Finite Group Databasebeta

← #481372

Subgroups — #481372

Order 512.

705 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60696C211yesmaximal, characteristic
2#33420C211yesmaximal, characteristic, complemented
3#17799C211yesmaximal, characteristic, complemented
4#11676C211yesmaximal, characteristic, complemented
5#11674C211yesmaximal, characteristic, complemented
6#11673C211yesmaximal, characteristic, complemented
7#11672C211yesmaximal, characteristic, complemented
8#11662C211yesmaximal, characteristic, complemented
9#11660C211yesmaximal, characteristic, complemented
10#11659C211yesmaximal, characteristic, complemented
11#11658C211yesmaximal, characteristic, complemented
12#11650C211yesmaximal, characteristic, complemented
13#11648C211yesmaximal, characteristic, complemented
14#11647C211yesmaximal, characteristic, complemented
15#11646C211yesmaximal, characteristic, complemented
16#3461—21no—
17#3178—21no—
18#3178—21no—
19#3177C411yescharacteristic
20#3177C411yescharacteristic
21#3175C411yescharacteristic
22#3175C411yescharacteristic
23#3172—21no—
24#3172—21no—
25#3170C2211yescharacteristic
26#3170C2211yescharacteristic
27#3168C2211yescharacteristic
28#3168C2211yescharacteristic
29#3164C411yescharacteristic
30#3156C411yescharacteristic
31#3137—21no—
32#3063—21no—
33#3061C411yescharacteristic
34#3059C411yescharacteristic
35#2989—21no—
36C4×#137—21no—
37C4×#135C2211yescharacteristic
38C2×C4×Q16C2211yescharacteristic
39C4×#131C2211yescharacteristic
40#2164—21no—
41#2164—21no—
42#2162C2211yescharacteristic
43#2162C2211yescharacteristic
44#2162C2211yescharacteristic
45#2162C2211yescharacteristic
46#2151—21no—
47#2113—21no—
48C4×M32C2211yescharacteristic
49C2×C4×C16C2211yescharacteristic
50#1569—21no—
51#1569—21no—
52#1568C2211yescharacteristic
53#1568C2211yescharacteristic
54#1567C2211yescharacteristic
55#1567C2211yescharacteristic
56#1565—21no—
57#1564C2211yescharacteristic
58C2×#335C2211yescharacteristic
59#1332C2211yescharacteristic
60#1332C2211yescharacteristic
61#1332C2211yescharacteristic
62#1332C2211yescharacteristic
63#1332C2211yescharacteristic
64#1332C2211yescharacteristic
65#1332C2211yescharacteristic
66#1332C2211yescharacteristic
67#1332C2211yescharacteristic
68#1332C2211yescharacteristic
69#1332C2211yescharacteristic
70#1332C2211yescharacteristic
71#1331C2211yescharacteristic
72#1331C2211yescharacteristic
73#1331C2211yescharacteristic
74#1331C2211yescharacteristic
75C2×#137—21no—
76C2×#135C2×C411yescharacteristic
77C22×Q16C2×C411yescharacteristic
78C2×#131C2×C411yescharacteristic
79C2×#131D411yescharacteristic
80#544—21no—
81#542—21no—
82#542—21no—
83#542—41no—
84#541—21no—
85#541—21no—
86#541—41no—
87#539—41no—
88#539—41no—
89#537—21no—
90#536—21no—
91#536—21no—
92#533—21no—
93#533—21no—
94#533—41no—
95#529—21no—
96#529—21no—
97#529—41no—
98#529—41no—
99#529—41no—
100C4×#41—21no—
101C4×#41D411yescharacteristic
102C2×C4×Q8—21no—
103C2×M32—21no—
104C22×C16—21no—
105#501—21no—
106#501—21no—
107#500C2×C411yescharacteristic
108#500C2×C411yescharacteristic
109#499C2×C411yescharacteristic
110#499C2×C411yescharacteristic
111#497—21no—
112#497—21no—
113#495C2×C411yescharacteristic
114#495C2×C411yescharacteristic
115#494C2×C411yescharacteristic
116#494C2×C411yescharacteristic
117#491—21no—
118#491—21no—
119#489—21no—
120#489—21no—
121#488—21no—
122#488—21no—
123#487—21no—
124#487—21no—
125#470—21no—
126#470—21no—
127#470—21no—
128#470—21no—
129#468—21no—
