Finite Group Databasebeta

← #6552052

Subgroups — #6552052

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38984C211yesmaximal, characteristic
2#34491C211yesmaximal, characteristic
3#30160C212yesmaximal
4#29831C211yesmaximal, characteristic
5#29549C212yesmaximal
6#29455C212yesmaximal
7#29287C211yesmaximal, characteristic
8#28653C211yesmaximal, characteristic
9#27512C212yesmaximal
10#27437C212yesmaximal
11#27128C212yesmaximal
12#27017C212yesmaximal
13#26317C211yesmaximal, characteristic
14#26283C211yesmaximal, characteristic
15#25936C211yesmaximal, characteristic
16#25837C211yesmaximal, characteristic
17#23789C212yesmaximal
18#23777C211yesmaximal, characteristic
19#23217C211yesmaximal, characteristic
20#23163C211yesmaximal, characteristic
21#23102C211yesmaximal, characteristic
22#23062C211yesmaximal, characteristic
23#22822C211yesmaximal, characteristic
24C2×Q82C2211yescharacteristic
25C2×#498C2211yescharacteristic
26C2×#487C2211yescharacteristic
27C2×#475C2211yescharacteristic
28C2×#462C2211yescharacteristic
29#2862C2211yescharacteristic
30#2767—22no—
31#2762—22no—
32#2645C2212yes—
33#2581—21no—
34#2568—21no—
35#2568C2211yescharacteristic
36#2546C2211yescharacteristic
37#2514—21no—
38#2511C2212yes—
39#2451C2212yes—
40#2446—21no—
41#2381—22no—
42#2369C2211yescharacteristic
43#2365—21no—
44#2365C2211yescharacteristic
45#2365C2212yes—
46#2365C2211yescharacteristic
47#2356—21no—
48#2356C2212yes—
49#2355C2211yescharacteristic
50#2355C2211yescharacteristic
51#2331—21no—
52#2331C2212yes—
53#2327—22no—
54#2320—21no—
55#2320C2212yes—
56#2320C2212yes—
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59#2295C2211yescharacteristic
60#2295C2211yescharacteristic
61#2101C2212yes—
62#2100C2211yescharacteristic
63#2097C2212yes—
64#2095C2211yescharacteristic
65#2088C2212yes—
66#2088C2212yes—
67#2087C2211yescharacteristic
68#2087C2211yescharacteristic
69#2085C2212yes—
70#2074C2212yes—
71#2068C2211yescharacteristic
72#2068C2211yescharacteristic
73#2066C2212yes—
74#2066C2212yes—
75#2066C2212yes—
76#2066C2212yes—
77#2058C2212yes—
78#2058C2212yes—
79#2057C2211yescharacteristic
80#2056C2212yes—
81#2056C2212yes—
82#2024C2211yescharacteristic
83#2024C2212yes—
84#2024C2212yes—
85#1990C2211yescharacteristic
86#1988C2212yes—
87#1988C2211yescharacteristic
88#1987C2211yescharacteristic
89#1987C2211yescharacteristic
90#1984C2211yescharacteristic
91#1984C2211yescharacteristic
92#1981C2211yescharacteristic
93#1971C2212yes—
94#1956C2211yescharacteristic
95#1948C2211yescharacteristic
96#1948C2211yescharacteristic
97#1948C2212yes—
98#1944C2212yes—
99#1933C2212yes—
100#1933C2212yes—
101#1928C2212yes—
102#1928C2212yes—
103#1927C2212yes—
104#1927C2212yes—
105#1927C2212yes—
106#1927C2212yes—
107#1927C2212yes—
108#1921C2212yes—
109#1920C2212yes—
110#1920C2212yes—
111#1919C2212yes—
112#1919C2212yes—
113#1917C2212yes—
114#1916C2212yes—
115#1878C2212yes—
116#1878C2212yes—
117#1878C2212yes—
