Finite Group Databasebeta

← #7541645

Subgroups — #7541645

Order 512.

1278 conjugacy classes of subgroups.

The subgroup lattice has 755 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59443C211yesmaximal, characteristic
2#36285C211yesmaximal, characteristic
3#32434C212yesmaximal
4#32429C212yesmaximal
5#29210C211yesmaximal, characteristic
6#29144C211yesmaximal, characteristic
7#27732C212yesmaximal
8#27141C212yesmaximal
9#27134C212yesmaximal
10#26320C211yesmaximal, characteristic
11#26317C211yesmaximal, characteristic
12#26095C212yesmaximal
13#25294C211yesmaximal, characteristic
14#25294C211yesmaximal, characteristic
15#24997C211yesmaximal, characteristic
16#24994C211yesmaximal, characteristic
17#24687C212yesmaximal
18#23149C211yesmaximal, characteristic
19#22809C212yesmaximal
20#22740C211yesmaximal, characteristic
21#22739C211yesmaximal, characteristic
22C4×#479C211yesmaximal, characteristic
23C4×#462C211yesmaximal, characteristic
24C2×#479C2211yescharacteristic
25C2×#462C2211yescharacteristic
26C2×C8×Q8C2211yescharacteristic
27C2×C8×Q8C2211yescharacteristic
28C2×C4×Q16C2212yes—
29#2863—21no—
30#2859C2211yescharacteristic
31#2776—22no—
32#2606—22no—
33#2590—22no—
34#2582C2211yescharacteristic
35#2582C2211yescharacteristic
36#2570—21no—
37#2570C2212yes—
38#2568C2211yescharacteristic
39#2568C2211yescharacteristic
40#2565C2212yes—
41#2565C2212yes—
42#2533—22no—
43#2514—21no—
44#2514C2212yes—
45#2511C2212yes—
46#2507—21no—
47#2446—21no—
48#2446—21no—
49#2370C2212yes—
50#2356—21no—
51#2356C2212yes—
52#2356C2212yes—
53#2355C2211yescharacteristic
54#2355C2211yescharacteristic
55Q8×C4⋊C4C2211yescharacteristic
56Q8×C4⋊C4C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×#125C2211yescharacteristic
60#2295—21no—
61C42×Q8C2211yescharacteristic
62#2101C2212yes—
63#2099C2212yes—
64#2097C2212yes—
65#2096C2212yes—
66#2074C2212yes—
67#2074C2212yes—
68#2072C2212yes—
69#2072C2212yes—
70#2068C2212yes—
71#2068C2212yes—
72#2059C2212yes—
73#2059C2212yes—
74#2056C2212yes—
75#2056C2212yes—
76#2026C2212yes—
77#2026C2212yes—
78#1991C2212yes—
79#1989C2211yescharacteristic
80#1984C2211yescharacteristic
81#1984C2211yescharacteristic
82#1981C2211yescharacteristic
83#1969C2212yes—
84#1969C2212yes—
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1949C2211yescharacteristic
88#1949C2212yes—
89#1949C2212yes—
90#1948C2211yescharacteristic
91#1948C2211yescharacteristic
92#1947C2212yes—
93#1947C2211yescharacteristic
94#1947C2211yescharacteristic
95#1947C2212yes—
96#1947C2211yescharacteristic
97#1939C2212yes—
98#1933C2212yes—
99#1933C2212yes—
100#1928C2212yes—
101#1928C2212yes—
102#1927C2212yes—
103#1927C2212yes—
104#1927C2212yes—
105#1927C2212yes—
106#1927C2212yes—
107#1920C2211yescharacteristic
108#1920C2212yes—
109#1919C2212yes—
110#1918C2211yescharacteristic
111#1916C2212yes—
112#1916C2212yes—
113#1878C2212yes—
114#1865C2211yescharacteristic
115#1863C2211yescharacteristic
116#1846C2212yes—
117#1835C2212yes—
118#1834C2212yes—
119#1834C2211yescharacteristic
120#1834C2211yescharacteristic
121#1805C2211yescharacteristic
122#1805C2212yes—
123#1805C2212yes—
124#1803C2211yescharacteristic
125C4×#107C2212yes—
126C4×#107C2212yes—
127C4×#107C2211yescharacteristic
128C4×#106C2211yescharacteristic
129C8×C4⋊C4C2212yes—
130C4×C4⋊C8C2211yescharacteristic
131C4×C4⋊C8C2211yescharacteristic
132C4×#103C2212yes—
133C4×#103C2211yescharacteristic
134C4×#103C2211yescharacteristic
135C22×Q16C2312yes—
136C2×C4⋊Q8D411yescharacteristic
137C2×C4⋊Q8C2311yescharacteristic
138C2×#125C2311yescharacteristic
