Finite Group Databasebeta

← #6583270

Subgroups — #6583270

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3386C211yesmaximal, characteristic
2C2×#3381C211yesmaximal, characteristic
3#36593C211yesmaximal, characteristic, complemented
4#36276C211yesmaximal, characteristic
5#33314C211yesmaximal, characteristic, complemented
6#33202C211yesmaximal, characteristic, complemented
7#33195C211yesmaximal, characteristic
8#28820C214yesmaximal, complemented
9#28684C211yesmaximal, characteristic, complemented
10#28675C211yesmaximal, characteristic, complemented
11#26059C214yesmaximal
12#26055C214yesmaximal, complemented
13#25840C211yesmaximal, characteristic, complemented
14#25838C211yesmaximal, characteristic
15#25836C211yesmaximal, characteristic, complemented
16#25835C211yesmaximal, characteristic
17#22678C214yesmaximal
18#21714C211yesmaximal, characteristic, complemented
19C4×#440C211yesmaximal, characteristic
20C2×#557C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×D4×Q8C2211yescharacteristic
24#3386—28no—
25#3381—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#500C2211yescharacteristic
29C2×#499C2212yes—
30C2×#498C2211yescharacteristic
31C2×#498C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#477C2212yes—
35C2×#475C2211yescharacteristic
36C2×#478C2212yes—
37C2×#478C2212yes—
38C2×#474C2211yescharacteristic
39C2×#474C2211yescharacteristic
40C2×#446C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×#440C2212yes—
43C2×#440C2212yes—
44C2×C4×SD16C2212yes—
45C2×C4×SD16C2211yescharacteristic
46#2868—21no—
47#2868C2211yescharacteristic
48#2862C2211yescharacteristic
49#2857—21no—
50#2567C2211yescharacteristic
51#2552—24no—
52#2552C2214yes—
53#2550C2214yes—
54#2549—24no—
55#2546—21no—
56#2546—21no—
57#2546C2211yescharacteristic
58#2546C2211yescharacteristic
59#2546C2211yescharacteristic
60#2502—21no—
61#2434—21no—
62Q8×C4⋊C4C2214yes—
63#2349C2214yes—
64C4×C4⋊Q8—21no—
65C4×C4⋊Q8C2211yescharacteristic
66C2×#391—21no—
67C42×Q8C2211yescharacteristic
68C42×D4C2211yescharacteristic
69#2088C2214yes—
70#2086C2214yes—
71#2059C2214yes—
72#2059C2214yes—
73#2059C2214yes—
74#2057C2214yes—
75#1988C2211yescharacteristic
76#1988C2211yescharacteristic
77#1988C2211yescharacteristic
78#1987C2211yescharacteristic
79#1984C2211yescharacteristic
80#1984C2211yescharacteristic
81#1949C2214yes—
82#1947C2214yes—
83#1943C2214yes—
84#1942C2214yes—
85#1940C2214yes—
86#1865C2214yes—
87#1864C2214yes—
88#1863C2214yes—
89#1862C2214yes—
90#1847C2214yes—
91#1847C2214yes—
92#1842C2211yescharacteristic
93#1805C2214yes—
94#1802C2214yes—
95#1776C2212yes—
96C4×#107C2211yescharacteristic
97#1771C2214yes—
98#1763C2212yes—
99#1762C2212yes—
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102C4×#97C2211yescharacteristic
103C23×Q8—21no—
104C22×SD16C2311yescharacteristic
105#557—24no—
106#557—24no—
107#557—42no—
108#557—42no—
109#554—24no—
110#554—42no—
111D4×Q8—24no—
112D4×Q8—42no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8D411yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×C4⋊Q8C2311yescharacteristic
125C2×C4⋊Q8C2312yes—
126C2×#125—22no—
127C2×#125—22no—
128C2×#125C2312yes—
129C2×#125C2312yes—
130C2×#125C2312yes—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8—21no—
133C2×C22⋊Q8—22no—
134C2×C22⋊Q8—22no—
135C2×C4×Q8—21no—
136C2×C4×Q8—21no—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2311yescharacteristic
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2311yescharacteristic
142C2×C4×Q8C2311yescharacteristic
143C2×C4×D4C2312yes—
144C2×C4×D4C2312yes—
145C2×C4×D4C2311yescharacteristic
146C2×C4×D4C2312yes—
147C22×C4⋊C4—21no—
148C22×C42—21no—
149#501—28no—
150#500—24no—
151#500—42no—
152#499—28no—
153#498—24no—
154#498—42no—
155#479—28no—
156#478—28no—
157#478—28no—
158#477—24no—
159#477—28no—
160#475—24no—
161#474—24no—
162#474—24no—
