Finite Group Databasebeta

← #6439191

Subgroups — #6439191

Order 512.

2252 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3084C211yesmaximal, characteristic, complemented
2C22×#428C211yesmaximal, characteristic, complemented
3#35224C211yesmaximal, characteristic, complemented
4#34201C211yesmaximal, characteristic
5#26367C214yesmaximal, complemented
6#26361C214yesmaximal, complemented
7#26359C214yesmaximal, complemented
8#26353C214yesmaximal, complemented
9#23737C211yesmaximal, characteristic, complemented
10#23667C211yesmaximal, characteristic, complemented
11#23611C211yesmaximal, characteristic, complemented
12#23555C211yesmaximal, characteristic, complemented
13#21174C211yesmaximal, characteristic, complemented
14#21038C211yesmaximal, characteristic, complemented
15#20970C211yesmaximal, characteristic
16#20909C211yesmaximal, characteristic
17#20882C211yesmaximal, characteristic, complemented
18#20845C211yesmaximal, characteristic, complemented
19#20835C211yesmaximal, characteristic, complemented
20C23×M16C2211yescharacteristic
21C2×#544C2211yescharacteristic
22C2×#525C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C22×#117C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26#3084—28no—
27C2×#479C2212yescomplemented
28C2×#475C2212yescomplemented
29C2×#478C2212yescomplemented
30C2×#476C2212yescomplemented
31C2×#428—28no—
32C2×#428C2211yescharacteristic
33C2×#428C2214yes—
34C2×#428C2214yes—
35C2×#428C2211yescharacteristic
36C22×#107C2211yescharacteristic
37C22×#107C2211yescharacteristic
38C22×#106C2211yescharacteristic
39C22×#106C2211yescharacteristic
40C2×#424C2211yescharacteristic
41C2×#424C2211yescharacteristic
42C2×#417C2211yescharacteristic
43C2×#417C2211yescharacteristic
44C2×#416C2211yescharacteristic
45C2×#416C2211yescharacteristic
46C2×#407C2211yescharacteristic
47#2848—22no—
48#2627—22no—
49#2442—22no—
50#2442C2214yes—
51#2440—22no—
52#2440C2214yes—
53#2363C2214yes—
54#2362C2214yes—
55#2354—22no—
56#2352—22no—
57#2321—22no—
58#2317C2211yescharacteristic
59#2317C2211yescharacteristic
60#2317C2211yescharacteristic
61#2316C2211yescharacteristic
62#2311C2211yescharacteristic
63C4×#123—22no—
64#2280C2211yescharacteristic
65#2280C2211yescharacteristic
66#2274C2211yescharacteristic
67#2097C2212yescomplemented
68#2095C2212yescomplemented
69#2093C2212yescomplemented
70#2092C2212yescomplemented
71#2087C2214yescomplemented
72#2086C2214yescomplemented
73#2068C2214yescomplemented
74#2067C2214yescomplemented
75#2059C2214yescomplemented
76#2057C2214yescomplemented
77#1949C2214yescomplemented
78#1948C2214yescomplemented
79#1920C2214yescomplemented
80#1919C2214yescomplemented
81#1895C2212yes—
82#1895C2212yes—
83#1894C2212yes—
84#1894C2212yes—
85#1874C2214yes—
86#1872C2214yes—
87#1865C2214yescomplemented
88#1864C2214yescomplemented
89#1854C2214yes—
90#1854C2214yes—
91#1851C2214yes—
92#1851C2214yes—
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1736C2211yescharacteristic
97#1735C2211yescharacteristic
98#1735C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C24×C4—21no—
103C24×C4C2311yescharacteristic
104C22×M16—24no—
105C22×M16—24no—
106C22×M16—24no—
107C22×M16C2312yes—
108C22×M16C2312yes—
109C22×M16C2311yescharacteristic
110C22×M16C2311yescharacteristic
111C23×C8C2311yescharacteristic
112C23×C8C2311yescharacteristic
113#544—24no—
114#544—42no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8C2312yes—
117C2×#125—21no—
118C2×#125C2312yes—
119#525—24no—
120#525Q818yescomplemented
121C2×#123—21no—
122C2×C22⋊Q8—22no—
123C2×C22⋊Q8—24no—
124C2×C22⋊Q8C2312yes—
125C2×#121C2312yes—
126C2×#120—21no—
127#520—24no—
128#520—42no—
129C2×C4×Q8C2312yes—
130C2×C4×D4—22no—
131C2×C4×D4—22no—