139C2×#125C2312yes—
140C2×C4×Q8—21no—
141C2×C4×Q8—21no—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2311yescharacteristic
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×Q8C2311yescharacteristic
147C2×C4×Q8C2311yescharacteristic
148#479—24no—
149#462—24no—
150C8×Q8—24no—
151C8×Q8—24no—
152C4×Q16—28no—
153C2×#107C2311yescharacteristic
154C2×#107C2312yes—
155C2×#107C2312yes—
156C2×#107C2312yes—
157C2×#107C2311yescharacteristic
158C2×#106C2311yescharacteristic
159C2×#106C2312yes—
160C2×#106C2312yes—
161C2×#106C2311yescharacteristic
162C2×C4⋊C8C2311yescharacteristic
163C2×C4⋊C8C2312yes—
164C2×C4⋊C8C2312yes—
165C2×C4⋊C8C2311yescharacteristic
166C2×C4⋊C8C2311yescharacteristic
167C2×C4⋊C8C2311yescharacteristic
168C2×#103C2311yescharacteristic
169C2×#103C2311yescharacteristic
170C2×#103C2312yes—
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#103C2312yes—
174C2×#103C2312yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2311yescharacteristic
178C2×#103C2311yescharacteristic
179C2×#103C2312yes—
180C2×C4×C8C2312yes—
181C2×C4×C8C2311yescharacteristic
182C2×C4×C8C2311yescharacteristic
183C2×C4×C8C2312yes—
184C2×C4×C8C2312yes—
185#401—22no—
186#400—22no—
187#400—22no—
188#400—22no—
189#400—22no—
190#400—22no—
191#400—22no—
192#398—22no—
193#398—22no—
194#398—22no—
195#398—22no—
196#398—22no—
197#398—22no—
198#398—22no—
199#391—21no—
200#391—21no—
201#391—21no—
202#391—21no—
203#391—21no—
204#391—21no—
205#391D411yescharacteristic
206#391D411yescharacteristic
207#391C2312yes—
208#391C2312yes—
209#391C2312yes—
210#391C2312yes—
211#391C2312yes—
212#391C2312yes—
213#391C2312yes—
214#389—21no—
215#389—21no—
216#389—21no—
217#389—22no—
218#389—21no—
219#389—22no—
220#389—22no—
221#389—21no—
222#389—21no—
223#389—21no—
224#389—21no—
225#389—21no—
226#389—21no—
227#389D412yes—
228#389D412yes—
229#389C2311yescharacteristic
230#389C2311yescharacteristic
231#389C2312yes—
232#389C2312yes—
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—21no—
243#387—21no—
244#387—22no—
245#387—22no—
246#387—22no—
247#387—22no—
248#387—22no—
249#387—22no—
250#387—22no—
251#387—22no—
252#384D411yescharacteristic
253#384C2311yescharacteristic
254#384C2312yes—
255#384C2312yes—
256#384C2311yescharacteristic
257#384C2311yescharacteristic
258#383—22no—
259#383—22no—
260#383—22no—
261#382—21no—
262#382—21no—
263#382C2311yescharacteristic
264#382C2312yes—
265#382C2312yes—
266#382C2312yes—
267C4×C4⋊C4—21no—
268C4×C4⋊C4—21no—
269C4×C4⋊C4—21no—
270C4×C4⋊C4—21no—
271C4×C4⋊C4—21no—
272C4×C4⋊C4—21no—
273C4×C4⋊C4—21no—
274C4×C4⋊C4—21no—
275C4×C4⋊C4—21no—
276C4×C4⋊C4—22no—
277C4×C4⋊C4C2311yescharacteristic
278C4×C4⋊C4C2311yescharacteristic
279C4×C4⋊C4C2311yescharacteristic
280C4×C4⋊C4C2311yescharacteristic
281C4×C4⋊C4C2312yes—
282C4×C4⋊C4C2312yes—
283C4×C4⋊C4C2312yes—
284C4×C4⋊C4C2312yes—
285C4×C4⋊C4C2311yescharacteristic
286C4×C4⋊C4C2312yes—
287C4×C4⋊C4C2312yes—
288#376—22no—
289#376—22no—
290C43—21no—
291C43C2311yescharacteristic
292#340C2312yes—
293#340C2312yes—
294#340C2312yes—
295#340C2312yes—
296#340C2312yes—
297#340C2312yes—
298#340C2312yes—
299#340C2312yes—
300#340C2312yes—
301#340C2311yescharacteristic
302#340C2311yescharacteristic
303#340C2311yescharacteristic
304#340C2311yescharacteristic
305#340C2311yescharacteristic
306#340C2311yescharacteristic
307#340C2312yes—
308#340C2311yescharacteristic
309#340C2311yescharacteristic
310#336C2312yes—
311#336C2312yes—
312#336C2312yes—
313#336C2312yes—
314#336C2312yes—
315#336C2311yescharacteristic
316#336C2311yescharacteristic
