Finite Group Databasebeta

← #6443404

Subgroups — #6443404

Order 512.

2150 conjugacy classes of subgroups.

The subgroup lattice has 697 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3081C211yesmaximal, characteristic
2C22×#478C211yesmaximal, characteristic
3#35640C211yesmaximal, characteristic, complemented
4#34248C211yesmaximal, characteristic
5#26059C214yesmaximal, complemented
6#26053C214yesmaximal, complemented
7#26049C214yesmaximal, complemented
8#26041C214yesmaximal, complemented
9#23667C211yesmaximal, characteristic, complemented
10#23666C211yesmaximal, characteristic, complemented
11#23556C211yesmaximal, characteristic, complemented
12#23553C211yesmaximal, characteristic, complemented
13#21174C211yesmaximal, characteristic, complemented
14#21147C211yesmaximal, characteristic, complemented
15#20876C211yesmaximal, characteristic
16#20873C211yesmaximal, characteristic
17#20803C211yesmaximal, characteristic, complemented
18#20650C211yesmaximal, characteristic
19#20416C211yesmaximal, characteristic
20C2×#544C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×#518C2211yescharacteristic
23C2×#518C2211yescharacteristic
24C22×C4×D4C2211yescharacteristic
25#3081—28no—
26C2×#478—28no—
27C2×#478C2214yes—
28C2×#478C2214yes—
29C2×#478C2212yes—
30C2×#478C2212yes—
31C2×#476C2212yes—
32C2×#474C2212yes—
33C2×#428C2211yescharacteristic
34C2×#428C2211yescharacteristic
35C2×#428C2211yescharacteristic
36C2×#427C2211yescharacteristic
37C22×#107C2211yescharacteristic
38C22×#106C2211yescharacteristic
39C2×#424C2211yescharacteristic
40C2×#424C2211yescharacteristic
41C22×C4⋊C8C2211yescharacteristic
42C2×#418C2211yescharacteristic
43C2×#418C2211yescharacteristic
44C22×#102C2211yescharacteristic
45C22×#102C2211yescharacteristic
46C2×#405C2211yescharacteristic
47#2583—22no—
48#2552—22no—
49#2552C2214yescomplemented
50#2550—22no—
51#2550C2214yescomplemented
52#2502—22no—
53#2349—22no—
54#2349C2214yes—
55D4×C4⋊C4—22no—
56D4×C4⋊C4C2214yes—
57#2317C2211yescharacteristic
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2317C2211yescharacteristic
61#2315—22no—
62#2296C2211yescharacteristic
63#2296C2211yescharacteristic
64#2296C2211yescharacteristic
65C2×#385—22no—
66#2274C2211yescharacteristic
67#2088C2212yescomplemented
68#2088C2212yescomplemented
69#2087C2212yescomplemented
70#2087C2214yescomplemented
71#2086C2212yescomplemented
72#2086C2214yescomplemented
73#2059C2214yescomplemented
74#2059C2214yescomplemented
75#2059C2214yescomplemented
76#2058C2214yescomplemented
77#1949C2214yescomplemented
78#1948C2214yescomplemented
79#1940C2214yescomplemented
80#1939C2214yescomplemented
81#1864C2214yes—
82#1864C2214yes—
83#1862C2214yes—
84#1862C2214yes—
85#1847C2212yescomplemented
86#1847C2212yescomplemented
87#1846C2212yescomplemented
88#1845C2212yescomplemented
89#1804C2214yes—
90#1802C2214yes—
91#1762C2214yes—
92C4×#102C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1728C2211yescharacteristic
101C23×D4C2311yescharacteristic
102C24×C4—21no—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C23×C8C2311yescharacteristic
106C23×C8C2311yescharacteristic
107#544—24no—
108#544—42no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C2312yes—
111C2×#125—21no—
112C2×#125—24no—
113C2×#125—22no—
114C2×#125—42no—
115C2×#125C2314yes—
116C2×#125C2312yes—
117C2×C22⋊Q8—22no—
118#518—24no—
119#518—24no—
120#518—42no—
121#518Q818yescomplemented
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—24no—
126C2×C4×D4—22no—
127C2×C4×D4—42no—
128C2×C4×D4C2314yes—
129C2×C4×D4C2312yes—
130C2×C4×D4C2312yes—
131C2×C4×D4C2314yes—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—21no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4—24no—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2311yescharacteristic
144C22×C22⋊C4—22no—
145C22×C22⋊C4—21no—
146C22×C22⋊C4—22no—
147C22×C42C2311yescharacteristic
148#478—216no—
149#478—216no—
150#478—28no—
151#478—28no—
152#476—28no—
153#474—28no—
154#428—24no—
155#428—42no—
156#428—42no—
157#427—24no—
158C2×#107—24no—
159C2×#107C2312yes—
160C2×#107C2311yescharacteristic
161C2×#107C2312yes—
162C2×#107C2312yes—
163C2×#107C2311yescharacteristic
164C2×#107C2312yes—
