Finite Group Databasebeta

← #6443387

Subgroups — #6443387

Order 512.

2050 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3084C211yesmaximal, characteristic
2C2×#3083C211yesmaximal, characteristic
3#35330C211yesmaximal, characteristic, complemented
4#34183C211yesmaximal, characteristic
5#26055C214yesmaximal, complemented
6#26054C214yesmaximal, complemented
7#26053C214yesmaximal, complemented
8#26052C214yesmaximal, complemented
9#23668C211yesmaximal, characteristic, complemented
10#23667C211yesmaximal, characteristic, complemented
11#23556C211yesmaximal, characteristic, complemented
12#23555C211yesmaximal, characteristic, complemented
13#21171C211yesmaximal, characteristic, complemented
14#21144C211yesmaximal, characteristic, complemented
15#20882C211yesmaximal, characteristic
16#20881C211yesmaximal, characteristic
17C2×#1847C211yesmaximal, characteristic, complemented
18#20649C211yesmaximal, characteristic
19#20411C211yesmaximal, characteristic
20C2×#544C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#520C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×C4×#41C2211yescharacteristic
25#3084—28no—
26#3083—28no—
27C2×#479C2212yes—
28C2×#479C2212yes—
29C2×#477C2212yes—
30C2×#475C2212yes—
31C2×#478C2212yes—
32C2×#478C2212yes—
33C2×#476C2212yes—
34C2×#474C2212yes—
35C2×#428C2211yescharacteristic
36C2×#428C2211yescharacteristic
37C22×#107C2211yescharacteristic
38C22×#107C2211yescharacteristic
39C22×#106C2211yescharacteristic
40C22×#106C2211yescharacteristic
41C2×#424C2211yescharacteristic
42C2×#424C2211yescharacteristic
43C2×#423C2211yescharacteristic
44C2×#417C2211yescharacteristic
45C2×#417C2211yescharacteristic
46C2×#416C2211yescharacteristic
47C2×#416C2211yescharacteristic
48C22×#97C2211yescharacteristic
49#2862—21no—
50#2859—21no—
51#2660—21no—
52#2660—21no—
53#2550C2214yes—
54#2550C2214yes—
55#2549—24no—
56Q8×C4⋊C4C2214yes—
57D4×C4⋊C4—24no—
58D4×C4⋊C4C2214yes—
59#2317—21no—
60#2317—21no—
61#2317C2211yescharacteristic
62#2317C2211yescharacteristic
63#2317C2211yescharacteristic
64#2317C2211yescharacteristic
65#2285—21no—
66#2285C2211yescharacteristic
67C2×#384—21no—
68C2×#384C2211yescharacteristic
69C2×#382C2211yescharacteristic
70C4×#117C2211yescharacteristic
71#2088C2214yes—
72#2088C2214yes—
73#2087C2214yes—
74#2086C2214yes—
75#2059C2214yes—
76#2059C2214yes—
77#2058C2214yes—
78#2057C2214yes—
79#1948C2214yes—
80#1947C2214yes—
81#1940C2214yes—
82#1940C2214yes—
83#1865C2214yes—
84#1864C2214yes—
85#1863C2214yes—
86#1862C2214yes—
87#1847C2214yes—
88#1847C2214yes—
89#1805C2214yes—
90#1804C2214yes—
91#1763C2212yes—
92#1763C2212yes—
93#1762C2212yes—
94#1762C2212yes—
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1736C2211yescharacteristic
98#1736C2211yescharacteristic
99#1735C2211yescharacteristic
100#1735C2211yescharacteristic
101#1729C2211yescharacteristic
102#1729C2211yescharacteristic
103C22×#41C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#544—24no—
109#544—24no—
110#544—42no—
111#544—42no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2312yes—
116C2×#125—21no—
117C2×#125—21no—
118C2×#125C2312yes—
119C2×#125C2312yes—
120C2×C22⋊Q8—22no—
121C2×C22⋊Q8—22no—
122#520—24no—
123#520—24no—