317C22×Q8C2×D411yescharacteristic
318C22×Q8C2×D411yescharacteristic
319C22×Q8C2412yes—
320C22×Q8C2411yescharacteristic
321C2×Q16—24no—
322C2×Q16—24no—
323C2×Q16—42no—
324C22×C8C2412yes—
325C22×C8C2411yescharacteristic
326C22×C8C2411yescharacteristic
327C22×C8C2411yescharacteristic
328C22×C8C2411yescharacteristic
329C22×C8C2412yes—
330C4⋊Q8—24no—
331C4⋊Q8—22no—
332C4⋊Q8—22no—
333#125—22no—
334#125—22no—
335#125—42no—
336#125—42no—
337C4×Q8—22no—
338C4×Q8—22no—
339C4×Q8—22no—
340C4×Q8—22no—
341C4×Q8—22no—
342C4×Q8—24no—
343C4×Q8—22no—
344C4×Q8—22no—
345C4×Q8—24no—
346C4×Q8—41no—
347C4×Q8—41no—
348C4×Q8—44no—
349C4×Q8—44no—
350C4×Q8#4114yes—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4C2×D411yescharacteristic
376C2×C4⋊C4C2×D411yescharacteristic
377C2×C4⋊C4C2×D412yes—
378C2×C4⋊C4C2×D412yes—
379C2×C4⋊C4C2×D412yes—
380C2×C4⋊C4C2×D412yes—
381C2×C4⋊C4C2×D412yes—
382C2×C4⋊C4C2×D412yes—
383C2×C4⋊C4C2412yes—
384C2×C4⋊C4C2411yescharacteristic
385C2×C4⋊C4C2411yescharacteristic
386C2×C4⋊C4C2411yescharacteristic
387C2×C4⋊C4C2411yescharacteristic
388C2×C4⋊C4C2412yes—
389C2×C4⋊C4C2412yes—
390C2×C4⋊C4C2411yescharacteristic
391C2×C4⋊C4C2411yescharacteristic
392C2×C4⋊C4C2411yescharacteristic
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—22no—
396C2×C42—22no—
397C2×C42—21no—
398C2×C42—21no—
399C2×C42—21no—
400C2×C42—22no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42—21no—
404C2×C42—22no—
405C2×C42—22no—
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D411yescharacteristic
409C2×C42C2×D411yescharacteristic
410C2×C42C2411yescharacteristic
411C2×C42C2412yes—
412C2×C42C2412yes—
413C2×C42C2411yescharacteristic
414C2×C42C2411yescharacteristic
415#107—24no—
416#107—22no—
417#107—42no—
418#107—42no—
419#107—41no—
420#106—22no—
421#106—42no—
422#106—42no—
423#106—41no—
424C4⋊C8—24no—
425C4⋊C8—24no—
426C4⋊C8—22no—
427C4⋊C8—22no—
428C4⋊C8—22no—
429C4⋊C8#4114yes—
430#103—24no—
431#103—22no—
432#103—22no—
433#103—24no—
434#103—24no—
435#103—42no—
436#103—41no—
437#103—41no—
438#103—42no—
439#103—42no—
440#103—42no—
441#103—42no—
442C4×C8—24no—
443C4×C8—24no—
444C4×C8—24no—
445C4×C8—22no—
446C4×C8—22no—
447#95—21no—
448#95—22no—
449#95—22no—
450#95—22no—
451#95—22no—
452#95—22no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—22no—
457#95—21no—
458#95—22no—
459#95—21no—
460#95—22no—
461#95—21no—
462#95—22no—
463#95—22no—
464#95—22no—
465#95—22no—
466#95—22no—
467#95—22no—
468#95—21no—
469#95—21no—
470#95—22no—
471#95—22no—
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#95C2×D412yes—
483#95C2×D412yes—
484#95C2×D412yes—
485#95C2×D412yes—
486C2×Q8—22no—
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8—22no—
490C2×Q8—22no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—22no—
494C2×Q8—22no—
495C2×Q8—21no—
496C2×Q8—22no—
497C2×Q8—22no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—22no—
501C2×Q8—22no—
502C2×Q8#13514yes—
503C2×Q8C2×#4112yes—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—21no—
512C22×C4—22no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—22no—
516C22×C4C22×D412yes—
517C22×C4C22×D412yes—
518C22×C4C22×D412yes—
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D412yes—
525C22×C4C22×D411yescharacteristic
526C22×C4C2511yescharacteristic
527Q16—48no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—22no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—22no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—22no—
545C2×C8C2×#4114yes—
546C4⋊C4—24no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—24no—
