Finite Group Databasebeta

← #6583268

Subgroups — #6583268

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3385C211yesmaximal, characteristic
2C2×#3381C211yesmaximal, characteristic
3#36474C211yesmaximal, characteristic, complemented
4#36180C211yesmaximal, characteristic
5#33314C211yesmaximal, characteristic, complemented
6#33201C211yesmaximal, characteristic
7#33198C211yesmaximal, characteristic, complemented
8#28820C214yesmaximal, complemented
9#28684C211yesmaximal, characteristic, complemented
10#28676C211yesmaximal, characteristic, complemented
11#26057C214yesmaximal, complemented
12#26052C214yesmaximal
13#25841C211yesmaximal, characteristic, complemented
14#25839C211yesmaximal, characteristic
15#25836C211yesmaximal, characteristic, complemented
16#25835C211yesmaximal, characteristic
17#22677C214yesmaximal
18#21714C211yesmaximal, characteristic, complemented
19C4×#440C211yesmaximal, characteristic
20C2×Q82C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#554C2211yescharacteristic
24#3385—28no—
25#3381—28no—
26C2×#501C2212yes—
27C2×#500C2211yescharacteristic
28C2×#500C2211yescharacteristic
29C2×#499C2212yes—
30C2×#498C2211yescharacteristic
31C2×#498C2211yescharacteristic
32C2×#479C2212yes—
33C2×#479C2212yes—
34C2×#477C2211yescharacteristic
35C2×#477C2211yescharacteristic
36C2×#478C2212yes—
37C2×#476C2211yescharacteristic
38C2×#474C2211yescharacteristic
39C2×#474C2212yes—
40C2×#446C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×#440C2212yes—
43C2×#440C2212yes—
44C2×C4×SD16C2212yes—
45C2×C4×SD16C2211yescharacteristic
46#2868C2211yescharacteristic
47#2862—21no—
48#2862C2211yescharacteristic
49#2857—21no—
50#2567—21no—
51#2567C2211yescharacteristic
52#2567C2211yescharacteristic
53#2567C2211yescharacteristic
54#2552C2214yes—
55#2550—24no—
56#2550C2214yes—
57#2546—21no—
58#2546C2211yescharacteristic
59#2502—21no—
60#2502—21no—
61#2440—24no—
62#2351C2214yes—
63D4×C4⋊C4C2214yes—
64C4×C4⋊Q8—21no—
65C4×C4⋊Q8C2211yescharacteristic
66C2×#384—21no—
67C42×Q8C2211yescharacteristic
68C42×D4C2211yescharacteristic
69#2088C2214yes—
70#2087C2214yes—
71#2059C2214yes—
72#2059C2214yes—
73#2059C2214yes—
74#2057C2214yes—
75#1988C2211yescharacteristic
76#1987C2211yescharacteristic
77#1987C2211yescharacteristic
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#1804C2214yes—
94#1803C2214yes—
95#1776C2212yes—
96C4×#107C2211yescharacteristic
97#1771C2214yes—
98#1763C2212yes—
99#1762C2212yes—
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102C4×#97C2211yescharacteristic
103C22×SD16C2311yescharacteristic
104Q82—24no—
105Q82—42no—
106#557—24no—
107#557—42no—
108#554—24no—
109#554—24no—
110#554—42no—
111#554—42no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—22no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8D411yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2311yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2312yes—
123C2×#125—22no—
124C2×#125C2312yes—
125C2×#125C2312yes—
126C2×#125C2312yes—
127C2×C22⋊Q8—21no—
128C2×C22⋊Q8—21no—
129C2×C22⋊Q8—22no—
130C2×C22⋊Q8—22no—
131C2×C4×Q8—21no—
132C2×C4×Q8—21no—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2312yes—
135C2×C4×Q8C2311yescharacteristic
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2311yescharacteristic
138C2×C4×Q8C2311yescharacteristic
139C2×C4×D4C2312yes—
140C2×C4×D4C2312yes—
141C2×C4×D4C2311yescharacteristic
142C2×C4×D4C2312yes—
143C22×C4⋊C4—21no—
144C22×C4⋊C4—21no—
145C22×C4⋊C4—24no—
146C22×C42—21no—
147#501—28no—
148#500—24no—
149#500—42no—
150#499—28no—
151#498—24no—
152#498—42no—
153#479—28no—
154#479—28no—
155#478—28no—
156#477—24no—
157#477—24no—
158#476—24no—
159#474—24no—
160#474—28no—
161#446—24no—
162C8×Q8—24no—
163#440—28no—
