Finite Group Databasebeta

← #7620467

Subgroups — #7620467

Order 512.

1235 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#45179C212yesmaximal
2#31980C212yesmaximal
3#31825C212yesmaximal
4#29550C212yesmaximal
5#28015C211yesmaximal, characteristic
6#28012C211yesmaximal, characteristic
7#27995C211yesmaximal, characteristic
8#27992C211yesmaximal, characteristic
9#27580C212yesmaximal
10#27444C212yesmaximal
11#26772C212yesmaximal
12#26714C212yesmaximal
13#26483C211yesmaximal, characteristic
14#26483C211yesmaximal, characteristic
15#26255C211yesmaximal, characteristic
16#26255C211yesmaximal, characteristic
17#23184C211yesmaximal, characteristic
18#23160C211yesmaximal, characteristic
19#22808C211yesmaximal, characteristic
20#22685C211yesmaximal, characteristic
21#21676C211yesmaximal, characteristic
22#21281C211yesmaximal, characteristic
23#21281C211yesmaximal, characteristic
24C2×#499C2211yescharacteristic
25C2×#494C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27C2×C4×Q16C2211yescharacteristic
28C2×C4×Q16C2211yescharacteristic
29C2×C4×Q16C2211yescharacteristic
30#2776—22no—
31#2767—22no—
32#2767—22no—
33#2766—22no—
34#2590—22no—
35#2524—22no—
36#2524—22no—
37#2514C2212yes—
38#2511C2212yes—
39#2508—22no—
40#2508C2212yes—
41#2507C2212yes—
42#2453C2212yes—
43#2453C2212yes—
44#2446C2212yes—
45#2446C2212yes—
46#2402C2211yescharacteristic
47#2370C2212yes—
48#2365C2212yes—
49#2355C2212yes—
50#2351C2212yes—
51#2320C2212yes—
52#2320C2212yes—
53#2101C2212yes—
54#2100C2212yes—
55#2088C2212yes—
56#2087C2212yes—
57#2074C2212yes—
58#2072C2212yes—
59#2068C2212yes—
60#2066C2212yes—
61#2066C2212yes—
62#2066C2212yes—
63#2056C2212yes—
64#2056C2212yes—
65#2043C2212yes—
66#2043C2212yes—
67#2026C2212yes—
68#2026C2212yes—
69#1989C2212yes—
70#1981C2212yes—
71#1969C2211yescharacteristic
72#1969C2211yescharacteristic
73#1969C2211yescharacteristic
74#1969C2211yescharacteristic
75#1969C2211yescharacteristic
76#1969C2211yescharacteristic
77#1969C2211yescharacteristic
78#1969C2211yescharacteristic
79#1942C2211yescharacteristic
80#1942C2211yescharacteristic
81#1942C2211yescharacteristic
82#1942C2211yescharacteristic
83#1942C2211yescharacteristic
84#1942C2211yescharacteristic
85#1942C2211yescharacteristic
86#1942C2211yescharacteristic
87#1940C2211yescharacteristic
88#1940C2211yescharacteristic
89#1935C2212yes—
90#1935C2212yes—
91#1935C2212yes—
92#1935C2212yes—
93#1928C2212yes—
94#1928C2212yes—
95#1928C2212yes—
96#1927C2212yes—
97#1921C2212yes—
98#1920C2212yes—
99#1920C2212yes—
100#1919C2212yes—
101#1917C2211yescharacteristic
102#1917C2211yescharacteristic
103#1917C2211yescharacteristic
104#1917C2211yescharacteristic
105#1878C2212yes—
106#1878C2212yes—
107#1878C2212yes—
108#1878C2212yes—
109#1865C2212yes—
110#1865C2212yes—
111#1865C2212yes—
112#1865C2212yes—
113#1846C2211yescharacteristic
114#1845C2211yescharacteristic
115#1832C2211yescharacteristic
116#1805C2212yes—
117#1803C2212yes—
118C4×#107C2211yescharacteristic
119C4×#107C2211yescharacteristic
120C4×#107C2211yescharacteristic
121C4×#107C2211yescharacteristic
122C8×C4⋊C4C2211yescharacteristic
123C8×C4⋊C4C2211yescharacteristic
124C8×C4⋊C4C2211yescharacteristic
125C8×C4⋊C4C2211yescharacteristic
126C8×C4⋊C4C2211yescharacteristic
127C8×C4⋊C4C2211yescharacteristic
128#1763C2212yes—
129#1763C2212yes—
130#1726C2211yescharacteristic
131C22×Q16C2311yescharacteristic
132C22×Q16C2311yescharacteristic
133C22×Q16C2312yes—
134C2×C4⋊Q8C2312yes—
135C2×#125C2312yes—
136C2×C4×Q8C2312yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2312yes—
140#499—24no—
141#494—24no—
142C4×Q16—24no—
143C4×Q16—24no—
144C4×Q16—24no—
145C4×Q16—24no—
146C2×#107C2311yescharacteristic
147C2×#107C2311yescharacteristic
