Finite Group Databasebeta

← #7503559

Subgroups — #7503559

Order 512.

2042 conjugacy classes of subgroups.

The subgroup lattice has 768 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3097C211yesmaximal, characteristic
2C2×#3082C211yesmaximal, characteristic, complemented
3#35667C211yesmaximal, characteristic
4#35532C211yesmaximal, characteristic
5#31122C214yesmaximal, complemented
6#31120C214yesmaximal, complemented
7#31119C214yesmaximal, complemented
8#31117C214yesmaximal, complemented
9#23658C211yesmaximal, characteristic, complemented
10#23613C211yesmaximal, characteristic, complemented
11#23507C211yesmaximal, characteristic, complemented
12#23487C211yesmaximal, characteristic
13#23484C211yesmaximal, characteristic
14#23457C211yesmaximal, characteristic, complemented
15C2×#1956C211yesmaximal, characteristic, complemented
16#21151C211yesmaximal, characteristic, complemented
17#21046C211yesmaximal, characteristic
18#20913C211yesmaximal, characteristic
19#20877C211yesmaximal, characteristic, complemented
20C2×#544C2211yescharacteristic
21C2×#536C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C22×C4⋊Q8C2211yescharacteristic
24C2×#520C2211yescharacteristic
25C2×#519C2211yescharacteristic
26#3097—28no—
27C2×#484C2212yes—
28C2×#485C2212yes—
29C2×#485C2212yes—
30C2×#483C2212yes—
31#3082—28no—
32C2×#479C2212yes—
33C2×#479C2212yes—
34C2×#477C2212yes—
35C2×#475C2212yes—
36C2×#428C2211yescharacteristic
37C2×#428C2211yescharacteristic
38C2×#427C2211yescharacteristic
39C2×#427C2211yescharacteristic
40C2×#424C2211yescharacteristic
41C2×#419C2211yescharacteristic
42C2×#419C2211yescharacteristic
43C2×#419C2211yescharacteristic
44C2×#419C2211yescharacteristic
45C22×#103C2211yescharacteristic
46C22×#103C2211yescharacteristic
47C22×#103C2211yescharacteristic
48C22×#103C2211yescharacteristic
49C2×#408C2211yescharacteristic
50C2×#408C2211yescharacteristic
51C22×C4×C8C2211yescharacteristic
52#2567—21no—
53#2566—21no—
54#2565—21no—
55#2546—21no—
56#2546—21no—
57#2545—21no—
58#2545—21no—
59#2542—21no—
60#2449—24no—
61#2441C2214yes—
62#2440C2214yes—
63#2429C2214yes—
64#2429C2214yes—
65#2413—24no—
66#2317C2211yescharacteristic
67#2311C2211yescharacteristic
68#2310C2211yescharacteristic
69#2297C2211yescharacteristic
70#2296C2211yescharacteristic
71#2296C2211yescharacteristic
72C2×#384C2211yescharacteristic
73#2084C2214yes—
74#2083C2214yes—
75#2068C2214yes—
76#2066C2214yes—
77#2043C2214yes—
78#2043C2214yes—
79#2036C2214yes—
80#2036C2214yes—
81#2035C2214yes—
82#2035C2214yes—
83#2026C2214yes—
84#2024C2214yes—
85#1956C2214yes—
86#1956C2214yes—
87#1942C2212yes—
88#1942C2212yes—
89#1941C2212yes—
90#1941C2212yes—
91#1923C2214yes—
92#1923C2214yes—
93#1873C2214yes—
94#1872C2214yes—
95#1865C2214yes—
96#1863C2214yes—
97#1736C2211yescharacteristic
98#1736C2211yescharacteristic
99#1735C2211yescharacteristic
100#1735C2211yescharacteristic
101#1729C2211yescharacteristic
102#1728C2211yescharacteristic
103C22×#41C2311yescharacteristic
104C23×Q8C2311yescharacteristic
105C23×Q8C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C22×M16C2311yescharacteristic
108C23×C8C2311yescharacteristic
109C23×C8C2311yescharacteristic
110#544—24no—
111#544—42no—
112#536—24no—
113#536—24no—
114#536—42no—
115#536—42no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—24no—
120C2×C4⋊Q8—21no—
121C2×C4⋊Q8—21no—
122C2×C4⋊Q8—42no—
123C2×C4⋊Q8C2314yes—
124C2×C4⋊Q8C2312yes—
125C2×#125—21no—
126C2×#125C2312yes—
127C2×#124—21no—
128C2×#124—21no—
129C2×#123—22no—
130C2×#123—22no—
131C2×C22⋊Q8—22no—
132C2×C22⋊Q8—24no—
133C2×C22⋊Q8C2312yes—
134C2×C22⋊Q8C2312yes—
135C2×C22⋊Q8C2314yes—
136#520—24no—
137#520—42no—
138#519—24no—