163#446—24no—
164C8×Q8—24no—
165#440—28no—
166#440Q8116yescomplemented
167C4×SD16—28no—
168C4×SD16D418yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#107C2312yes—
173C2×#107C2312yes—
174C2×#107C2314yes—
175C2×#107C2312yes—
176C2×#106C2312yes—
177C2×#106C2312yes—
178C2×#106C2312yes—
179C2×#106C2314yes—
180C2×#106C2311yescharacteristic
181C2×C4⋊C8C2312yes—
182C2×C4⋊C8C2312yes—
183C2×C4⋊C8C2311yescharacteristic
184C2×C4⋊C8C2311yescharacteristic
185C2×C4⋊C8C2314yes—
186C2×#103C2312yes—
187C2×#103C2312yes—
188C2×#103C2311yescharacteristic
189C2×#103C2312yes—
190C2×#102C2312yes—
191C2×#102C2312yes—
192C2×#102C2311yescharacteristic
193C2×#102C2312yes—
194C2×#97C2312yes—
195C2×#97C2312yes—
196C2×#97C2312yes—
197C2×C4×C8C2311yescharacteristic
198C2×C4×C8C2312yes—
199C2×C4×C8C2311yescharacteristic
200#398—24no—
201#398—22no—
202#398—22no—
203#398—22no—
204#398—24no—
205#398—24no—
206#395—22no—
207#395—24no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—21no—
213#391—21no—
214#391—22no—
215#391—24no—
216#391D411yescharacteristic
217#391D411yescharacteristic
218#389—24no—
219#389—24no—
220#389—24no—
221#389—24no—
222#389D414yes—
223#389C2314yes—
224#389C2314yes—
225#389C2314yes—
226#389C2314yes—
227#385—24no—
228#385—24no—
229#384—21no—
230#384—24no—
231#384—24no—
232#384—22no—
233#384D411yescharacteristic
234#384C2314yes—
235#382C2314yes—
236#380—24no—
237C4×C4⋊C4—24no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2311yescharacteristic
244C4×C4⋊C4C2314yes—
245C4×C22⋊C4—21no—
246C4×C22⋊C4—22no—
247C2×#95—24no—
248C43—21no—
249C43C2311yescharacteristic
250#340C2312yes—
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#340C2312yes—
255#340C2314yes—
256#340C2312yes—
257#336C2314yes—
258#336C2314yes—
259C22×Q8—21no—
260C22×Q8—21no—
261C22×Q8—21no—
262C22×Q8—22no—
263C22×Q8—21no—
264C22×Q8—44no—
265C22×Q8—44no—
266C22×Q8—82no—
267C22×Q8C2×D411yescharacteristic
268C22×Q8C2×D411yescharacteristic
269C22×Q8C2411yescharacteristic
270C22×D4C2411yescharacteristic
271C23×C4—21no—
272C23×C4—22no—
273C23×C4—22no—
274C23×C4—22no—
275C2×SD16C2×D414yes—
276C2×SD16C2×Q818yes—
277C22×C8C2412yes—
278C22×C8C2412yes—
279C22×C8C2411yescharacteristic
280C22×C8C2412yes—
281C22×C8C2411yescharacteristic
282C4⋊Q8—24no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—42no—
292C4⋊Q8—42no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—44no—
296C4⋊Q8—82no—
297C4⋊Q8—82no—
298C4⋊Q8—82no—
299C4⋊Q8—82no—
300#125—24no—
301#125—24no—
302#125—24no—
303#125—42no—
304#125—42no—
305#125—42no—
306#125—48no—
307#125—48no—
308C22⋊Q8—48no—
309C22⋊Q8—48no—
310C22⋊Q8—44no—
311C22⋊Q8—44no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×Q8—22no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—42no—
321C4×Q8—42no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×Q8C2×D414yes—
325C4×Q8C2×Q818yescomplemented
326C4×D4—24no—
327C4×D4—24no—
328C4×D4—22no—
329C4×D4—24no—
330C4×D4—42no—
331C4×D4—42no—
332C4×D4C2×D414yes—
333C4×D4C2×Q818yes—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—24no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—44no—
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D412yes—
359C2×C4⋊C4C2×D412yes—
360C2×C4⋊C4C2×D414yes—
361C2×C4⋊C4C2×D414yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2411yescharacteristic
367C2×C4⋊C4C2411yescharacteristic
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—21no—
379C2×C42—21no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—82no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D412yes—
386C2×C42C2412yes—
387C2×C42C2412yes—
388C2×C42C2411yescharacteristic
389C2×C42C2412yes—
390#107—24no—
391#107—24no—
392#107—24no—
393#107—24no—
394#107—24no—
395#107—44no—
396#107C2×Q818yescomplemented
397#106—24no—
398#106—24no—
399#106—24no—
400#106—44no—