132C2×C4×D4C2312yes—
133C2×#117—22no—
134C2×#117—21no—
135C2×#117—24no—
136C2×#117—42no—
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2314yes—
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C22×C4⋊C4—24no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—22no—
146C22×C4⋊C4—21no—
147C22×C4⋊C4—42no—
148C22×C4⋊C4C2311yescharacteristic
149C22×C4⋊C4C2311yescharacteristic
150C22×C4⋊C4C2314yes—
151C22×C4⋊C4C2311yescharacteristic
152C22×C4⋊C4C2311yescharacteristic
153C22×C22⋊C4—21no—
154C22×C42—21no—
155#479—28no—
156#478—28no—
157#476—28no—
158#475—28no—
159#428—24no—
160#428—216no—
161#428—216no—
162#428—42no—
163C2×#107—24no—
164C2×#107—24no—
165C2×#107C2312yes—
166C2×#107C2314yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#106—24no—
171C2×#106D418yescomplemented
172C2×#106C2312yes—
173C2×#106C2314yes—
174C2×#106C2312yes—
175C2×#106C2312yes—
176C2×#106C2312yes—
177#424—24no—
178#424—24no—
179C2×C4⋊C8C2312yes—
180C2×C4⋊C8C2312yes—
181C2×C4⋊C8C2314yes—
182#417—24no—
183#417—42no—
184#416—24no—
185#416—24no—
186C2×#103C2311yescharacteristic
187C2×#103C2312yes—
188C2×#103C2311yescharacteristic
189C2×#103C2312yes—
190C2×#103C2312yes—
191C2×#102C2311yescharacteristic
192C2×#102C2312yes—
193C2×#102C2312yes—
194C2×#102C2311yescharacteristic
195C2×#102C2312yes—
196#407—24no—
197C2×C22⋊C8C2314yes—
198#400—24no—
199#398—24no—
200#398—24no—
201#395—24no—
202#391C2312yes—
203#390—24no—
204#390C2312yes—
205#389—22no—
206#389—22no—
207#389C2314yes—
208#389C2314yes—
209#389C2314yes—
210#388—24no—
211#388—24no—
212#388—24no—
213#387—24no—
214#386—22no—
215#386—24no—
216#385—22no—
217#385—22no—
218#385—22no—
219#385—24no—
220#385—24no—
221#384—21no—
222#384—21no—
223#384C2312yes—
224#382—21no—
225#382—21no—
226#382C2312yes—
227#381—21no—
228#381D412yes—
229#380—22no—
230#380—22no—
231#380—24no—
232#380D412yes—
233#380D412yes—
234#380C2312yes—
235#380C2312yes—
236#379—21no—
237#379D412yes—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—24no—
240C4×C4⋊C4C2314yes—
241C4×C22⋊C4—22no—
242C4×C22⋊C4—22no—
243C4×C22⋊C4—24no—
244C4×C22⋊C4C2314yes—
245#340C2312yes—
246#340C2314yes—
247#340C2312yes—
248#340C2314yes—
249#340C2312yes—
250#340C2314yes—
251#340C2312yes—
252#340C2314yes—
253#336C2314yes—
254#336C2312yes—
255#336C2312yes—
256C25C2×D411yescharacteristic
257C2×#41—22no—
258C2×#41—22no—
259C2×#41—22no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41C2×Q814yes—
263C22×Q8—22no—
264C22×Q8C2411yescharacteristic
265C22×D4—21no—
266C22×D4—22no—
267C22×D4C2411yescharacteristic
268C23×C4—22no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—24no—
276C23×C4—21no—
277C23×C4—21no—
278C23×C4—22no—
279C23×C4—44no—
280C23×C4—84no—
281C23×C4C2×D412yes—
282C23×C4C2×D411yescharacteristic
283C23×C4C2×Q814yes—
284C23×C4#4114yes—
285C23×C4C2412yes—
286C23×C4C2411yescharacteristic
287C2×M16—232no—
288C2×M16—24no—
289C2×M16—24no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—22no—
294C2×M16—22no—
295C2×M16—22no—
296C2×M16—24no—
297C2×M16—24no—
298C2×M16—24no—
299C2×M16—24no—
300C22×C8—22no—
301C22×C8—22no—
302C22×C8—22no—
303C22×C8—22no—
304C22×C8C2×D414yes—
305C22×C8#4114yes—
306C22×C8C2412yes—
307C22×C8C2412yes—
308C22×C8C2411yescharacteristic
309C22×C8C2411yescharacteristic
310C22×C8C2412yes—
311C4⋊Q8—24no—
312C4⋊Q8—42no—
313C4⋊Q8—44no—
314#125—24no—
315#125—42no—
316#125—44no—
317#123—28no—
318C22⋊Q8—24no—
319C22⋊Q8—48no—
320C22⋊Q8—88no—
321C22⋊Q8C2×Q818yes—
322#121—24no—
323#121C2×Q818yes—
324#120—28no—
325C4×Q8—24no—
326C4×Q8—42no—
327C4×D4—216no—
328C4×D4—24no—
329C4×D4—42no—
330C4×D4—48no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—22no—
339#117—22no—
340#117—28no—
341#117—44no—
342#117—44no—
343#117—48no—
