Finite Group Databasebeta

← #189297

Subgroups — #189297

Order 512.

854 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60208C211yesmaximal, characteristic
2#20825C211yesmaximal, characteristic, complemented
3#17787C211yesmaximal, characteristic, complemented
4#17573C211yesmaximal, characteristic
5#17571C211yesmaximal, characteristic
6#8681C211yesmaximal, characteristic, complemented
7#8680C211yesmaximal, characteristic, complemented
8#8679C211yesmaximal, characteristic, complemented
9#8678C211yesmaximal, characteristic, complemented
10#8677C211yesmaximal, characteristic, complemented
11#8676C211yesmaximal, characteristic, complemented
12#8675C211yesmaximal, characteristic, complemented
13#8674C211yesmaximal, characteristic, complemented
14#7958C211yesmaximal, characteristic, complemented
15#7956C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17#2971—21no—
18#2971C412yescomplemented
19#2970—21no—
20#2970C412yescomplemented
21#2969—21no—
22#2969—21no—
23#2969C412yescomplemented
24#2969C412yescomplemented
25#2960C2211yescharacteristic
26#2960C2211yescharacteristic
27#2931C2211yescharacteristic
28#2931C2211yescharacteristic
29#2930—21no—
30#2930—21no—
31#2923—21no—
32#2920C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#431C2211yescharacteristic
35#2875—21no—
36C2×C4×M16C2211yescharacteristic
37#2283C2211yescharacteristic
38#2283C2211yescharacteristic
39#1846C2211yescharacteristic
40#1846C2211yescharacteristic
41#1835C2211yescharacteristic
42#1835C2211yescharacteristic
43#1809—21no—
44#1809—21no—
45#1776—21no—
46#1776—21no—
47#1565—21no—
48#1565—21no—
49#1565—21no—
50#1565—21no—
51#1564C2211yescharacteristic
52#1564C2211yescharacteristic
53#1564C2211yescharacteristic
54#1564C2211yescharacteristic
55#1521—22no—
56#1518C2211yescharacteristic
57#1518C2211yescharacteristic
58#1518C2211yescharacteristic
59#1518C2211yescharacteristic
60#1517C2211yescharacteristic
61#1517C2211yescharacteristic
62#1517C2211yescharacteristic
63#1517C2211yescharacteristic
64#1516—22no—
65#1501—22no—
66#1500—22no—
67#1286C2211yescharacteristic
68#1286C2211yescharacteristic
69#1286C2211yescharacteristic
70#1286C2211yescharacteristic
71#1286C2211yescharacteristic
72#1286C2211yescharacteristic
73#1286C2211yescharacteristic
74#1286C2211yescharacteristic
75C2×#131—21no—
76C2×#131—21no—
77C22×M16D411yescharacteristic
78C22×M16D411yescharacteristic
79#533Q811yescharacteristic
80#533Q811yescharacteristic
81#533C2311yescharacteristic
82#532Q811yescharacteristic
83#532Q811yescharacteristic
84#532C2311yescharacteristic
85#529—21no—
86#529—21no—
87#529—21no—
88#529—21no—
89#529C2×C412yes—
90#529D412yes—
91#525—21no—
92#525—21no—
93#525D412yes—
94C4×#41—21no—
95C4×#41—21no—
96C4×#41C2×C412yes—
97C22×C42C2311yescharacteristic
98#446—21no—
99#446—21no—
100#446C2×C412yes—
101#446C2×C412yes—
102C8×Q8—21no—
103C8×Q8—21no—
104C8×Q8C2×C412yes—
105C8×Q8C2×C412yes—
106#436—22no—
107#436—22no—
108#436—22no—
109#436—22no—
110#436—21no—
111#436—21no—
112#436C2×C412yes—
113#436C2×C412yes—
114#435—22no—
115#435—22no—
116#435—22no—
117#435—22no—
118C8×D4—21no—
119C8×D4—21no—
120C8×D4C2×C412yes—