401#106C2×D414yes—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—22no—
405C4⋊C8—22no—
406C4⋊C8—28no—
407#103—24no—
408#103—24no—
409#103C2×D414yes—
410#103C2×Q818yescomplemented
411#102—24no—
412#102—24no—
413#102C2×D414yes—
414#102C2×Q818yes—
415#97—24no—
416#97—24no—
417#97C2×Q818yescomplemented
418C4×C8—24no—
419C4×C8—22no—
420C4×C8C2×D414yes—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—21no—
425#95—22no—
426#95—21no—
427#95—22no—
428#95—24no—
429#95—24no—
430#95—24no—
431#95C2×D414yes—
432#95C2×D414yes—
433C24—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—82no—
455C2×Q8—82no—
456C2×Q8—816no—
457C2×Q8C22×D412yes—
458C2×Q8C22×Q814yes—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4C22×D412yes—
462C2×D4C22×Q814yes—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—24no—
469C22×C4—28no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—82no—
482C22×C4—84no—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D411yescharacteristic
491C22×C4C2511yescharacteristic
492SD16C4⋊Q8116yescomplemented
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8C22×D412yes—
501C2×C8C22×D412yes—
502C2×C8C22×Q814yes—
503C2×C8C22×Q814yes—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—24no—
509C4⋊C4—28no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—48no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—84no—
539C4⋊C4—84no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4C22×D412yes—
544C4⋊C4C22×D412yes—
545C4⋊C4C22×Q814yes—
546C4⋊C4C22×Q814yes—
547C22⋊C4—24no—
548C22⋊C4—28no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C42—24no—
552C42—28no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—42no—
560C42—44no—
561C42—44no—
562C42—44no—
563C42—42no—
564C42—42no—
565C42—82no—
566C42C22×D412yes—
567C42C22×Q814yes—
568C42H(5,𝔽2)12yes—
569C422−1+414yes—
570C23—22no—
571C23—24no—
572C23—42no—
573C23—41no—
574C23—41no—
575C23C23×D411yescharacteristic
576Q8—22no—
577Q8—24no—
578Q8—44no—
579Q8—44no—
580Q8—88no—
581Q8—84no—
582Q8—84no—
583Q8—84no—
584Q8—162no—
585Q8C2×C4⋊Q814yes—
586D4—22no—
587D4—24no—
588D4C2×C4⋊Q814yes—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—48no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—88no—
630C2×C4—88no—
631C2×C4—164no—
632C2×C4D4212yes—
633C2×C4D4212yes—
634C2×C4D4×Q814yes—
635C2×C4D4×Q814yes—
636C2×C4#57512yes—
637C2×C4#57612yes—
638C2×C4C23×D411yescharacteristic
639C2×C4C23×Q812yes—
640C2×C4C2×H(5,𝔽2)11yescharacteristic
641C2×C4C2×2−1+412yes—
642C8—22no—
643C8—24no—
644C8—42no—
645C8—42no—
646C8C2×C4⋊Q814yes—
647C22—24no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—44no—
652C22C2×D4211yescharacteristic
653C22C2×D4×Q812yes—
654C22C2×#57511yescharacteristic
655C22C2×#57611yescharacteristic
656C22#358612yes—
657C4—24no—
658C4—22no—
659C4—24no—
660C4—24no—
661C4—42no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—44no—
668C4—44no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4#326814yes—
674C4C22×C4⋊Q811yescharacteristic
675C4#349911yescharacteristic
676C4#350312yes—
677C2—44no—
678C2D4×C4⋊Q811yescharacteristic
679C2C2×#326811yescharacteristic
680C2#6108311yescharacteristic
681C2#6136912yes—
682C2#6138112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
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. #36593C2semidirect
5. #33314C2semidirect
6. #33202C2semidirect
8. #28820C2semidirect
9. #28684C2semidirect
10. #28675C2semidirect
12. #26055C2semidirect
13. #25840C2semidirect
15. #25836C2semidirect
18. #21714C2semidirect
166. #440Q8semidirect
325. C4×Q8C2×Q8semidirect
396. #107C2×Q8semidirect
410. #103C2×Q8semidirect
417. #97C2×Q8semidirect
492. SD16C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#33862
C2×#33812
#365932
#362762
#333142
#332022
#331952
#288202
#286842
#286752
#260592
#260552
#258402
#258382
#258362
#258352
#226782
#217142
C4×#4402