Finite Group Databasebeta

← #6583264

Subgroups — #6583264

Order 512.

1932 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3384C211yesmaximal, characteristic
2C2×#3380C211yesmaximal, characteristic
3#36276C211yesmaximal, characteristic, complemented
4#36180C211yesmaximal, characteristic, complemented
5#33314C211yesmaximal, characteristic, complemented
6#33201C211yesmaximal, characteristic, complemented
7#33197C211yesmaximal, characteristic, complemented
8#28822C214yesmaximal, complemented
9#28682C211yesmaximal, characteristic, complemented
10#28677C211yesmaximal, characteristic, complemented
11#26059C214yesmaximal, complemented
12#26053C214yesmaximal, complemented
13#25839C211yesmaximal, characteristic, complemented
14#25839C211yesmaximal, characteristic, complemented
15#25835C211yesmaximal, characteristic, complemented
16#25834C211yesmaximal, characteristic, complemented
17#22676C214yesmaximal
18#21714C211yesmaximal, characteristic, complemented
19C4×#442C211yesmaximal, characteristic
20C2×#554C2211yescharacteristic
21C2×#554C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3384—28no—
25#3380—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#500C2211yescharacteristic
29C2×#499C2212yes—
30C2×#498C2211yescharacteristic
31C2×#498C2211yescharacteristic
32C2×#478C2212yes—
33C2×#478C2212yes—
34C2×#478C2212yes—
35C2×#476C2211yescharacteristic
36C2×#476C2211yescharacteristic
37C2×#476C2211yescharacteristic
38C2×#476C2212yes—
39C2×#474C2211yescharacteristic
40C2×#446C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×#442C2212yes—
43C2×#442C2212yes—
44C2×C4×D8C2212yes—
45C2×C4×D8C2211yescharacteristic
46#2868C2211yescharacteristic
47#2862C2211yescharacteristic
48#2857—21no—
49#2857—21no—
50#2567C2211yescharacteristic
51#2567C2211yescharacteristic
52#2567C2211yescharacteristic
53#2552C2214yes—
54#2550C2214yes—
55#2549—24no—
56#2546C2211yescharacteristic
57#2502—21no—
58#2502—21no—
59#2502—21no—
60#2440—24no—
61#2434—21no—
62#2349C2214yes—
63D4×C4⋊C4C2214yes—
64C4×C4⋊Q8C2211yescharacteristic
65C2×#391—21no—
66C2×#384—21no—
67C42×D4C2211yescharacteristic
68C42×D4C2211yescharacteristic
69#2088C2214yes—
70#2086C2214yes—
71#2059C2214yes—
72#2059C2214yes—
73#2059C2214yes—
74#2058C2214yes—
75#1988C2211yescharacteristic
76#1987C2211yescharacteristic
77#1987C2211yescharacteristic
78#1987C2211yescharacteristic
79#1984C2211yescharacteristic
80#1984C2211yescharacteristic
81#1949C2214yes—
82#1948C2214yes—
83#1944C2214yes—
84#1941C2214yes—
85#1940C2214yes—
86#1864C2214yes—
87#1864C2214yes—
88#1862C2214yes—
89#1862C2214yes—
90#1847C2214yes—
91#1847C2214yes—
92#1842C2211yescharacteristic
93#1804C2214yes—
94#1802C2214yes—
95#1776C2212yes—
96C4×#106C2211yescharacteristic
97#1771C2214yes—
98#1762C2212yes—
99#1762C2212yes—
100C4×#102C2211yescharacteristic
101C4×#102C2211yescharacteristic
102C4×#97C2211yescharacteristic
103C23×Q8—21no—
104C22×D8C2311yescharacteristic
105#554—24no—
106#554—24no—
107#554—24no—
108#554—42no—
109#554—42no—
110#554—42no—
111D4×Q8—24no—
112D4×Q8—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—22no—
115C2×C4⋊Q8D411yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2312yes—
121C2×#125C2312yes—
122C2×#125C2312yes—
123C2×#125C2312yes—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—21no—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—22no—
129C2×C22⋊Q8—22no—
130C2×C22⋊Q8—22no—
131C2×C22⋊Q8—22no—
132C2×C4×Q8C2311yescharacteristic
133C2×C4×Q8C2311yescharacteristic
134C2×C4×D4C2312yes—
135C2×C4×D4C2312yes—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×C4×D4C2311yescharacteristic
139C2×C4×D4C2311yescharacteristic
140C2×C4×D4C2312yes—
141C2×C4×D4C2312yes—
142C22×C4⋊C4—21no—
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—24no—
146C22×C42—21no—
147C22×C42—21no—
148#501—28no—
149#500—24no—
150#500—42no—
151#499—28no—
152#498—24no—
153#498—42no—