130#468—21no—
131#468—21no—
132#468—21no—
133#467—21no—
134#467—21no—
135#467C2×C411yescharacteristic
136#467C2×C411yescharacteristic
137#467C2×C411yescharacteristic
138#467C2×C411yescharacteristic
139#467C2×C411yescharacteristic
140#467C2×C411yescharacteristic
141#466C2×C411yescharacteristic
142#466C2×C411yescharacteristic
143#465—21no—
144#465—21no—
145#446—41no—
146#441—21no—
147#441—21no—
148#440—21no—
149#440—21no—
150C4×Q16—21no—
151C4×Q16—21no—
152C4×Q16—21no—
153C4×Q16—21no—
154C4×Q16C2×C411yescharacteristic
155C4×Q16C2×C411yescharacteristic
156C4×Q16C2×C411yescharacteristic
157C4×Q16C2311yescharacteristic
158C4×Q16C2311yescharacteristic
159C4×Q16C2311yescharacteristic
160C4×SD16—21no—
161C4×SD16—21no—
162C4×SD16—21no—
163C4×SD16—21no—
164C4×D8C2×C411yescharacteristic
165C4×D8C2311yescharacteristic
166#436—41no—
167#435—41no—
168#435—41no—
169#432—21no—
170#432—21no—
171#431—21no—
172#428—21no—
173#427C2×C411yescharacteristic
174C2×#107C2×C411yescharacteristic
175#423—21no—
176#423—21no—
177C2×C4⋊C8—21no—
178#419—21no—
179#417—21no—
180#416—21no—
181C2×#103—21no—
182#405C2×C411yescharacteristic
183#405C2×C411yescharacteristic
184#405C2×C411yescharacteristic
185#405D411yescharacteristic
186#405D411yescharacteristic
187#405D411yescharacteristic
188C4×M16D411yescharacteristic
189C4×M16D411yescharacteristic
190C4×M16D411yescharacteristic
191C2×C4×C8C2311yescharacteristic
192C2×C4×C8C2311yescharacteristic
193C2×C4×C8C2311yescharacteristic
194C4⋊C16—21no—
195C4⋊C16—21no—
196C4⋊C16—21no—
197C4⋊C16—21no—
198C22⋊C16—21no—
199C22⋊C16—21no—
200C22⋊C16—21no—
201C22⋊C16—21no—
202#346—21no—
203C4×C16—21no—
204C4×C16C2311yescharacteristic
205C4×C16C2311yescharacteristic
206C4×C16C2311yescharacteristic
207C4×C16C2311yescharacteristic
208#335—21no—
209#335—21no—
210#335—21no—
211#335C2311yescharacteristic
212#335C2311yescharacteristic
213#335C2311yescharacteristic
214#335C2311yescharacteristic
215#334—21no—
216#334—21no—
217#334—21no—
218C2×#41—21no—
219C2×#41C22⋊C411yescharacteristic
220C22×Q8—21no—
221#137—44no—
222#135—42no—
223C2×Q16—21no—
224C2×Q16—21no—
225C2×Q16—21no—
226C2×Q16—42no—
227C2×Q16C22×C411yescharacteristic
228C2×Q16C22×C411yescharacteristic
229C2×Q16C22×C411yescharacteristic
230C2×SD16—21no—
231C2×SD16—21no—
232C2×SD16—21no—
233C2×D8C22×C411yescharacteristic
234#131—42no—
235#131—42no—
236C2×M16—41no—
237C2×M16—41no—
238C2×M16C22⋊C411yescharacteristic
239C2×M16C22⋊C411yescharacteristic
240C2×M16C22⋊C411yescharacteristic
241C2×M16C22⋊C411yescharacteristic
242C2×M16C22⋊C411yescharacteristic
243C2×M16C22⋊C411yescharacteristic
244C22×C8—41no—
245C22×C8C22×C411yescharacteristic
246C22×C8C22×C411yescharacteristic
247C22×C8C22×C411yescharacteristic
248C22×C8C2×D411yescharacteristic
249C22×C8C2×D411yescharacteristic
250C22×C8C2×D411yescharacteristic
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—41no—
256C4⋊Q8—41no—
257#126—41no—
258#126—41no—
259#126—41no—
260#126—41no—
261#126—41no—
262#126—41no—
263#126—41no—
264#126—41no—
265#125—21no—
266#125—21no—
267#125—41no—
268#125—41no—
269#125—41no—
270#125—41no—
271#125—41no—
272#125—41no—
273#125—41no—
274#124—21no—
275#124—21no—
276#124—41no—
277#124—41no—
278#124—41no—
279#123—41no—
280#123—41no—
281#123—41no—
282#123—41no—
283#123—41no—
284#123—41no—
285#123—41no—
286#123—41no—
287C22⋊Q8—21no—
288C22⋊Q8—21no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C22⋊Q8—41no—
294C22⋊Q8—41no—