118#1865C2212yes—
119#1865C2212yes—
120#1863C2212yes—
121#1842C2211yescharacteristic
122#1832C2212yes—
123#1776C2212yes—
124C4×#107C2211yescharacteristic
125C4×#107C2211yescharacteristic
126C4×#106C2212yes—
127#1770C2211yescharacteristic
128C8×C4⋊C4C2211yescharacteristic
129#1768C2212yes—
130C4×C4⋊C8C2211yescharacteristic
131#1763C2212yes—
132C4×#103C2211yescharacteristic
133C4×#103C2212yes—
134C4×#103C2211yescharacteristic
135C22×Q16C2312yes—
136Q82—24no—
137Q82—24no—
138C2×C4⋊Q8—21no—
139C2×C4⋊Q8—22no—
140C2×C4⋊Q8—21no—
141C2×C4⋊Q8C2311yescharacteristic
142C2×C4⋊Q8C2311yescharacteristic
143C2×C4⋊Q8C2312yes—
144C2×C4⋊Q8C2311yescharacteristic
145C2×C4⋊Q8C2311yescharacteristic
146C2×C4×Q8—21no—
147C2×C4×Q8—21no—
148C2×C4×Q8—22no—
149C2×C4×Q8—21no—
150C2×C4×Q8—21no—
151C2×C4×Q8C2312yes—
152C2×C4×Q8C2312yes—
153C2×C4×Q8C2311yescharacteristic
154C2×C4×Q8C2311yescharacteristic
155#498—24no—
156#487—24no—
157#475—24no—
158#462—24no—
159C2×#107C2311yescharacteristic
160C2×#107C2311yescharacteristic
161C2×#107C2311yescharacteristic
162C2×#107C2311yescharacteristic
163C2×#106C2312yes—
164C2×#106C2312yes—
165C2×#106C2312yes—
166C2×#106C2312yes—
167C2×#106C2312yes—
168C2×C4⋊C8C2311yescharacteristic
169C2×C4⋊C8C2311yescharacteristic
170C2×C4⋊C8C2311yescharacteristic
171C2×C4⋊C8C2312yes—
172C2×C4⋊C8C2312yes—
173C2×C4⋊C8C2311yescharacteristic
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2311yescharacteristic
180C2×#103C2311yescharacteristic
181C2×#103C2312yes—
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2311yescharacteristic
185C2×#103C2311yescharacteristic
186C2×#97C2312yes—
187C2×C4×C8C2311yescharacteristic
188C2×C4×C8C2311yescharacteristic
189C2×C4×C8C2312yes—
190#401—22no—
191#400—22no—
192#400—22no—
193#400—22no—
194#400—22no—
195#400—22no—
196#400—22no—
197#398—22no—
198#398—22no—
199#395—22no—
200#391—21no—
201#391—21no—
202#391—21no—
203#391—21no—
204#391—22no—
205#391—21no—
206#391—21no—
207#391—21no—
208#391—21no—
209#391—22no—
210#391—22no—
211#391—22no—
212#391—22no—
213#391D411yescharacteristic
214#391D411yescharacteristic
215#391C2311yescharacteristic
216#391C2311yescharacteristic
217#391C2312yes—
218#391C2312yes—
219#391C2312yes—
220#391C2312yes—
221#391C2312yes—
222#391C2312yes—
223#389—21no—
224#389—21no—
225#389—21no—
226#389—21no—
227#389D411yescharacteristic
228#389D411yescharacteristic
229#389C2311yescharacteristic
230#389C2312yes—
231#389C2311yescharacteristic
232#389C2311yescharacteristic
233#389C2312yes—
234#389C2312yes—
235#389C2312yes—
236#389C2311yescharacteristic
237#389C2312yes—
238#387—22no—
239#387—22no—
240#387—22no—
241#387—22no—
242#387—22no—
243#387—21no—
244#387—21no—
245#387—22no—
246#387—22no—
247#387—22no—
248#387—22no—
249#387—22no—
250#387—22no—
251#387—22no—
252#387—22no—
253#387—22no—
254#387—22no—
255#387—22no—
256#387—22no—
257#387D412yes—
258#387D412yes—
259#384C2311yescharacteristic