344#117C2×Q814yes—
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—28no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—24no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—24no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—44no—
367C2×C4⋊C4—44no—
368C2×C4⋊C4—48no—
369C2×C4⋊C4—44no—
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2×D411yescharacteristic
372C2×C4⋊C4C2×D412yes—
373C2×C4⋊C4C2×D414yes—
374C2×C4⋊C4C2×D414yes—
375C2×C4⋊C4C2412yes—
376C2×C4⋊C4C2412yes—
377C2×C4⋊C4C2411yescharacteristic
378C2×C4⋊C4C2411yescharacteristic
379C2×C4⋊C4C2412yes—
380C2×C4⋊C4C2412yes—
381C2×C4⋊C4C2412yes—
382C2×C4⋊C4C2412yes—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4—22no—
386C2×C22⋊C4—24no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—22no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—22no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—44no—
394C2×C22⋊C4C2×D412yes—
395C2×C22⋊C4C2×D412yes—
396C2×C22⋊C4C2×D414yes—
397C2×C22⋊C4C2×D414yes—
398C2×C42—21no—
399C2×C42—24no—
400C2×C42—22no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—24no—
404C2×C42—44no—
405C2×C42C2412yes—
406C2×C42C2412yes—
407#107—24no—
408#107—24no—
409#107—28no—
410#107—24no—
411#107—42no—
412#106—28no—
413#106—24no—
414#106—42no—
415#106C2×D418yes—
416#106C2×D418yes—
417C4⋊C8—24no—
418C4⋊C8—24no—
419C4⋊C8—28no—
420#103—22no—
421#103—22no—
422#103—24no—
423#103—24no—
424#103—42no—
425#102—22no—
426#102—22no—
427#102—24no—
428#102—24no—
429#102—42no—
430C22⋊C8—28no—
431#95—24no—
432#95—22no—
433#95—22no—
434#95—24no—
435#95—24no—
436#95—24no—
437#95—24no—
438#95—24no—
439#95C2×D414yes—
440#95C2×D414yes—
441C24—22no—
442C24—22no—
443C24—21no—
444C24—42no—
445C24—42no—
446C24C22⋊Q814yes—
447C24C22×D412yes—
448C24C22×D411yescharacteristic
449#41—28no—
450#41—28no—
451#41—44no—
452#41—44no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—21no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—44no—
459C2×Q8—84no—
460C2×Q8C22×Q812yes—
461C2×D4—24no—
462C2×D4—28no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—82no—
471C2×D4—82no—
472C2×D4C22×Q812yes—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—28no—
476C22×C4—28no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—28no—
480C22×C4—28no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—22no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—21no—
497C22×C4—44no—
498C22×C4—48no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—48no—
502C22×C4—42no—
503C22×C4—42no—
504C22×C4—84no—
505C22×C4—82no—
506C22×C4—82no—
507C22×C4—82no—
508C22×C4—84no—
509C22×C4—82no—
510C22×C4—816no—
511C22×C4C22⋊Q818yes—
512C22×C4C22⋊Q814yes—
513C22×C4C22×D414yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D412yes—
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D412yes—
519C22×C4C22×Q814yes—
520C22×C4C22×Q812yes—
521C22×C4C2×#4114yes—
522C22×C4C2×#4112yes—
523C22×C4C2×#4112yes—
524C22×C4H(5,𝔽2)12yes—
525C22×C42−1+412yes—
526C22×C4C2511yescharacteristic
527M16—216no—
528M16—216no—
529M16—44no—
530M16—44no—
531C2×C8—28no—
532C2×C8—28no—
533C2×C8—22no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—22no—
539C2×C8—22no—
540C2×C8—22no—
541C2×C8—22no—
542C2×C8—22no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8C22×D412yes—
546C2×C8C22×D412yes—
547C2×C8C2×#4114yes—
548C4⋊C4—28no—
549C4⋊C4—24no—
550C4⋊C4—24no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—24no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—22no—
560C4⋊C4—22no—
561C4⋊C4—22no—
562C4⋊C4—22no—
563C4⋊C4—21no—
564C4⋊C4—22no—
565C4⋊C4—21no—
566C4⋊C4—22no—
567C4⋊C4—48no—
568C4⋊C4—44no—
569C4⋊C4—44no—
570C4⋊C4—44no—