295C22⋊Q8—41no—
296C22⋊Q8—41no—
297C22⋊Q8—41no—
298C22⋊Q8—41no—
299C22⋊Q8—41no—
300C22⋊Q8—41no—
301C22⋊Q8—41no—
302#121—21no—
303#121—21no—
304#121—41no—
305#121—41no—
306#121—41no—
307C4×Q8—21no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—41no—
314C4×Q8—41no—
315C4×Q8—41no—
316C4×Q8—41no—
317C4×Q8D811yescharacteristic
318C4×Q8SD1611yescharacteristic
319C4×D4—21no—
320C4×D4—21no—
321C4×D4—41no—
322C4×D4—41no—
323C4×D4—41no—
324C4×D4—41no—
325C4×D4D811yescharacteristic
326C4×D4D811yescharacteristic
327C4×D4D811yescharacteristic
328C4×D4SD1611yescharacteristic
329C4×D4SD1611yescharacteristic
330C4×D4SD1611yescharacteristic
331#117—21no—
332#117—41no—
333#117—41no—
334#117—41no—
335#117—41no—
336#117—41no—
337#117C22⋊C411yescharacteristic
338#117C22⋊C411yescharacteristic
339#117C22⋊C411yescharacteristic
340C2×C4⋊C4—21no—
341C2×C4⋊C4—41no—
342C2×C42—41no—
343C2×C42—41no—
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347M32—42no—
348C2×C16—21no—
349C2×C16—21no—
350C2×C16—21no—
351C2×C16—21no—
352C2×C16—42no—
353#107—21no—
354#107—21no—
355#107—21no—
356#107C22×C411yescharacteristic
357#107C22×C411yescharacteristic
358#107C22×C411yescharacteristic
359#107C22×C411yescharacteristic
360#106—21no—
361#106—21no—
362#106—21no—
363C4⋊C8—21no—
364C4⋊C8—21no—
365C4⋊C8—21no—
366C4⋊C8—21no—
367C4⋊C8—41no—
368C4⋊C8—41no—
369C4⋊C8—41no—
370#103—21no—
371#103—21no—
372#103—21no—
373#103—21no—
374#103—21no—
375#103—21no—
376#103—21no—
377#103—21no—
378#103—21no—
379#103—21no—
380#103—21no—
381#103—21no—
382#102—21no—
383#102—21no—
384#102—21no—
385#102—21no—
386C22⋊C8—41no—
387C22⋊C8—41no—
388C22⋊C8—41no—
389#97—41no—
390#97—41no—
391#97C22⋊C411yescharacteristic
392#97C22⋊C411yescharacteristic
393#97C22⋊C411yescharacteristic
394#97C22⋊C411yescharacteristic
395#97C22⋊C411yescharacteristic
396#97C22⋊C411yescharacteristic
397C4×C8—41no—
398C4×C8C22×C411yescharacteristic
399C4×C8C22×C411yescharacteristic
400C4×C8C22×C411yescharacteristic
401C4×C8C2×D411yescharacteristic
402C4×C8C2×D411yescharacteristic
403C4×C8C2×D411yescharacteristic
404C4×C8C2×D411yescharacteristic
405C4×C8C2×D411yescharacteristic
406C4×C8C2×D411yescharacteristic
407C4×C8C2411yescharacteristic
408#41—41no—
409#41—41no—
410#41—42no—
411#41—42no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—42no—
416C2×Q8—42no—
417C2×Q8—41no—
418C2×Q8—41no—
419C2×Q8—41no—
420C2×Q8#10211yescharacteristic
421C2×D4—21no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—41no—
425C2×D4#10211yescharacteristic
426C2×D4#10211yescharacteristic
427C2×D4#10211yescharacteristic
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4C2×C22⋊C411yescharacteristic
434C22×C4C2×C22⋊C411yescharacteristic
435C22×C4C2×C22⋊C411yescharacteristic
436C22×C4C2×D811yescharacteristic
437C22×C4C2×D811yescharacteristic
438C22×C4C2×D811yescharacteristic
439C22×C4C2×SD1611yescharacteristic
440C22×C4C2×SD1611yescharacteristic
441C22×C4C2×SD1611yescharacteristic
442Q16—22no—
443Q16—22no—
444Q16—22no—
445Q16—42no—
446Q16—42no—
447Q16—42no—
448SD16—42no—
449SD16—42no—
450SD16—42no—
451D8—22no—
452M16—22no—
453M16—22no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—161no—
459M16—161no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—161no—
471C2×C8C2×C22⋊C411yescharacteristic
472C2×C8C2×C22⋊C411yescharacteristic
473C2×C8C2×C22⋊C411yescharacteristic
474C2×C8C2×C22⋊C411yescharacteristic
475C2×C8C2×C22⋊C411yescharacteristic
