescharacteristic
696C2×C4C2×H(5,𝔽2)11yescharacteristic
697C2×C4C2×2−1+411yescharacteristic
698C8—24no—
699C8—42no—
700C8—42no—
701C8C2×C22⋊Q814yes—
702C8C2×C22⋊Q814yes—
703C22—24no—
704C22—22no—
705C22—22no—
706C22—24no—
707C22—42no—
708C22—42no—
709C22—44no—
710C22—44no—
711C22—82no—
712C22—82no—
713C22—81no—
714C22—81no—
715C22—81no—
716C22—81no—
717C22#245214yes—
718C22C2×#53211yescharacteristic
719C22C2×D4211yescharacteristic
720C22C2×D4×Q811yescharacteristic
721C22#358512yes—
722C22#358612yes—
723C4—44no—
724C4—42no—
725C4—42no—
726C4—42no—
727C4—42no—
728C4—44no—
729C4—44no—
730C4—42no—
731C4—42no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4#245214yes—
737C4C22×C22⋊Q811yescharacteristic
738C4C2×#54211yescharacteristic
739C4C2×#54811yescharacteristic
740C4C2×#55411yescharacteristic
741C2—24no—
742C2—82no—
743C2—82no—
744C2C2×#245211yescharacteristic
745C2#6072112yes—
746C2#6105811yescharacteristic
747C2#6106111yescharacteristic
748C2#6138012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#135C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#501C2semidirect
3. #35639C2semidirect
4. #35574C2semidirect
5. #26499C2semidirect
6. #26498C2semidirect
7. #26489C2semidirect
8. #26488C2semidirect
9. #23611C2semidirect
10. #23610C2semidirect
11. #23453C2semidirect
12. #23452C2semidirect
15. #21158C2semidirect
16. #21155C2semidirect
17. #20882C2semidirect
18. #20881C2semidirect
19. #20436C2semidirect
106. C2×#135Q8semidirect
291. C2×Q16C2×Q8semidirect
296. C2×SD16C2×Q8semidirect
297. C2×SD16C2×Q8semidirect
300. C2×D8C2×Q8semidirect
546. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#29512
#356392
#355742
#264992
#264982
#264892
#264882
#236112
#236102
#234532
#234522
#212292
#212282
#211582
#211552
#208822
#208812
#204362