571C4⋊C4—44no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4—84no—
575C4⋊C4—84no—
576C4⋊C4—84no—
577C4⋊C4—84no—
578C4⋊C4C22×D414yes—
579C4⋊C4C22×D414yes—
580C4⋊C4C22×Q814yes—
581C22⋊C4—24no—
582C22⋊C4—28no—
583C22⋊C4—24no—
584C22⋊C4—24no—
585C22⋊C4—28no—
586C22⋊C4—44no—
587C22⋊C4—44no—
588C22⋊C4—44no—
589C22⋊C4—48no—
590C22⋊C4—44no—
591C22⋊C4—44no—
592C22⋊C4—42no—
593C22⋊C4—42no—
594C22⋊C4—84no—
595C42—24no—
596C42—22no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—48no—
601C42—44no—
602C42—42no—
603C42—42no—
604C42—84no—
605C23—22no—
606C23—22no—
607C23—22no—
608C23—22no—
609C23—22no—
610C23—22no—
611C23—22no—
612C23—22no—
613C23—28no—
614C23—22no—
615C23—48no—
616C23—44no—
617C23—48no—
618C23—44no—
619C23—42no—
620C23—42no—
621C23—41no—
622C23—41no—
623C23C2×C22⋊Q814yes—
624C23C2×C22⋊Q812yes—
625C23D4212yes—
626C23#54712yes—
627C23C23×D411yescharacteristic
628Q8—22no—
629Q8—22no—
630Q8—41no—
631Q8—41no—
632Q8—164no—
633D4—22no—
634D4—22no—
635D4—48no—
636D4—41no—
637D4—41no—
638D4—88no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—28no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—216no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—48no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4—816no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—164no—
682C2×C4C2×C22⋊Q814yes—
683C2×C4C2×C22⋊Q812yes—
684C2×C4#52514yes—
685C2×C4#53914yes—
686C2×C4D4212yes—
687C2×C4#54712yes—
688C2×C4D4×Q814yes—
689C2×C4#55414yes—
690C2×C4C23×D411yescharacteristic
691C2×C4C23×Q811yescharacteristic
692C2×C4C22×#4111yescharacteristic
693C2×C4C22×#4111yescharacteristic
694C2×C4C2×H(5,𝔽2)11yescharacteristic
695C2×C4C2×2−1+411yescharacteristic
696C8—22no—
697C8—22no—
698C8—24no—
699C8—42no—
700C8—42no—
701C22—24no—
702C22—24no—
703C22—22no—
704C22—22no—
705C22—24no—
706C22—24no—
707C22—24no—
708C22—416no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—44no—
714C22—42no—
715C22—82no—
716C22—81no—
717C22—81no—
718C22#244914yes—
719C22C22×C22⋊Q811yescharacteristic
720C22C2×D4211yescharacteristic
721C22C2×#54711yescharacteristic
722C22#358512yes—
723C22#358612yes—
724C4—24no—
725C4—44no—
726C4—44no—
727C4—42no—
728C4—42no—
729C4—44no—
730C4—42no—
731C4—44no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—81no—
736C4—81no—
737C4#244914yes—
738C4C2×#52511yescharacteristic
739C4C2×#53911yescharacteristic
740C4C2×D4×Q811yescharacteristic
741C4C2×#55411yescharacteristic
742C2—24no—
743C2—44no—
744C2—82no—
745C2C2×#244911yescharacteristic
746C2#6049312yes—
747C2#6105911yescharacteristic
748C2#6106011yescharacteristic
749C2#6126512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3084C2semidirect
2. C22×#428C2semidirect
3. #35224C2semidirect
5. #26367C2semidirect
6. #26361C2semidirect
7. #26359C2semidirect
8. #26353C2semidirect
9. #23737C2semidirect
10. #23667C2semidirect
11. #23611C2semidirect
12. #23555C2semidirect
13. #21174C2semidirect
14. #21038C2semidirect
17. #20882C2semidirect
18. #20845C2semidirect
19. #20835C2semidirect
27. C2×#479C22semidirect
28. C2×#475C22semidirect
29. C2×#478C22semidirect
30. C2×#476C22semidirect
67. #2097C22semidirect
68. #2095C22semidirect
69. #2093C22semidirect
70. #2092C22semidirect
71. #2087C22semidirect
72. #2086C22semidirect
73. #2068C22semidirect
74. #2067C22semidirect
75. #2059C22semidirect
76. #2057C22semidirect
77. #1949C22semidirect
78. #1948C22semidirect
79. #1920C22semidirect
80. #1919C22semidirect
87. #1865C22semidirect
88. #1864C22semidirect
120. #525Q8semidirect
171. C2×#106D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30842
C22×#4282
#352242
#342012
#263672
#263612
#263592
#263532
#237372
#236672
#236112
#235552
#211742
#210382
#209702
#209092
#208822
#208452
#208352