124#520—42no—
125#520—42no—
126C4×#41—24no—
127C4×#41Q818yescomplemented
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×D4—22no—
132C2×C4×D4—22no—
133C2×C4×D4—22no—
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×#117—21no—
138C2×#117—21no—
139C2×#117—21no—
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C2×#117C2311yescharacteristic
145C22×C4⋊C4—22no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—21no—
148C22×C4⋊C4—24no—
149C22×C4⋊C4C2311yescharacteristic
150C22×C4⋊C4C2311yescharacteristic
151C22×C4⋊C4C2311yescharacteristic
152C22×C4⋊C4C2311yescharacteristic
153C22×C42—21no—
154C22×C42C2311yescharacteristic
155#479—28no—
156#479—28no—
157#478—28no—
158#478—28no—
159#477—28no—
160#476—28no—
161#475—28no—
162#474—28no—
163#428—24no—
164#428—24no—
165C2×#107—24no—
166C2×#107—42no—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2314yes—
172C2×#106—24no—
173C2×#106—42no—
174C2×#106C2312yes—
175C2×#106C2312yes—
176C2×#106C2312yes—
177C2×#106C2312yes—
178C2×#106C2314yes—
179#424—24no—
180#424—24no—
181#423—24no—
182C2×C4⋊C8C2312yes—
183C2×C4⋊C8C2312yes—
184C2×C4⋊C8C2312yes—
185C2×C4⋊C8C2312yes—
186C2×C4⋊C8C2314yes—
187#417—24no—
188#417—24no—
189#416D418yes—
190#416Q818yescomplemented
191C2×#103C2312yes—
192C2×#103C2312yes—
193C2×#103C2311yescharacteristic
194C2×#103C2311yescharacteristic
195C2×#103C2312yes—
196C2×#102C2312yes—
197C2×#102C2312yes—
198C2×#102C2311yescharacteristic
199C2×#102C2311yescharacteristic
200C2×#102C2312yes—
201C2×#97—24no—
202C2×#97C2312yes—
203C2×#97C2312yes—
204#398—24no—
205#398—24no—
206#395—24no—
207#395—24no—
208#391—24no—
209#391—24no—
210#389—24no—
211#389—24no—
212#389—24no—
213#389—24no—
214#389C2314yes—
215#389C2314yes—
216#389C2314yes—
217#389C2314yes—
218#385—22no—
219#385—22no—
220#385—22no—
221#385—22no—
222#385—24no—
223#385—24no—
224#384—21no—
225#384—21no—
226#384—21no—
227#384—21no—
228#384—24no—
229#384—21no—
230#384—21no—
231#384D411yescharacteristic
232#384D411yescharacteristic
233#384C2314yes—
234#382—21no—
235#382—21no—
236#382—21no—
237#382—21no—
238#382—21no—
239#382—21no—
240#382D411yescharacteristic
241#382D411yescharacteristic
242#382C2314yes—
243#380—24no—
244#380D414yes—
245C4×C4⋊C4—24no—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2314yes—
249C4×C22⋊C4—22no—
250#376—21no—
251C2×#95—22no—
252C43—21no—
253#340C2312yes—
254#340C2314yes—
255#340C2312yes—
256#340C2314yes—
257#340C2312yes—
258#340C2314yes—
259#340C2312yes—
260#340C2314yes—
261#336C2314yes—
262#336C2314yes—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41C2×D414yes—
267C2×#41C2×Q814yes—
268C2×#41C2×Q814yes—
269C22×Q8—22no—
270C22×Q8—22no—
271C22×Q8C2411yescharacteristic
272C22×D4—21no—
273C22×D4C2411yescharacteristic
274C23×C4—21no—
275C23×C4—22no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—21no—
279C23×C4—21no—
280C23×C4—22no—
281C23×C4—22no—
282C23×C4C2×D412yes—
283C23×C4C2411yescharacteristic
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8C2×D414yes—
295C22×C8C2×Q814yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2411yescharacteristic
299C22×C8C2411yescharacteristic
300C22×C8C2412yes—
301C4⋊Q8—24no—
302C4⋊Q8—24no—
303C4⋊Q8—42no—
304C4⋊Q8—42no—