165C2×#107C2312yes—
166C2×#106—24no—
167C2×#106C2314yes—
168C2×#106C2311yescharacteristic
169C2×#106C2312yes—
170C2×#106C2311yescharacteristic
171C2×#106C2312yes—
172C2×#106C2312yes—
173#424—24no—
174#424—24no—
175C2×C4⋊C8—24no—
176C2×C4⋊C8C2312yes—
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2312yes—
179C2×C4⋊C8C2312yes—
180C2×C4⋊C8C2314yes—
181#418—24no—
182#418Q818yescomplemented
183C2×#102—24no—
184C2×#102D418yes—
185C2×#102C2312yes—
186C2×#102C2314yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2314yes—
191#405—24no—
192C2×#97C2312yes—
193C2×C4×C8C2312yes—
194#398—24no—
195#398—24no—
196#398—24no—
197#395—24no—
198#391—24no—
199#389—22no—
200#389—22no—
201#389—24no—
202#389—24no—
203#389D412yes—
204#389D412yes—
205#389C2314yes—
206#389C2312yes—
207#389C2312yes—
208#389C2314yes—
209#389C2312yes—
210#389C2312yes—
211#385—21no—
212#385—21no—
213#385—21no—
214#385—21no—
215#385—24no—
216#385—24no—
217#385—22no—
218#385—22no—
219#385—22no—
220#385—22no—
221#385—24no—
222#385—24no—
223#385D412yes—
224#385D412yes—
225#384—21no—
226#384—21no—
227#384—24no—
228#384C2312yes—
229#384C2312yes—
230#382—21no—
231#382—21no—
232#382C2312yes—
233#382C2312yes—
234#380—24no—
235#380—24no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4C2314yes—
240C4×C4⋊C4C2314yes—
241C4×C22⋊C4—22no—
242C2×#95—24no—
243#340C2312yes—
244#340C2314yes—
245#340C2312yes—
246#340C2314yes—
247#340C2312yes—
248#340C2312yes—
249#340C2314yes—
250#340C2312yes—
251#340C2312yes—
252#336C2314yes—
253#336C2314yes—
254C25—21no—
255C22×Q8—22no—
256C22×D4—21no—
257C22×D4—22no—
258C22×D4—22no—
259C22×D4—22no—
260C22×D4C2×D414yes—
261C22×D4C2×Q814yes—
262C22×D4C2×Q814yes—
263C22×D4C2412yes—
264C23×C4—21no—
265C23×C4—24no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—24no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—44no—
272C23×C4—84no—
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2411yescharacteristic
276C2×M16—22no—
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—22no—
280C2×M16—22no—
281C2×M16C2×Q814yes—
282C22×C8—22no—
283C22×C8—22no—
284C22×C8—22no—
285C22×C8—22no—
286C22×C8—22no—
287C22×C8C2×D414yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C22×C8C2411yescharacteristic
291C22×C8C2411yescharacteristic
292C22×C8C2412yes—
293C4⋊Q8—24no—
294C4⋊Q8—42no—
295C4⋊Q8—44no—
296#125—28no—
297#125—24no—
298#125—44no—
299#125—42no—
300#125—48no—
301#125—44no—
302C22⋊Q8—48no—
303C4×D4—216no—
304C4×D4—28no—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—28no—
308C4×D4—44no—
309C4×D4—42no—
310C4×D4—42no—
311C4×D4—48no—
312C4×D4—44no—
313C4×D4—44no—
314C4×D4C2×Q8116yes—
315#117—22no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117C2×Q814yes—
324#117C2×Q814yes—
325#117C2×Q814yes—
326#117#4114yes—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—44no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—88no—
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D414yes—
355C2×C4⋊C4C2×D414yes—
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2411yescharacteristic
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—24no—
369C2×C22⋊C4—216no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—24no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—48no—
376C2×C22⋊C4—44no—
377C2×C22⋊C4C2×D412yes—
378C2×C22⋊C4C2×D412yes—
379C2×C42—21no—
380C2×C42—21no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—21no—
384C2×C42—21no—
385C2×C42—24no—
386C2×C42C2×D412yes—
387C2×C42C2412yes—
388C2×C42C2412yes—
389C2×C42C2412yes—
390#107—24no—
391#107—24no—
392#107—22no—
393#107—22no—
394#107—24no—
395#107—42no—
396#107—42no—
397#106—22no—
398#106—22no—
399#106—28no—
400#106—24no—
401#106—42no—
402#106—42no—
403C4⋊C8—24no—
404C4⋊C8—24no—
405C4⋊C8—24no—
406C4⋊C8—24no—
407C4⋊C8—28no—
408#102—28no—
409#102—24no—
410#102—24no—
411#102C2×D418yes—
412#102C2×D418yes—
413#102C2×Q8116yes—
414#97—24no—
415C4×C8—24no—
416#95—22no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—24no—
424#95—24no—
425#95C2×D414yes—
426#95C2×D414yes—
427C24—22no—
428C24—24no—
429C24—24no—
430C24—24no—
431C24—82no—
432C24—82no—
433C24C22×D411yescharacteristic
434C2×Q8—44no—
435C2×Q8—84no—
436C2×D4—24no—