154#478—28no—
155#478—28no—
156#478—28no—
157#476—24no—
158#476—24no—
159#476—24no—
160#476—28no—
161#474—24no—
162#446—24no—
163C8×Q8—24no—
164#442—28no—
165#442Q8116yescomplemented
166C4×D8—28no—
167C4×D8D418yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2314yes—
172C2×#107C2311yescharacteristic
173C2×#106C2312yes—
174C2×#106C2312yes—
175C2×#106C2312yes—
176C2×#106C2312yes—
177C2×#106C2312yes—
178C2×#106C2314yes—
179C2×#106C2312yes—
180C2×C4⋊C8C2312yes—
181C2×C4⋊C8C2312yes—
182C2×C4⋊C8C2314yes—
183C2×C4⋊C8C2311yescharacteristic
184C2×C4⋊C8C2311yescharacteristic
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2311yescharacteristic
190C2×#102C2311yescharacteristic
191C2×#102C2312yes—
192C2×#102C2312yes—
193C2×#97C2312yes—
194C2×#97C2312yes—
195C2×#97C2312yes—
196C2×C4×C8C2311yescharacteristic
197C2×C4×C8C2312yes—
198C2×C4×C8C2311yescharacteristic
199#398—24no—
200#398—22no—
201#398—24no—
202#398—24no—
203#395—22no—
204#395—22no—
205#395—22no—
206#395—24no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—22no—
214#391—24no—
215#391D411yescharacteristic
216#391D411yescharacteristic
217#389—24no—
218#389—24no—
219#389D414yes—
220#389C2314yes—
221#389C2314yes—
222#389C2314yes—
223#389C2314yes—
224#385—24no—
225#385—24no—
226#385—24no—
227#385—24no—
228#384—21no—
229#384—21no—
230#384—21no—
231#384—24no—
232#384—21no—
233#384—21no—
234#384D411yescharacteristic
235#384C2314yes—
236#382C2314yes—
237#380—24no—
238#380—24no—
239C4×C4⋊C4C2311yescharacteristic
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2314yes—
244C4×C22⋊C4—21no—
245C4×C22⋊C4—22no—
246C4×C22⋊C4—21no—
247C4×C22⋊C4—22no—
248C2×#95—24no—
249C43C2311yescharacteristic
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#340C2314yes—
254#340C2312yes—
255#340C2314yes—
256#340C2312yes—
257#336C2314yes—
258#336C2314yes—
259C22×Q8—21no—
260C22×Q8—22no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8—44no—
265C22×Q8—44no—
266C22×Q8—82no—
267C22×Q8C2×D411yescharacteristic
268C22×Q8C2×D411yescharacteristic
269C22×D4C2411yescharacteristic
270C22×D4C2411yescharacteristic
271C23×C4—21no—
272C23×C4—22no—
273C23×C4—21no—
274C23×C4—22no—
275C23×C4—22no—
276C23×C4—22no—
277C23×C4—22no—
278C23×C4—22no—
279C2×D8C2×D414yes—
280C2×D8C2×Q818yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2411yescharacteristic
284C22×C8C2412yes—
285C22×C8C2411yescharacteristic
286C4⋊Q8—24no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—42no—
296C4⋊Q8—42no—
297C4⋊Q8—84no—
298C4⋊Q8—82no—
299#125—24no—
300#125—24no—
301#125—24no—
302#125—42no—
303#125—42no—
304#125—42no—
305C22⋊Q8—48no—
306C22⋊Q8—48no—
307C22⋊Q8—48no—
308C22⋊Q8—48no—
309C22⋊Q8—44no—
310C22⋊Q8—44no—
311C22⋊Q8—44no—
312C22⋊Q8—44no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×D4—24no—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4—24no—
324C4×D4—24no—
325C4×D4—42no—
326C4×D4—42no—
327C4×D4—42no—
328C4×D4—42no—
329C4×D4C2×D414yes—
330C4×D4C2×D414yes—
331C4×D4C2×Q818yes—
332C4×D4C2×Q818yescomplemented
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4—44no—
352C2×C4⋊C4—44no—
353C2×C4⋊C4—88no—
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D412yes—
357C2×C4⋊C4C2×D412yes—
358C2×C4⋊C4C2×D414yes—
359C2×C4⋊C4C2×D414yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2411yescharacteristic
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—22no—
380C2×C42—44no—
381C2×C42—82no—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D412yes—
385C2×C42C2412yes—
386C2×C42C2412yes—
387C2×C42C2411yescharacteristic
388C2×C42C2412yes—
389#107—24no—
390#107—24no—
391#107—24no—
392#107—44no—
393#107C2×D414yes—
394#106—24no—
395#106—24no—
396#106—24no—
397#106—24no—
398#106—24no—
399#106—44no—
400#106C2×Q818yescomplemented
401C4⋊C8—24no—
402C4⋊C8—24no—
403C4⋊C8—22no—
404C4⋊C8—22no—
405C4⋊C8—28no—
406#102—24no—
407#102—24no—
408#102—24no—
409#102—24no—