121C8×D4C2×C412yes—
122#433—22no—
123#433—22no—
124#432—21no—
125#432—22no—
126#432—22no—
127#432C2×C412yes—
128#431—21no—
129#431—21no—
130#431—21no—
131#431—21no—
132#431—21no—
133#431—21no—
134#431—21no—
135#431C2×C412yes—
136#431C2311yescharacteristic
137#431C2311yescharacteristic
138#431C2311yescharacteristic
139#431C2311yescharacteristic
140C2×#107—21no—
141C2×#107—21no—
142C2×#106—21no—
143C2×#106—21no—
144#424—21no—
145#424—21no—
146#423C2311yescharacteristic
147#423C2311yescharacteristic
148C2×C4⋊C8—21no—
149C2×C4⋊C8—21no—
150#408—21no—
151#408—21no—
152#407—21no—
153C2×C22⋊C8—21no—
154C2×C22⋊C8—21no—
155#405—21no—
156#405—21no—
157#405—22no—
158#405—22no—
159C4×M16—21no—
160C4×M16—22no—
161C4×M16C2×C411yescharacteristic
162C4×M16C2×C411yescharacteristic
163C4×M16C2×C411yescharacteristic
164C4×M16C2×C411yescharacteristic
165C4×M16D411yescharacteristic
166C4×M16D411yescharacteristic
167C4×M16D411yescharacteristic
168C4×M16D411yescharacteristic
169C2×#97—22no—
170C2×#97D411yescharacteristic
171C2×#97D411yescharacteristic
172C2×C4×C8—21no—
173C2×C4×C8—21no—
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176#383—21no—
177#383—21no—
178#382C2311yescharacteristic
179#382C2311yescharacteristic
180#382C2311yescharacteristic
181#382C2311yescharacteristic
182#381—21no—
183#381—21no—
184#380—21no—
185#380—21no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190#376—21no—
191#376—21no—
192#340—21no—
193#340—21no—
194#340—21no—
195#340—21no—
196#335—21no—
197#335—21no—
198#335—21no—
199#335—21no—
200#334—21no—
201#334—21no—
202#334—21no—
203#334—21no—
204#328—22no—
205#328—22no—
206#328—22no—
207#328—22no—
208#327—22no—
209#327—22no—
210#327—22no—
211#327—22no—
212C2×#41—21no—
213C2×#41—21no—
214C2×#41C4⋊C412yes—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4#4111yescharacteristic
218#131—42no—
219#131—42no—
220#131—42no—
221#131—42no—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—22no—
225C2×M16—22no—
226C2×M16—22no—
227C2×M16—22no—
228C2×M16—41no—
229C2×M16—41no—
230C2×M16C22⋊C411yescharacteristic
231C2×M16C22⋊C411yescharacteristic
232C2×M16C22⋊C411yescharacteristic
233C2×M16C22⋊C411yescharacteristic
234C2×M16C22⋊C411yescharacteristic
235C2×M16C22⋊C411yescharacteristic
236C2×M16C22⋊C411yescharacteristic
237C2×M16C22⋊C411yescharacteristic
238C22×C8—22no—
239C22×C8—22no—
240C22×C8—41no—
241C22×C8—41no—
242C22×C8C2×D411yescharacteristic
243C22×C8C2×D411yescharacteristic
244C22×C8C2×D411yescharacteristic
245C22×C8C2×D411yescharacteristic
246C4⋊Q8C2×Q811yescharacteristic
247C4⋊Q8C2×Q811yescharacteristic
248C4⋊Q8C2×Q811yescharacteristic
249#127C2×Q811yescharacteristic
250#126—22no—
251#126—22no—
252#125—21no—
253#125—21no—
254#125C4⋊C412yes—
255#124—21no—
256#124—21no—
257#124C4⋊C412yes—
258#123—22no—
259#123—22no—
260#123—22no—
261C22⋊Q8—21no—
262C22⋊Q8—21no—
263C22⋊Q8—21no—
264C22⋊Q8—21no—
265C22⋊Q8C4⋊C412yes—
266C22⋊Q8C2×D412yes—
267#121—21no—
268#121—21no—
269#121—21no—
270#121—21no—
271#121C4⋊C412yes—
272#121C2×D412yes—
273#120—22no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8—21no—
278C4×Q8C4⋊C412yes—