164#440Q8116yescomplemented
165C4×SD16—28no—
166C4×SD16D418yes—
167C2×#107C2312yes—
168C2×#107C2312yes—
169C2×#107C2312yes—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#107C2314yes—
173C2×#107C2312yes—
174C2×#106C2312yes—
175C2×#106C2312yes—
176C2×#106C2312yes—
177C2×#106C2314yes—
178C2×#106C2311yescharacteristic
179C2×C4⋊C8C2312yes—
180C2×C4⋊C8C2311yescharacteristic
181C2×C4⋊C8C2311yescharacteristic
182C2×C4⋊C8C2312yes—
183C2×C4⋊C8C2314yes—
184C2×#103C2312yes—
185C2×#103C2312yes—
186C2×#103C2311yescharacteristic
187C2×#103C2312yes—
188C2×#102C2312yes—
189C2×#102C2312yes—
190C2×#102C2311yescharacteristic
191C2×#102C2312yes—
192C2×#97C2312yes—
193C2×#97C2312yes—
194C2×#97C2312yes—
195C2×C4×C8C2311yescharacteristic
196C2×C4×C8C2312yes—
197C2×C4×C8C2311yescharacteristic
198#398—24no—
199#398—22no—
200#398—24no—
201#398—24no—
202#395—24no—
203#395—22no—
204#395—22no—
205#395—22no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391—21no—
212#391—22no—
213#391—24no—
214#391D411yescharacteristic
215#391D411yescharacteristic
216#389—24no—
217#389—24no—
218#389—24no—
219#389—24no—
220#389D414yes—
221#389C2314yes—
222#389C2314yes—
223#389C2314yes—
224#389C2314yes—
225#385—24no—
226#385—24no—
227#384—21no—
228#384—21no—
229#384—21no—
230#384—24no—
231#384—21no—
232#384—21no—
233#384D411yescharacteristic
234#384C2314yes—
235#382—24no—
236#382C2314yes—
237#380—24no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4—24no—
241C4×C4⋊C4C2311yescharacteristic
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2312yes—
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2314yes—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—21no—
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×Q8C2×D411yescharacteristic
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2411yescharacteristic
267C22×D4C2411yescharacteristic
268C23×C4—21no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—22no—
272C2×SD16C2×D414yes—
273C2×SD16C2×Q818yes—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2411yescharacteristic
277C22×C8C2412yes—
278C22×C8C2411yescharacteristic
279C4⋊Q8—24no—
280C4⋊Q8—22no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—42no—
289C4⋊Q8—44no—
290C4⋊Q8—42no—
291C4⋊Q8—44no—
292C4⋊Q8—48no—
293C4⋊Q8—44no—
294C4⋊Q8—84no—
295#125—24no—
296#125—24no—
297#125—24no—
298#125—42no—
299#125—42no—
300#125—42no—
301#125—48no—
302C22⋊Q8—48no—
303C22⋊Q8—48no—
304C22⋊Q8—44no—
305C22⋊Q8—44no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—22no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—42no—
315C4×Q8—42no—
316C4×Q8—44no—
317C4×Q8—44no—
318C4×Q8C2×D414yes—
319C4×Q8C2×Q818yes—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—22no—
323C4×D4—24no—
324C4×D4—42no—
325C4×D4—42no—
326C4×D4C2×D414yes—
327C4×D4C2×Q818yes—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—24no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—24no—
345C2×C4⋊C4—24no—
346C2×C4⋊C4—24no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—44no—
350C2×C4⋊C4—44no—
351C2×C4⋊C4—88no—
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D412yes—
355C2×C4⋊C4C2×D412yes—
356C2×C4⋊C4C2×D414yes—
357C2×C4⋊C4C2×D414yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2411yescharacteristic
363C2×C4⋊C4C2411yescharacteristic
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C42—21no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—21no—
375C2×C42—22no—
376C2×C42—21no—
377C2×C42—22no—
378C2×C42—44no—
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D412yes—
382C2×C42C2412yes—
383C2×C42C2412yes—
384C2×C42C2411yescharacteristic
385C2×C42C2412yes—
386#107—24no—
387#107—24no—
388#107—24no—
389#107—24no—
390#107—24no—
391#107—44no—