410#102C2×D414yes—
411#102C2×D414yes—
412#102C2×Q818yes—
413#102C2×Q818yescomplemented
414#97—24no—
415#97—24no—
416#97C2×Q818yescomplemented
417C4×C8—24no—
418C4×C8—22no—
419C4×C8C2×D414yes—
420#95—24no—
421#95—24no—
422#95—24no—
423#95—21no—
424#95—21no—
425#95—22no—
426#95—22no—
427#95—24no—
428#95—24no—
429#95—24no—
430#95C2×D414yes—
431#95C2×D414yes—
432C24—21no—
433C24—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—44no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—82no—
449C2×Q8—82no—
450C2×Q8—82no—
451C2×Q8—82no—
452C2×Q8—816no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4C22×D412yes—
458C2×D4C22×D412yes—
459C2×D4C22×Q814yes—
460C2×D4C22×Q814yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—28no—
468C22×C4—28no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—82no—
485C22×C4—82no—
486C22×C4—82no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4—84no—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C2511yescharacteristic
499D8C4⋊Q8116yescomplemented
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8C22×D412yes—
508C2×C8C22×D412yes—
509C2×C8C22×Q814yes—
510C2×C8C22×Q814yes—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—48no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C4⋊C4—84no—
546C4⋊C4C22×D412yes—
547C4⋊C4C22×D412yes—
548C4⋊C4C22×Q814yes—
549C4⋊C4C22×Q814yes—
550C22⋊C4—24no—
551C22⋊C4—24no—
552C22⋊C4—28no—
553C22⋊C4—28no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—42no—
565C42—44no—
566C42—42no—
567C42—42no—
568C42—82no—
569C42C22×D412yes—
570C42C22×Q814yes—
571C42H(5,𝔽2)12yes—
572C422−1+414yes—
573C23—22no—
574C23—22no—
575C23—24no—
576C23—24no—
577C23—42no—
578C23—42no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23C23×D411yescharacteristic
584Q8—44no—
585Q8—44no—
586Q8—88no—
587Q8—84no—
588Q8—162no—
589Q8—162no—
590Q8—162no—
591D4—22no—
592D4—22no—
593D4—24no—
594D4—24no—
595D4C2×C4⋊Q814yes—
596D4C2×C4⋊Q814yes—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—416no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—48no—
626C2×C4—48no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4—164no—
639C2×C4—164no—
640C2×C4—164no—
641C2×C4D4212yes—
642C2×C4D4212yes—
643C2×C4D4×Q814yes—
644C2×C4D4×Q814yes—
645C2×C4#57512yes—
646C2×C4#57712yes—
647C2×C4C23×D411yescharacteristic
648C2×C4C23×Q812yes—
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×2−1+412yes—
651C8—22no—
652C8—24no—
653C8—42no—
654C8—42no—
655C8C2×C4⋊Q814yes—
656C22—24no—
657C22—24no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22C2×D4211yescharacteristic
667C22C2×D4×Q812yes—
668C22C2×#57511yescharacteristic
669C22C2×#57711yescharacteristic
670C22#358512yes—
671C4—22no—
672C4—24no—
673C4—24no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—44no—
681C4—44no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4#326714yes—
687C4C22×C4⋊Q811yescharacteristic
688C4#349911yescharacteristic
689C4#350312yes—
690C2—44no—
691C2—44no—
692C2D4×C4⋊Q811yescharacteristic
693C2C2×#326711yescharacteristic
694C2#6108211yescharacteristic
695C2#6136712yes—
696C2#6138212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D8C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #36276C2semidirect
4. #36180C2semidirect
5. #33314C2semidirect
6. #33201C2semidirect
7. #33197C2semidirect
8. #28822C2semidirect
9. #28682C2semidirect
10. #28677C2semidirect
11. #26059C2semidirect
12. #26053C2semidirect
13. #25839C2semidirect
14. #25839C2semidirect
15. #25835C2semidirect
16. #25834C2semidirect
18. #21714C2semidirect
165. #442Q8semidirect
332. C4×D4C2×Q8semidirect
400. #106C2×Q8semidirect
413. #102C2×Q8semidirect
416. #97C2×Q8semidirect
499. D8C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#33842
C2×#33802
#362762
#361802
#333142
#332012
#331972
#288222
#286822
#286772
#260592
#260532
#258392
#258352
#258342
#226762
#217142
C4×#4422