260#384C2311yescharacteristic
261#383—22no—
262#383—22no—
263#383—22no—
264#382—21no—
265#382—21no—
266#382—21no—
267#382—21no—
268#382—22no—
269#382—22no—
270#382C2312yes—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—21no—
274C4×C4⋊C4—21no—
275C4×C4⋊C4—21no—
276C4×C4⋊C4—21no—
277C4×C4⋊C4—21no—
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2312yes—
280C4×C4⋊C4C2311yescharacteristic
281C4×C4⋊C4C2312yes—
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2312yes—
285C4×C4⋊C4C2311yescharacteristic
286C4×C4⋊C4C2311yescharacteristic
287C4×C4⋊C4C2312yes—
288#376—21no—
289#376—22no—
290#376C2312yes—
291C43C2311yescharacteristic
292#340C2312yes—
293#340C2312yes—
294#340C2311yescharacteristic
295#340C2311yescharacteristic
296#340C2311yescharacteristic
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2312yes—
301#340C2311yescharacteristic
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#340C2312yes—
305#340C2311yescharacteristic
306#340C2311yescharacteristic
307#340C2312yes—
308#340C2312yes—
309#340C2312yes—
310#336C2311yescharacteristic
311#336C2312yes—
312#336C2312yes—
313#336C2312yes—
314#336C2312yes—
315#336C2312yes—
316#336C2311yescharacteristic
317#336C2312yes—
318C22×Q8—22no—
319C22×Q8—21no—
320C22×Q8—21no—
321C22×Q8—21no—
322C22×Q8C2×D411yescharacteristic
323C22×Q8C2412yes—
324C22×Q8C2411yescharacteristic
325C2×Q16—24no—
326C2×Q16—42no—
327C2×Q16—42no—
328C22×C8C2412yes—
329C22×C8C2411yescharacteristic
330C22×C8C2411yescharacteristic
331C22×C8C2411yescharacteristic
332C22×C8C2412yes—
333C22×C8C2411yescharacteristic
334C4⋊Q8—216no—
335C4⋊Q8—28no—
336C4⋊Q8—24no—
337C4⋊Q8—24no—
338C4⋊Q8—22no—
339C4⋊Q8—22no—
340C4⋊Q8—22no—
341C4⋊Q8—22no—
342C4⋊Q8—22no—
343C4⋊Q8—22no—
344C4⋊Q8—41no—
345C4⋊Q8—82no—
346C4⋊Q8#4114yes—
347C4×Q8—28no—
348C4×Q8—24no—
349C4×Q8—24no—
350C4×Q8—22no—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—24no—
354C4×Q8—42no—
355C4×Q8—82no—
356C4×Q8—82no—
357C4×Q8—84no—
358C4×Q8—82no—
359C4×Q8#4114yes—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4C2×D411yescharacteristic
383C2×C4⋊C4C2×D412yes—
384C2×C4⋊C4C2×D411yescharacteristic
385C2×C4⋊C4C2×D412yes—
386C2×C4⋊C4C2×D411yescharacteristic
387C2×C4⋊C4C2×D412yes—
388C2×C4⋊C4C2411yescharacteristic
389C2×C4⋊C4C2412yes—
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2412yes—
392C2×C4⋊C4C2411yescharacteristic
393C2×C4⋊C4C2411yescharacteristic
394C2×C4⋊C4C2411yescharacteristic
395C2×C4⋊C4C2412yes—
396C2×C4⋊C4C2411yescharacteristic
397C2×C4⋊C4C2411yescharacteristic
398C2×C42—21no—
399C2×C42—22no—
400C2×C42—22no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—22no—
404C2×C42—21no—
405C2×C42—22no—
406C2×C42—22no—
407C2×C42—22no—
408C2×C42—22no—
409C2×C42C2×D411yescharacteristic
410C2×C42C2×D411yescharacteristic
411C2×C42C2×D411yescharacteristic
412C2×C42C2×D411yescharacteristic
413C2×C42C2412yes—
414C2×C42C2412yes—
415C2×C42C2411yescharacteristic
416C2×C42C2411yescharacteristic