279C4×Q8C22×C412yes—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4C4⋊C412yes—
291C4×D4C22×C412yes—
292#117—21no—
293#117—21no—
294#117—22no—
295#117—22no—
296#117—22no—
297#117C4⋊C412yes—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—41no—
304C2×C42—22no—
305C2×C42—41no—
306C2×C42—41no—
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×Q811yescharacteristic
314C2×C42C2×Q811yescharacteristic
315C2×C42#4111yescharacteristic
316C2×C42#4111yescharacteristic
317C2×C42#4111yescharacteristic
318C2×C42C2411yescharacteristic
319#107—42no—
320#107—42no—
321#106—42no—
322#106—42no—
323C4⋊C8—22no—
324C4⋊C8—22no—
325C4⋊C8—22no—
326C4⋊C8—22no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—21no—
330C4⋊C8—21no—
331C4⋊C8—21no—
332C4⋊C8—21no—
333C4⋊C8—41no—
334C4⋊C8—41no—
335C4⋊C8C22×C412yes—
336C4⋊C8C22×C412yes—
337C22⋊C8—22no—
338C22⋊C8—22no—
339C22⋊C8—22no—
340C22⋊C8—22no—
341C22⋊C8—22no—
342C22⋊C8—22no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—21no—
347C22⋊C8—41no—
348C22⋊C8—41no—
349#97—22no—
350#97—22no—
351#97—22no—
352#97C22⋊C411yescharacteristic
353#97C22⋊C411yescharacteristic
354#97C22⋊C411yescharacteristic
355#97C22⋊C411yescharacteristic
356#97C22⋊C411yescharacteristic
357#97C22⋊C411yescharacteristic
358#97C22⋊C411yescharacteristic
359#97C22⋊C411yescharacteristic
360C4×C8—22no—
361C4×C8—22no—
362C4×C8—22no—
363C4×C8—21no—
364C4×C8—21no—
365C4×C8—21no—
366C4×C8—21no—
367C4×C8—41no—
368C4×C8—41no—
369C4×C8C22×C411yescharacteristic
370C4×C8C22×C411yescharacteristic
371C4×C8C22×C411yescharacteristic
372C4×C8C22×C411yescharacteristic
373C4×C8C2×D411yescharacteristic
374C4×C8C2×D411yescharacteristic
375C4×C8C2×D411yescharacteristic
376C4×C8C2×D411yescharacteristic
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—41no—
386#95—41no—
387#95—41no—
388#95—41no—
389C24#12111yescharacteristic
390#41—41no—
391#41—41no—
392#41—41no—
393#41—41no—
394#41—82no—
395#41—82no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8C2×C4⋊C412yes—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4C2×C4⋊C412yes—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C4⋊C411yescharacteristic
421C22×C4C2×C4⋊C411yescharacteristic
422C22×C4#11711yescharacteristic
423C22×C4#11711yescharacteristic
424C22×C4#12111yescharacteristic
425C22×C4#12111yescharacteristic
426C22×C4#12111yescharacteristic
427C22×C4#12111yescharacteristic
428C22×C4#12111yescharacteristic
429C22×C4C22⋊Q811yescharacteristic
430C22×C4C22⋊Q811yescharacteristic
431C22×C4C22⋊Q811yescharacteristic
432C22×C4C22⋊Q811yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2×#4111yescharacteristic
436M16—22no—
437M16—22no—
438M16—22no—
439M16—22no—
440M16—41no—
441M16—41no—
442M16—41no—
443M16—41no—
444M16—44no—
445M16—41no—
446M16—41no—
447M16—41no—
448M16—41no—
449M16—44no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—44no—
459C2×C8—44no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—82no—
465C2×C8—82no—
466C2×C8C2×C22⋊C411yescharacteristic
467C2×C8C2×C22⋊C411yescharacteristic
468C2×C8C2×C22⋊C411yescharacteristic
469C2×C8C2×C22⋊C411yescharacteristic
470C2×C8C2×C22⋊C411yescharacteristic