305C4⋊Q8—44no—
306C4⋊Q8—44no—
307#125—24no—
308#125—24no—
309#125—42no—
310#125—42no—
311#125—44no—
312#125—44no—
313C22⋊Q8—48no—
314C22⋊Q8—48no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—24no—
318C4×Q8—42no—
319C4×Q8—42no—
320C4×Q8C2×Q818yes—
321C4×D4—216no—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—24no—
325C4×D4—42no—
326C4×D4—42no—
327C4×D4—48no—
328C4×D4—48no—
329C4×D4C2×Q818yes—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—28no—
338#117—44no—
339#117—44no—
340#117C2×D414yes—
341#117C2×Q814yes—
342#117C2×Q814yes—
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—22no—
351C2×C4⋊C4—22no—
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—24no—
366C2×C4⋊C4—24no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—44no—
371C2×C4⋊C4—44no—
372C2×C4⋊C4—88no—
373C2×C4⋊C4—84no—
374C2×C4⋊C4C2×D411yescharacteristic
375C2×C4⋊C4C2×D411yescharacteristic
376C2×C4⋊C4C2×D411yescharacteristic
377C2×C4⋊C4C2×D411yescharacteristic
378C2×C4⋊C4C2×D414yes—
379C2×C4⋊C4C2×D414yes—
380C2×C4⋊C4C2412yes—
381C2×C4⋊C4C2412yes—
382C2×C4⋊C4C2412yes—
383C2×C4⋊C4C2412yes—
384C2×C4⋊C4C2411yescharacteristic
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2412yes—
387C2×C4⋊C4C2412yes—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—22no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4C2×D412yes—
395C2×C22⋊C4C2×D412yes—
396C2×C42—21no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—22no—
400C2×C42—28no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—22no—
404C2×C42—22no—
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42C2×Q814yes—
408C2×C42#4114yes—
409C2×C42C2412yes—
410C2×C42C2412yes—
411C2×C42C2412yes—
412#107—24no—
413#107—24no—
414#107—24no—
415#107—24no—
416#107—44no—
417#106—24no—
418#106—24no—
419#106—24no—
420#106—24no—
421#106—44no—
422C4⋊C8—24no—
423C4⋊C8—24no—
424C4⋊C8—24no—
425C4⋊C8—24no—
426C4⋊C8—28no—
427#103—24no—
428#103—24no—
429#103C2×D414yes—
430#103C2×D414yes—
431#103C2×Q818yes—
432#102—24no—
433#102—24no—
434#102C2×D414yes—
435#102C2×D414yes—
436#102C2×Q818yes—
437#97—24no—
438#97—24no—
439#95—24no—
440#95—24no—
441#95—22no—
442#95—22no—
443#95—24no—
444#95—24no—
445#95—24no—
446#95—24no—
447#95C2×D414yes—
448#95C2×D414yes—
449C24—21no—
450C24C22×D411yescharacteristic
451#41—28no—
452#41—28no—
453#41—28no—
454#41C4⋊Q8116yes—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—44no—
459C2×Q8—44no—
460C2×Q8—84no—
461C2×Q8—84no—
462C2×Q8C22×D412yes—
463C2×Q8C22×Q812yes—
464C2×Q8C22×Q812yes—
465C2×D4—24no—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4C22×D412yes—
472C2×D4C22×Q812yes—
473C2×D4C22×Q812yes—
474C22×C4—21no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—28no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—84no—
498C22×C4—84no—
499C22×C4—82no—
500C22×C4—82no—
501C22×C4—82no—
502C22×C4—82no—
503C22×C4C22⋊Q818yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D412yes—
512C22×C4C22×Q812yes—
513C22×C4C22×Q812yes—
514C22×C4C2×#4112yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C42−1+412yes—
517C22×C4C2511yescharacteristic
518M16—44no—
519M16—44no—
520C2×C8—28no—
521C2×C8—28no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8C22×D412yes—