476C2×C8C2×C22⋊C411yescharacteristic
477C2×C8C23×C411yescharacteristic
478C2×C8C22×D411yescharacteristic
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
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⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4#10211yescharacteristic
507C4⋊C4#10211yescharacteristic
508C4⋊C4#10211yescharacteristic
509C4⋊C4#10211yescharacteristic
510C4⋊C4#10211yescharacteristic
511C4⋊C4#10211yescharacteristic
512C22⋊C4—41no—
513C22⋊C4—41no—
514C22⋊C4—41no—
515C22⋊C4—41no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C22⋊C4—41no—
519C22⋊C4—41no—
520C22⋊C4—41no—
521C22⋊C4—41no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4#10211yescharacteristic
528C22⋊C4#10211yescharacteristic
529C22⋊C4#10211yescharacteristic
530C22⋊C4#10211yescharacteristic
531C22⋊C4#10211yescharacteristic
532C22⋊C4#10211yescharacteristic
533C42—41no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—41no—
539C42—41no—
540C42C2×C22⋊C411yescharacteristic
541C42C2×C22⋊C411yescharacteristic
542C42C2×C22⋊C411yescharacteristic
543C42C2×D811yescharacteristic
544C42C2×D811yescharacteristic
545C42C2×D811yescharacteristic
546C42C2×SD1611yescharacteristic
547C42C2×SD1611yescharacteristic
548C42C2×SD1611yescharacteristic
549C42C22×D411yescharacteristic
550C16—41no—
551C16—41no—
552C16—41no—
553C16—41no—
554C23—41no—
555C23C2×#10211yescharacteristic
556C23C2×#10211yescharacteristic
557C23C2×#10211yescharacteristic
558Q8—41no—
559Q8—41no—
560Q8—41no—
561Q8—41no—
562Q8—41no—
563Q8—41no—
564Q8—41no—
565Q8—82no—
566Q8—161no—
567Q8—161no—
568D4—41no—
569D4—41no—
570D4—41no—
571D4—41no—
572D4—41no—
573D4—82no—
574D4—161no—
575D4—161no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—81no—
601C2×C4—81no—
602C2×C4—161no—
603C2×C4—161no—
604C2×C4C2×#10211yescharacteristic
605C2×C4C2×#10211yescharacteristic
606C2×C4C2×#10211yescharacteristic
607C2×C4C2×#10211yescharacteristic
608C2×C4C2×#10211yescharacteristic
609C2×C4C2×#10211yescharacteristic
610C2×C4C2×#10211yescharacteristic
611C2×C4C2×#10211yescharacteristic
612C2×C4C2×#10211yescharacteristic
613C2×C4C22×C22⋊C411yescharacteristic
614C2×C4C22×D811yescharacteristic
615C2×C4C22×SD1611yescharacteristic
616C8—21no—
617C8—21no—
618C8—21no—
619C8—21no—
620C8—21no—
621C8—21no—
622C8—81no—
623C8—81no—
624C8—81no—
625C8—81no—
626C8#42112yes—
627C8#56812yes—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22C22×#10211yescharacteristic
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4#287911yescharacteristic
659C4C2×#42111yescharacteristic
660C4#341611yescharacteristic
661C4#341711yescharacteristic
662C2—41no—
663C2—41no—
664C2—41no—
665C2—161no—
666C2#1748511yescharacteristic
667C2#3338211yescharacteristic
668C2#3338511yescharacteristic
subgroupG/H
centerC2×C4C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
socleC22C22×#102
prime radical#3061C4
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33420C2semidirect
3. #17799C2semidirect
4. #11676C2semidirect
5. #11674C2semidirect
6. #11673C2semidirect
7. #11672C2semidirect
8. #11662C2semidirect
9. #11660C2semidirect
10. #11659C2semidirect
11. #11658C2semidirect
12. #11650C2semidirect
13. #11648C2semidirect
14. #11647C2semidirect
15. #11646C2semidirect

Maximal subgroups

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

subgroupindex
#606962
#334202
#177992
#116762
#116742
#116732
#116722
#116622
#116602
#116592
#116582
#116502
#116482
#116472
#116462