417C2×C42C2411yescharacteristic
418#107—22no—
419#107—22no—
420#107—41no—
421#107—41no—
422#106—24no—
423#106—24no—
424#106—24no—
425#106—42no—
426#106—42no—
427C4⋊C8—22no—
428C4⋊C8—22no—
429C4⋊C8—22no—
430C4⋊C8—41no—
431C4⋊C8—42no—
432C4⋊C8—42no—
433#103—24no—
434#103—22no—
435#103—24no—
436#103—24no—
437#103—24no—
438#103—24no—
439#103—22no—
440#103—42no—
441#103—42no—
442#103—42no—
443#103—41no—
444#103—41no—
445#97—42no—
446C4×C8—22no—
447C4×C8—22no—
448C4×C8—42no—
449#95—21no—
450#95—21no—
451#95—22no—
452#95—22no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—22no—
457#95—22no—
458#95—21no—
459#95—21no—
460#95—22no—
461#95—22no—
462#95—22no—
463#95—21no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—22no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—22no—
473#95—22no—
474#95—22no—
475#95—22no—
476#95—22no—
477#95—22no—
478#95—22no—
479#95—22no—
480#95—22no—
481#95—22no—
482#95—22no—
483#95—22no—
484#95C2×D412yes—
485#95C2×D412yes—
486#95C2×D412yes—
487#95C2×D412yes—
488#95C2×D412yes—
489#95C2×D412yes—
490#95C2×D412yes—
491C2×Q8—28no—
492C2×Q8—22no—
493C2×Q8—24no—
494C2×Q8—22no—
495C2×Q8—22no—
496C2×Q8—21no—
497C2×Q8—22no—
498C2×Q8—22no—
499C2×Q8—21no—
500C2×Q8—21no—
501C2×Q8—22no—
502C2×Q8—22no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—22no—
506C2×Q8—22no—
507C2×Q8—42no—
508C2×Q8—42no—
509C2×Q8—42no—
510C2×Q8—42no—
511C2×Q8—42no—
512C2×Q8—81no—
513C2×Q8—81no—
514C2×Q8—82no—
515C2×Q8—81no—
516C2×Q8—82no—
517C2×Q8C2×#4112yes—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—22no—
521C22×C4—21no—
522C22×C4—21no—
523C22×C4—22no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—22no—
527C22×C4—21no—
528C22×C4—22no—
529C22×C4—21no—
530C22×C4C22×D412yes—
531C22×C4C22×D411yescharacteristic
532C22×C4C22×D411yescharacteristic
533C22×C4C22×D411yescharacteristic
534C22×C4C22×D412yes—
535C22×C4C22×D412yes—
536C22×C4C22×D412yes—
537C22×C4C22×D411yescharacteristic
538C22×C4C22×D411yescharacteristic
539C22×C4C22×D411yescharacteristic
540C22×C4C2511yescharacteristic
541Q16—84no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—22no—
548C2×C8—22no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—22no—
554C2×C8—22no—
555C2×C8—21no—
556C2×C8—22no—
557C2×C8—21no—
558C2×C8—22no—
559C2×C82−1+412yes—
560C4⋊C4—28no—
561C4⋊C4—28no—
562C4⋊C4—24no—
563C4⋊C4—24no—
564C4⋊C4—24no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—24no—
568C4⋊C4—22no—
569C4⋊C4—22no—
570C4⋊C4—21no—
571C4⋊C4—21no—
572C4⋊C4—22no—
573C4⋊C4—22no—
574C4⋊C4—22no—
575C4⋊C4—21no—
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—21no—
579C4⋊C4—21no—
580C4⋊C4—21no—
581C4⋊C4—21no—
582C4⋊C4—22no—
583C4⋊C4—21no—
584C4⋊C4—21no—
585C4⋊C4—21no—
586C4⋊C4—42no—
587C4⋊C4—44no—
588C4⋊C4—42no—
589C4⋊C4—42no—
590C4⋊C4—42no—
591C4⋊C4—42no—
592C4⋊C4—42no—
593C4⋊C4—42no—
594C4⋊C4—42no—
595C4⋊C4—81no—
596C4⋊C4—81no—
597C4⋊C4—81no—
598C4⋊C4—81no—
599C4⋊C4—82no—