437C2×D4—28no—
438C2×D4—24no—
439C2×D4—28no—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4C4⋊Q8116yes—
446C2×D4C22×D414yes—
447C2×D4C22×Q814yes—
448C2×D4C22×Q814yes—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—216no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—84no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—816no—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D414yes—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×Q812yes—
487C22×C4C22×Q812yes—
488C22×C4C2×#4112yes—
489C22×C4H(5,𝔽2)12yes—
490C22×C42−1+412yes—
491C22×C4C2511yescharacteristic
492M16—28no—
493M16—44no—
494C2×C8—28no—
495C2×C8—21no—
496C2×C8—22no—
497C2×C8—21no—
498C2×C8—22no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—44no—
508C2×C8C22×D412yes—
509C2×C8C22×D412yes—
510C2×C8C22×Q814yes—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—84no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—84no—
543C4⋊C4—82no—
544C4⋊C4—84no—
545C4⋊C4—84no—
546C4⋊C4C22⋊Q818yes—
547C4⋊C4C22×D414yes—
548C4⋊C4C22×Q814yes—
549C4⋊C4C22×Q814yes—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—24no—
553C22⋊C4—216no—
554C22⋊C4—48no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4C22⋊Q818yes—
560C42—24no—
561C42—24no—
562C42—24no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—42no—
568C42—42no—
569C42—84no—
570C42C22×Q814yes—
571C42C2×#4114yes—
572C23—22no—
573C23—22no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—216no—
578C23—44no—
579C23—44no—
580C23—84no—
581C23—88no—
582C23—84no—
583C23—88no—
584C23C2×C22⋊Q812yes—
585C23D4212yes—
586C23D4×Q812yes—
587C23C23×D411yescharacteristic
588Q8—164no—
589D4—24no—
590D4—216no—
591D4—24no—
592D4—24no—
593D4C2×C4⋊Q818yes—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—816no—
629C2×C4—84no—
630C2×C4—816no—
631C2×C4—84no—
632C2×C4—84no—
633C2×C4—164no—
634C2×C4—164no—
635C2×C4C2×C22⋊Q818yes—
636C2×C4C2×C22⋊Q812yes—
637C2×C4C2×C4⋊Q814yes—
638C2×C4#54414yes—
639C2×C4D4212yes—
640C2×C4#54714yes—
641C2×C4D4×Q812yes—
642C2×C4#55414yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C23×Q811yescharacteristic
645C2×C4C23×Q811yescharacteristic
646C2×C4C22×#4111yescharacteristic
647C2×C4C2×H(5,𝔽2)11yescharacteristic
648C2×C4C2×2−1+411yescharacteristic
649C8—22no—
650C8—22no—
651C8—24no—
652C8—42no—
653C8—42no—
654C22—24no—
655C22—22no—
656C22—22no—
657C22—24no—
658C22—28no—
659C22—48no—
660C22—44no—
661C22—44no—
662C22—48no—
663C22—816no—
664C22#244014yes—
665C22C22×C22⋊Q811yescharacteristic
666C22C2×D4211yescharacteristic
667C22C2×D4×Q811yescharacteristic
668C22#358412yes—
669C22#358512yes—
670C4—28no—
671C4—44no—
672C4—42no—
673C4—42no—
674C4—44no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—81no—
679C4—81no—
680C4—81no—
681C4—82no—
682C4—82no—
683C4—81no—
684C4#244014yes—
685C4C22×C4⋊Q811yescharacteristic
686C4C2×#54411yescharacteristic
687C4C2×#54711yescharacteristic
688C4C2×#55411yescharacteristic
689C2—24no—
690C2—48no—
691C2C2×#244011yescharacteristic
692C2#6049612yes—
693C2#6105111yescharacteristic
694C2#6105911yescharacteristic
695C2#6138512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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
3. #35640C2semidirect
5. #26059C2semidirect
6. #26053C2semidirect
7. #26049C2semidirect
8. #26041C2semidirect
9. #23667C2semidirect
10. #23666C2semidirect
11. #23556C2semidirect
12. #23553C2semidirect
13. #21174C2semidirect
14. #21147C2semidirect
17. #20803C2semidirect
49. #2552C22semidirect
51. #2550C22semidirect
67. #2088C22semidirect
68. #2088C22semidirect
69. #2087C22semidirect
70. #2087C22semidirect
71. #2086C22semidirect
72. #2086C22semidirect
73. #2059C22semidirect
74. #2059C22semidirect
75. #2059C22semidirect
76. #2058C22semidirect
77. #1949C22semidirect
78. #1948C22semidirect
79. #1940C22semidirect
80. #1939C22semidirect
85. #1847C22semidirect
86. #1847C22semidirect
87. #1846C22semidirect
88. #1845C22semidirect
121. #518Q8semidirect
182. #418Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30812
C22×#4782
#356402
#342482
#260592
#260532
#260492
#260412
#236672
#236662
#235562
#235532
#211742
#211472
#208762
#208732
#208032
#206502
#204162