148C2×#107C2311yescharacteristic
149C2×#107C2311yescharacteristic
150C2×#107C2311yescharacteristic
151C2×#107C2312yes—
152C2×#107C2311yescharacteristic
153C2×#106C2312yes—
154C2×#106C2312yes—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2311yescharacteristic
157C2×C4⋊C8C2311yescharacteristic
158C2×C4⋊C8C2311yescharacteristic
159C2×C4⋊C8C2311yescharacteristic
160C2×C4⋊C8C2311yescharacteristic
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2312yes—
168C2×#103C2312yes—
169C2×#103C2312yes—
170C2×#103C2312yes—
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#97C2311yescharacteristic
174C2×#97C2311yescharacteristic
175C2×#97C2311yescharacteristic
176C2×C4×C8C2311yescharacteristic
177C2×C4×C8C2311yescharacteristic
178C2×C4×C8C2311yescharacteristic
179C2×C4×C8C2311yescharacteristic
180C2×C4×C8C2311yescharacteristic
181C2×C4×C8C2311yescharacteristic
182C2×C4×C8C2311yescharacteristic
183#401—22no—
184#400—22no—
185#400—22no—
186#400—22no—
187#400—22no—
188#400—22no—
189#400—22no—
190#400—22no—
191#400—22no—
192#400D411yescharacteristic
193#400D411yescharacteristic
194#400D411yescharacteristic
195#400D411yescharacteristic
196#398—22no—
197#398—22no—
198#398—22no—
199#398—22no—
200#398—22no—
201#398—22no—
202#398—22no—
203#398D411yescharacteristic
204#398D411yescharacteristic
205#398D411yescharacteristic
206#398D411yescharacteristic
207#391C2312yes—
208#391C2312yes—
209#391C2312yes—
210#391C2312yes—
211#391C2312yes—
212#391C2312yes—
213#391C2312yes—
214#391C2312yes—
215#389—22no—
216#389—22no—
217#389—22no—
218#389—22no—
219#389—22no—
220#389—22no—
221#389—22no—
222#389C2312yes—
223#389C2312yes—
224#389C2312yes—
225#389C2312yes—
226#389C2312yes—
227#389C2312yes—
228#389C2312yes—
229#389C2312yes—
230#387—22no—
231#387—22no—
232#387—22no—
233#387—22no—
234#387—22no—
235#387—22no—
236#387—22no—
237#387—22no—
238#387—22no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—22no—
244#387—22no—
245#387—22no—
246#387—22no—
247#387—22no—
248#387—22no—
249#387—22no—
250#387—22no—
251#384C2312yes—
252#383—22no—
253#383—22no—
254#383—22no—
255#382—22no—
256#382—22no—
257#382—22no—
258#382—22no—
259#382—22no—
260#382C2312yes—
261C4×C4⋊C4—22no—
262C4×C4⋊C4—22no—
263C4×C4⋊C4—22no—
264C4×C4⋊C4—22no—
265C4×C4⋊C4C2312yes—
266C4×C4⋊C4C2312yes—
267C4×C4⋊C4C2312yes—
268C4×C4⋊C4C2312yes—
269C4×C4⋊C4C2311yescharacteristic
270C4×C4⋊C4C2311yescharacteristic
271C4×C4⋊C4C2311yescharacteristic
272C4×C4⋊C4C2311yescharacteristic
273C4×C4⋊C4C2311yescharacteristic
274C4×C4⋊C4C2311yescharacteristic
275#376C2311yescharacteristic
276#340C2312yes—
277#340C2312yes—
278#340C2312yes—
279#340C2312yes—
280#340C2312yes—
281#340C2312yes—
282#340C2312yes—
283#340C2312yes—
284#340C2312yes—
285#340C2312yes—
286#340C2312yes—
287#340C2312yes—
288#336C2311yescharacteristic
289#336C2311yescharacteristic
290#336C2311yescharacteristic
291#336C2311yescharacteristic
292#336C2311yescharacteristic
293#336C2311yescharacteristic
294#336C2311yescharacteristic
295#336C2311yescharacteristic
296#336C2311yescharacteristic
297#336C2311yescharacteristic
298#336C2311yescharacteristic
299#336C2311yescharacteristic
300C22×Q8C2412yes—
301C22×Q8C2412yes—
302C2×Q16—24no—
303C2×Q16—22no—
304C2×Q16—22no—
305C2×Q16—22no—
306C2×Q16—22no—
307C2×Q16—22no—
308C2×Q16—42no—
309C2×Q16—42no—
310C2×Q16#4114yes—
311C22×C8C2411yescharacteristic
312C22×C8C2411yescharacteristic
313C22×C8C2411yescharacteristic
314C22×C8C2411yescharacteristic
315C22×C8C2411yescharacteristic
316C22×C8C2411yescharacteristic
317C22×C8C2411yescharacteristic
318C22×C8C2411yescharacteristic
319C4⋊Q8—24no—
320C4⋊Q8—42no—
321#125—24no—
322#125—42no—
323C4×Q8—24no—
324C4×Q8—24no—
325C4×Q8—24no—
326C4×Q8—24no—
327C4×Q8—42no—