600C4⋊C4—81no—
601C4⋊C4—82no—
602C4⋊C4—82no—
603C4⋊C4—82no—
604C4⋊C4—82no—
605C4⋊C4C2×#4114yes—
606C4⋊C4C2×#4112yes—
607C4⋊C42−1+412yes—
608C42—28no—
609C42—24no—
610C42—22no—
611C42—22no—
612C42—22no—
613C42—22no—
614C42—22no—
615C42—21no—
616C42—22no—
617C42—21no—
618C42—22no—
619C42—21no—
620C42—21no—
621C42—41no—
622C42—44no—
623C42—42no—
624C42—44no—
625C42—42no—
626C42—42no—
627C42—82no—
628C42—82no—
629C42—82no—
630C42—82no—
631C42C2×#4112yes—
632C42C2×#4112yes—
633C23C23×D411yescharacteristic
634Q8—22no—
635Q8—22no—
636Q8—44no—
637Q8—41no—
638Q8—41no—
639Q8—42no—
640Q8—42no—
641Q8—42no—
642Q8—42no—
643Q8—84no—
644Q8—84no—
645Q8—88no—
646Q8—84no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—22no—
657C2×C4—22no—
658C2×C4—24no—
659C2×C4—22no—
660C2×C4—22no—
661C2×C4—21no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—21no—
673C2×C4—21no—
674C2×C4—22no—
675C2×C4—22no—
676C2×C4—42no—
677C2×C4—41no—
678C2×C4—41no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—42no—
682C2×C4—41no—
683C2×C4—41no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—41no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—41no—
691C2×C4—41no—
692C2×C4—42no—
693C2×C4—41no—
694C2×C4—41no—
695C2×C4—41no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—41no—
699C2×C4—41no—
700C2×C4—41no—
701C2×C4—41no—
702C2×C4—41no—
703C2×C4—41no—
704C2×C4—42no—
705C2×C4—41no—
706C2×C4—42no—
707C2×C4—42no—
708C2×C4#52512yes—
709C2×C4#52512yes—
710C2×C4#57712yes—
711C2×C4#57712yes—
712C2×C4#57812yes—
713C2×C4C22×#4111yescharacteristic
714C2×C4C22×#4111yescharacteristic
715C2×C4C2×2−1+411yescharacteristic
716C2×C4C2×2−1+411yescharacteristic
717C2×C4#58511yescharacteristic
718C2×C4#58511yescharacteristic
719C8—22no—
720C8—42no—
721C8—42no—
722C8—41no—
723C8—41no—
724C8—41no—
725C22C2×#52511yescharacteristic
726C22#348911yescharacteristic
727C22#349711yescharacteristic
728C22C2×#57711yescharacteristic
729C22C2×#57711yescharacteristic
730C22C2×#57811yescharacteristic
731C22#358611yescharacteristic
732C4—22no—
733C4—22no—
734C4—42no—
735C4—41no—
736C4—42no—
737C4—44no—
738C4—42no—
739C4—41no—
740C4—42no—
741C4—41no—
742C4—42no—
743C4—42no—
744C4—42no—
745C4—82no—
746C4—82no—
747C4—81no—
748C4—81no—
749C4—81no—
750C4—82no—
751C4—81no—
752C4—81no—
753C4#312114yes—
754C4#350911yescharacteristic
755C4#351011yescharacteristic
756C4#351611yescharacteristic
757C4#351811yescharacteristic
758C2#3581711yescharacteristic
759C2C2×#312111yescharacteristic
760C2#6059111yescharacteristic
761C2#6219211yescharacteristic
762C2#6220211yescharacteristic
763C2#6238211yescharacteristic
764C2#6240711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2295 ◁ 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
#389842
#344912
#301602
#298312
#295492
#294552
#292872
#286532
#275122
#274372
#271282
#270172
#263172
#262832
#259362
#258372
#237892
#237772
#232172
#231632
#231022
#230622
#228222