535C2×C8C22×D412yes—
536C2×C8C22×Q814yes—
537C4⋊C4—28no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—44no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—44no—
564C4⋊C4—48no—
565C4⋊C4—42no—
566C4⋊C4—42no—
567C4⋊C4—42no—
568C4⋊C4—42no—
569C4⋊C4—84no—
570C4⋊C4—84no—
571C4⋊C4—84no—
572C4⋊C4—84no—
573C4⋊C4—84no—
574C4⋊C4C22×D412yes—
575C4⋊C4C22×D412yes—
576C4⋊C4C22×Q814yes—
577C4⋊C4C22×Q814yes—
578C22⋊C4—24no—
579C22⋊C4—24no—
580C22⋊C4—24no—
581C22⋊C4—24no—
582C22⋊C4—24no—
583C22⋊C4—28no—
584C22⋊C4—44no—
585C22⋊C4—44no—
586C42—24no—
587C42—28no—
588C42—24no—
589C42—24no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—44no—
595C42—44no—
596C42—82no—
597C42—82no—
598C42C22⋊Q814yes—
599C42C22⋊Q814yes—
600C42C22×Q814yes—
601C42C2×#4114yes—
602C23—22no—
603C23—22no—
604C23—22no—
605C23—22no—
606C23—22no—
607C23—42no—
608C23—42no—
609C23C2×C22⋊Q812yes—
610C23D4212yes—
611C23D4×Q812yes—
612C23C23×D411yescharacteristic
613Q8—22no—
614Q8—22no—
615Q8—22no—
616Q8—164no—
617Q8—164no—
618Q8C2×C4⋊Q814yes—
619D4—216no—
620D4—22no—
621D4—22no—
622D4—22no—
623D4C2×C4⋊Q814yes—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—216no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—24no—
644C2×C4—48no—
645C2×C4—416no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—164no—
661C2×C4—164no—
662C2×C4—162no—
663C2×C4—162no—
664C2×C4—162no—
665C2×C4—162no—
666C2×C4C2×C22⋊Q814yes—
667C2×C4C2×C22⋊Q814yes—
668C2×C4C2×C22⋊Q812yes—
669C2×C4C2×C4⋊Q814yes—
670C2×C4#54414yes—
671C2×C4D4212yes—
672C2×C4#54714yes—
673C2×C4D4×Q812yes—
674C2×C4#55414yes—
675C2×C4C23×D411yescharacteristic
676C2×C4C23×Q811yescharacteristic
677C2×C4C23×Q811yescharacteristic
678C2×C4C22×#4111yescharacteristic
679C2×C4C2×H(5,𝔽2)11yescharacteristic
680C2×C4C2×2−1+411yescharacteristic
681C8—22no—
682C8—22no—
683C8—24no—
684C8—42no—
685C8—42no—
686C22—24no—
687C22—22no—
688C22—22no—
689C22—24no—
690C22—24no—
691C22—44no—
692C22—42no—
693C22—42no—
694C22—44no—
695C22#244014yes—
696C22C22×C22⋊Q811yescharacteristic
697C22C2×D4211yescharacteristic
698C22C2×D4×Q811yescharacteristic
699C22#358512yes—
700C22#358612yes—
701C4—24no—
702C4—24no—
703C4—24no—
704C4—44no—
705C4—42no—
706C4—42no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—42no—
711C4—44no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4#244014yes—
717C4C22×C4⋊Q811yescharacteristic
718C4C2×#54411yescharacteristic
719C4C2×#54711yescharacteristic
720C4C2×#55411yescharacteristic
721C2—24no—
722C2—44no—
723C2C2×#244011yescharacteristic
724C2#6049212yes—
725C2#6108211yescharacteristic
726C2#6108311yescharacteristic
727C2#6138012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #35330C2semidirect
5. #26055C2semidirect
6. #26054C2semidirect
7. #26053C2semidirect
8. #26052C2semidirect
9. #23668C2semidirect
10. #23667C2semidirect
11. #23556C2semidirect
12. #23555C2semidirect
13. #21171C2semidirect
14. #21144C2semidirect
17. C2×#1847C2semidirect
127. C4×#41Q8semidirect
190. #416Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30842
C2×#30832
#353302
#341832
#260552
#260542
#260532
#260522
#236682
#236672
#235562
#235552
#211712
#211442
#208822
#208812
C2×#18472
#206492
#204112