392#107C2×Q818yes—
393#106—24no—
394#106—24no—
395#106—24no—
396#106—44no—
397#106C2×D414yes—
398C4⋊C8—24no—
399C4⋊C8—24no—
400C4⋊C8—22no—
401C4⋊C8—22no—
402C4⋊C8—28no—
403#103—24no—
404#103—24no—
405#103C2×D414yes—
406#103C2×Q818yes—
407#102—24no—
408#102—24no—
409#102C2×D414yes—
410#102C2×Q818yes—
411#97—24no—
412#97—24no—
413#97C2×Q818yes—
414C4×C8—24no—
415C4×C8—22no—
416C4×C8C2×D414yes—
417#95—21no—
418#95—22no—
419#95—21no—
420#95—22no—
421#95—24no—
422#95—24no—
423#95—24no—
424#95—24no—
425#95—24no—
426#95—24no—
427#95C2×D414yes—
428#95C2×D414yes—
429C24—21no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—21no—
437C2×Q8—22no—
438C2×Q8—44no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—82no—
446C2×Q8—82no—
447C2×Q8—82no—
448C2×Q8—84no—
449C2×Q8C22×D412yes—
450C2×Q8C22×Q814yes—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4C22×D412yes—
454C2×D4C22×Q814yes—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—24no—
461C22×C4—28no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C2511yescharacteristic
483SD16C4⋊Q8116yescomplemented
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8C22×D412yes—
492C2×C8C22×D412yes—
493C2×C8C22×Q814yes—
494C2×C8C22×Q814yes—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—28no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—48no—
525C4⋊C4—48no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C4⋊C4—84no—
534C4⋊C4C22×D412yes—
535C4⋊C4C22×D412yes—
536C4⋊C4C22×Q814yes—
537C4⋊C4C22×Q814yes—
538C22⋊C4—24no—
539C22⋊C4—28no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C42—24no—
543C42—28no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—42no—
551C42—44no—
552C42—44no—
553C42—44no—
554C42—42no—
555C42—42no—
556C42—82no—
557C42C22×D412yes—
558C42C22×Q814yes—
559C42H(5,𝔽2)12yes—
560C422−1+414yes—
561C23—22no—
562C23—24no—
563C23—42no—
564C23—41no—
565C23—41no—
566C23C23×D411yescharacteristic
567Q8—22no—
568Q8—24no—
569Q8—44no—
570Q8—44no—
571Q8—84no—
572Q8—162no—
573Q8—162no—
574Q8—162no—
575Q8—164no—
576Q8C2×C4⋊Q814yes—
577D4—22no—
578D4—24no—
579D4C2×C4⋊Q814yes—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—416no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—48no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—164no—
620C2×C4—164no—
621C2×C4D4212yes—
622C2×C4D4212yes—
623C2×C4D4×Q814yes—
624C2×C4D4×Q814yes—
625C2×C4#57512yes—
626C2×C4#57812yes—
627C2×C4C23×D411yescharacteristic
628C2×C4C23×Q812yes—
629C2×C4C2×H(5,𝔽2)11yescharacteristic
630C2×C4C2×2−1+412yes—
631C8—22no—
632C8—24no—
633C8—42no—
634C8—42no—
635C8C2×C4⋊Q814yes—
636C22—24no—
637C22—42no—
638C22—42no—
639C22—44no—
640C22—44no—
641C22C2×D4211yescharacteristic
642C22C2×D4×Q812yes—
643C22C2×#57511yescharacteristic
644C22C2×#57811yescharacteristic
645C22#358512yes—
646C4—24no—
647C4—22no—
648C4—24no—
649C4—24no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4—42no—
656C4—44no—
657C4—44no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4#326914yes—
663C4C22×C4⋊Q811yescharacteristic
664C4#349911yescharacteristic
665C4#350312yes—
666C2—44no—
667C2D4×C4⋊Q811yescharacteristic
668C2C2×#326911yescharacteristic
669C2#6108211yescharacteristic
670C2#6136712yes—
671C2#6138412yes—
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. #36474C2semidirect
5. #33314C2semidirect
7. #33198C2semidirect
8. #28820C2semidirect
9. #28684C2semidirect
10. #28676C2semidirect
11. #26057C2semidirect
13. #25841C2semidirect
15. #25836C2semidirect
18. #21714C2semidirect
164. #440Q8semidirect
483. SD16C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#33852
C2×#33812
#364742
#361802
#333142
#332012
#331982
#288202
#286842
#286762
#260572
#260522
#258412
#258392
#258362
#258352
#226772
#217142
C4×#4402