550C4⋊C4—24no—
551C4⋊C4—24no—
552C4⋊C4—24no—
553C4⋊C4—21no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—22no—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—22no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—22no—
566C4⋊C4—22no—
567C4⋊C4—22no—
568C4⋊C4—22no—
569C4⋊C4—21no—
570C4⋊C4—21no—
571C4⋊C4—21no—
572C4⋊C4—42no—
573C4⋊C4—42no—
574C4⋊C4—42no—
575C4⋊C4—42no—
576C4⋊C4—44no—
577C4⋊C4—44no—
578C4⋊C4—42no—
579C4⋊C4—42no—
580C4⋊C4—42no—
581C4⋊C4—44no—
582C4⋊C4—42no—
583C4⋊C4—42no—
584C4⋊C4—44no—
585C4⋊C4—44no—
586C4⋊C4—42no—
587C4⋊C4—42no—
588C4⋊C4—42no—
589C4⋊C4—81no—
590C4⋊C4—81no—
591C4⋊C4—81no—
592C4⋊C4—81no—
593C4⋊C4—82no—
594C4⋊C4—82no—
595C4⋊C4—82no—
596C4⋊C4—82no—
597C4⋊C4C2×#4112yes—
598C42—24no—
599C42—22no—
600C42—22no—
601C42—22no—
602C42—22no—
603C42—22no—
604C42—22no—
605C42—21no—
606C42—21no—
607C42—41no—
608C42—41no—
609C42—42no—
610C42—42no—
611C42—44no—
612C42—44no—
613C42—42no—
614C42—44no—
615C42—42no—
616C42—42no—
617C42—81no—
618C42—81no—
619C42—82no—
620C42—82no—
621C42C2×#4112yes—
622C42C2×#4112yes—
623C422−1+412yes—
624C422−1+412yes—
625C23C23×D411yescharacteristic
626Q8—28no—
627Q8—22no—
628Q8—22no—
629Q8—44no—
630Q8—41no—
631Q8—41no—
632Q8—42no—
633Q8—42no—
634Q8—42no—
635Q8—42no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—22no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—22no—
644C2×C4—21no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—21no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—22no—
659C2×C4—21no—
660C2×C4—22no—
661C2×C4—41no—
662C2×C4—41no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—41no—
670C2×C4—41no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—41no—
674C2×C4—42no—
675C2×C4—41no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—41no—
681C2×C4—41no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—41no—
685C2×C4—41no—
686C2×C4—42no—
687C2×C4—41no—
688C2×C4—42no—
689C2×C4—41no—
690C2×C4—41no—
691C2×C4—42no—
692C2×C4—41no—
693C2×C4—41no—
694C2×C4—42no—
695C2×C4#54712yes—
696C2×C4#54712yes—
697C2×C4#54812yes—
698C2×C4#54812yes—
699C2×C4C2×#13514yes—
700C2×C4C2×#13512yes—
701C2×C4#57812yes—
702C2×C4C22×#4111yescharacteristic
703C2×C4C22×#4111yescharacteristic
704C2×C4C2×2−1+411yescharacteristic
705C2×C4C2×2−1+411yescharacteristic
706C2×C4#58511yescharacteristic
707C2×C4#58511yescharacteristic
708C8—24no—
709C8—41no—
710C8—41no—
711C8—42no—
712C8—41no—
713C8—41no—
714C22C2×#54711yescharacteristic
715C22C2×#54811yescharacteristic
716C22#348711yescharacteristic
717C22C22×#13511yescharacteristic
718C22C2×#57811yescharacteristic
719C22#358411yescharacteristic
720C22#358611yescharacteristic
721C4—24no—
722C4—22no—
723C4—22no—
724C4—22no—
725C4—42no—
726C4—42no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—41no—
731C4—41no—
732C4—81no—
733C4—82no—
734C4—81no—
735C4—81no—
736C4—82no—
737C4—81no—
738C4—81no—
739C4—82no—
740C4—81no—
741C4—82no—
742C4#333614yes—
743C4#347511yescharacteristic
744C4#350711yescharacteristic
745C4#351811yescharacteristic
746C4#355511yescharacteristic
747C2#5906511yescharacteristic
748C2C2×#333611yescharacteristic
749C2#6128011yescharacteristic
750C2#6130311yescharacteristic
751C2#6134211yescharacteristic
752C2#6188711yescharacteristic
753C2#6192911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ 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
#594432
#362852
#324342
#324292
#292102
#291442
#277322
#271412
#271342
#263202
#263172
#260952
#252942
#249972
#249942
#246872
#231492
#228092
#227402
#227392
C4×#4792
C4×#4622