471C2×C8C2×C22⋊C411yescharacteristic
472C2×C8C2×C22⋊C411yescharacteristic
473C2×C8C2×C22⋊C411yescharacteristic
474C2×C8#13512yes—
475C2×C8#13512yes—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4C2×C4⋊C412yes—
495C4⋊C4C2×C4⋊C412yes—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—22no—
505C22⋊C4—82no—
506C22⋊C4—82no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—82no—
511C42—82no—
512C42C2×C22⋊C411yescharacteristic
513C42C2×C22⋊C411yescharacteristic
514C42C2×C4⋊C411yescharacteristic
515C42C2×C4⋊C411yescharacteristic
516C42#11711yescharacteristic
517C42#11711yescharacteristic
518C42#12111yescharacteristic
519C42#12111yescharacteristic
520C42C22⋊Q811yescharacteristic
521C42C22⋊Q811yescharacteristic
522C42C22⋊Q811yescharacteristic
523C42C22⋊Q811yescharacteristic
524C42C23×C411yescharacteristic
525C42C22×D411yescharacteristic
526C42C22×Q811yescharacteristic
527C42C2×#4111yescharacteristic
528C23—21no—
529C23—21no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23#38011yescharacteristic
534C23#38011yescharacteristic
535C23C2×#12111yescharacteristic
536Q8—42no—
537Q8—42no—
538Q8—41no—
539Q8—41no—
540D4—42no—
541D4—42no—
542D4—42no—
543D4—42no—
544D4—41no—
545D4—41no—
546D4—82no—
547D4—82no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4#38011yescharacteristic
590C2×C4#38011yescharacteristic
591C2×C4#38011yescharacteristic
592C2×C4#38011yescharacteristic
593C2×C4#38011yescharacteristic
594C2×C4#38011yescharacteristic
595C2×C4C22×C22⋊C411yescharacteristic
596C2×C4C22×C4⋊C411yescharacteristic
597C2×C4C2×#11711yescharacteristic
598C2×C4C2×#12111yescharacteristic
599C2×C4C2×C22⋊Q811yescharacteristic
600C2×C4C2×C22⋊Q811yescharacteristic
601C2×C4C2×#13511yescharacteristic
602C2×C4C2×#13511yescharacteristic
603C2×C4#57511yescharacteristic
604C2×C4#57511yescharacteristic
605C8—21no—
606C8—21no—
607C8—21no—
608C8—21no—
609C8—21no—
610C8—21no—
611C8—42no—
612C8—42no—
613C8—81no—
614C8—81no—
615C8—81no—
616C8—81no—
617C8#41612yes—
618C8#41612yes—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—81no—
631C22—81no—
632C22C2×#38011yescharacteristic
633C22#303511yescharacteristic
634C22#303511yescharacteristic
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—161no—
649C4—161no—
650C4—161no—
651C4—161no—
652C4C2×#41611yescharacteristic
653C4#289611yescharacteristic
654C4#307611yescharacteristic
655C4#307611yescharacteristic
656C2—21no—
657C2—21no—
658C2—41no—
659C2—41no—
660C2—82no—
661C2#2064511yescharacteristic
662C2#2067111yescharacteristic
663C2#2067511yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20825C2semidirect
3. #17787C2semidirect
6. #8681C2semidirect
7. #8680C2semidirect
8. #8679C2semidirect
9. #8678C2semidirect
10. #8677C2semidirect
11. #8676C2semidirect
12. #8675C2semidirect
13. #8674C2semidirect
14. #7958C2semidirect
15. #7956C2semidirect
18. #2971C4semidirect
20. #2970C4semidirect
23. #2969C4semidirect
24. #2969C4semidirect

Maximal subgroups

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

subgroupindex
#602082
#208252
#177872
#175732
#175712
#86812
#86802
#86792
#86782
#86772
#86762
#86752
#86742
#79582
#79562