328C4×Q8—42no—
329C4×Q8—42no—
330C4×Q8—42no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
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—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4C2×D411yescharacteristic
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382#107—22no—
383#107—22no—
384#107—22no—
385#107—22no—
386#107—22no—
387#107—42no—
388#107#4114yes—
389#106—24no—
390#106—42no—
391C4⋊C8—41no—
392C4⋊C8—41no—
393C4⋊C8—41no—
394C4⋊C8—41no—
395C4⋊C8—41no—
396C4⋊C8—41no—
397#103—24no—
398#103—24no—
399#103—24no—
400#103—24no—
401#103—42no—
402#103—42no—
403#103—42no—
404#103—42no—
405#103—42no—
406#103—42no—
407#103—42no—
408#103—42no—
409#97—41no—
410#97—41no—
411#97—41no—
412C4×C8—22no—
413C4×C8—22no—
414C4×C8—22no—
415C4×C8—22no—
416C4×C8—22no—
417C4×C8—22no—
418C4×C8—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446#95—22no—
447#95—22no—
448#95—22no—
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453#95C2×D411yescharacteristic
454#95C2×D411yescharacteristic
455#95C2×D411yescharacteristic
456#95C2×D411yescharacteristic
457#95C2×D411yescharacteristic
458#95C2×D411yescharacteristic
459#95C2×D411yescharacteristic
460#95C2×D411yescharacteristic
461#95C2×D411yescharacteristic
462#95C2×D411yescharacteristic
463#95C2×D411yescharacteristic
464#95C2×D411yescharacteristic
465C2×Q8—22no—
466C2×Q8—22no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—22no—
470C2×Q8—22no—
471C2×Q8—22no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—22no—
475C2×Q8—22no—
476C2×Q8C2×#4114yes—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C2511yescharacteristic
500Q16—28no—
501Q16—44no—
502Q16—84no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8C2×#4112yes—
525C2×C8C2×#4112yes—
526C2×C82−1+412yes—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—41no—
547C4⋊C4—41no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4C2×#4114yes—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—21no—
573C42—21no—
574C42—21no—
575C42—21no—
576C42—21no—
577C42—21no—
578C42—21no—
579C42—21no—
580C42—21no—
581C42—21no—
582C42—21no—
583C42—44no—
584C42—44no—
585C42—44no—
586C42—44no—
587C42—82no—
588C42—82no—
589C42—82no—
590C42—82no—
591C23C23×D411yescharacteristic
592Q8—24no—
593Q8—24no—
594Q8—42no—
595Q8—42no—
596Q8—42no—
597Q8—42no—
598Q8—42no—
599Q8—42no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4#57812yes—
665C2×C4#57812yes—
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C2×2−1+411yescharacteristic
669C2×C4#58511yescharacteristic
670C2×C4#58511yescharacteristic
671C2×C4#58511yescharacteristic
672C8—41no—
673C8—41no—
674C8—41no—
675C8—41no—
676C8—41no—
677C8—41no—
678C8—41no—
679C8#55214yes—
680C22D4×#4111yescharacteristic
681C22D4×#4111yescharacteristic
682C22#349711yescharacteristic
683C22C2×#57811yescharacteristic
684C22C2×#57811yescharacteristic
685C22#358411yescharacteristic
686C22#358411yescharacteristic
687C4—22no—
688C4—22no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—41no—
696C4—44no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—44no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4C2×#55211yescharacteristic
710C4#352111yescharacteristic
711C4#352411yescharacteristic
712C4#355711yescharacteristic
713C2#5984211yescharacteristic
714C2#6147311yescharacteristic
715C2#6147311yescharacteristic
716C2#6147711yescharacteristic
717C2#6147811yescharacteristic
718C2#6237411yescharacteristic
719C2#6237711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2402 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#451792
#319802
#318252
#295502
#280152
#280122
#279952
#279922
#275802
#274442
#267722
#267142
#264832
#262552
#231842
#231602
#228082
#226852
#216762
#212812