139#519—42no—
140C2×C4×Q8—21no—
141C2×C4×Q8—21no—
142C2×C4×Q8—21no—
143C2×C4×Q8C2312yes—
144C2×C4×Q8C2312yes—
145C2×C4×D4—21no—
146C2×C4×D4—22no—
147C2×#117—22no—
148C2×#117C2311yescharacteristic
149C2×#117C2311yescharacteristic
150C2×#117C2312yes—
151C2×#117C2311yescharacteristic
152C2×#117C2311yescharacteristic
153C2×#117C2311yescharacteristic
154C22×C4⋊C4—22no—
155C22×C4⋊C4—22no—
156C22×C4⋊C4C2311yescharacteristic
157C22×C4⋊C4C2311yescharacteristic
158C22×C4⋊C4C2312yes—
159C22×C42C2311yescharacteristic
160#485—28no—
161#485—28no—
162#484—28no—
163#483—28no—
164#479—28no—
165#479—28no—
166#477—28no—
167#475—28no—
168#428—24no—
169#428—24no—
170#427—42no—
171#427D418yes—
172C2×#107C2312yes—
173C2×#107C2312yes—
174C2×#107C2312yes—
175C2×#107C2311yescharacteristic
176C2×#107C2311yescharacteristic
177C2×#106C2312yes—
178C2×#106C2312yes—
179C2×#106C2312yes—
180C2×#106C2311yescharacteristic
181C2×#106C2311yescharacteristic
182#424—24no—
183C2×C4⋊C8C2312yes—
184C2×C4⋊C8C2312yes—
185#419—24no—
186#419—24no—
187#419—24no—
188#419—42no—
189C2×#103—24no—
190C2×#103—24no—
191C2×#103—42no—
192C2×#103—42no—
193C2×#103C2312yes—
194C2×#103C2312yes—
195C2×#103C2312yes—
196C2×#103C2312yes—
197C2×#103C2312yes—
198C2×#103C2312yes—
199C2×#103C2312yes—
200C2×#103C2312yes—
201C2×#103C2314yes—
202C2×#103C2314yes—
203C2×#103C2314yes—
204C2×#103C2314yes—
205#408—24no—
206#408—42no—
207C2×C22⋊C8C2312yes—
208C2×C22⋊C8C2312yes—
209C2×C22⋊C8C2314yes—
210C2×C4×C8—42no—
211C2×C4×C8C2312yes—
212C2×C4×C8C2312yes—
213#399—24no—
214#398—24no—
215#398—24no—
216#398—24no—
217#396—24no—
218#396—24no—
219#396—24no—
220#395—24no—
221#391—21no—
222#391—21no—
223#391—21no—
224#391D411yescharacteristic
225#391D411yescharacteristic
226#391C2314yes—
227#390—21no—
228#390D411yescharacteristic
229#390D411yescharacteristic
230#389—24no—
231#389—24no—
232#389—24no—
233#389C2314yes—
234#389C2312yes—
235#389C2312yes—
236#389C2312yes—
237#389C2312yes—
238#388—24no—
239#388—24no—
240#387—24no—
241#386—22no—
242#386D414yes—
243#385—22no—
244#385—22no—
245#385—24no—
246#385—22no—
247#384—21no—
248#384—21no—
249#384—21no—
250#384C2314yes—
251#382—21no—
252#381—22no—
253#380C2314yes—
254C4×C4⋊C4—21no—
255C4×C4⋊C4—21no—
256C4×C4⋊C4—21no—
257C4×C4⋊C4—21no—
258C4×C22⋊C4C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2312yes—
262#340C2312yes—
263#340C2312yes—
264#340C2312yes—
265#336C2314yes—
266#336C2314yes—
267C2×#41—22no—
268C2×#41—22no—
269C2×#41—22no—
270C2×#41—22no—
271C2×#41—22no—
272C2×#41—22no—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—22no—
276C22×Q8—22no—
277C22×Q8—22no—
278C22×Q8—22no—
279C22×Q8—22no—
280C22×Q8—22no—
281C22×Q8—22no—
282C22×Q8—22no—
283C22×Q8—22no—
284C22×Q8—22no—
285C22×Q8—22no—
286C22×Q8—22no—
287C22×Q8C2×D411yescharacteristic
288C22×Q8#4114yes—
289C22×Q8C2412yes—
290C22×Q8C2412yes—
291C22×D4C2×D412yes—
292C22×D4C2×D411yescharacteristic
293C23×C4—21no—
294C23×C4—21no—
295C23×C4—21no—
296C23×C4—21no—
297C23×C4—22no—
298C23×C4C2×D412yes—
299C23×C4C2412yes—
300C23×C4C2411yescharacteristic
301C2×M16—22no—
302C2×M16—22no—
303C2×M16—22no—
304C2×M16—22no—
305C2×M16—22no—
306C2×M16—22no—
307C22×C8—22no—
308C22×C8—22no—
309C22×C8—22no—
310C22×C8—22no—
311C22×C8—22no—
312C22×C8C2×D414yes—
313C22×C8C2412yes—
314C22×C8C2412yes—
315C22×C8C2412yes—
316C22×C8C2411yescharacteristic
317C22×C8C2411yescharacteristic
318C4⋊Q8—24no—
319C4⋊Q8—28no—
320C4⋊Q8—42no—
321C4⋊Q8—44no—
322C4⋊Q8—44no—
323C4⋊Q8—44no—
324C4⋊Q8—44no—
325C4⋊Q8—44no—
326C4⋊Q8—44no—
327#125—24no—
328#125—42no—
329#125—44no—
330#124—44no—
331#124—44no—
332#123—48no—
333#123—48no—
334C22⋊Q8—28no—
335C22⋊Q8—24no—
336C22⋊Q8—24no—
337C22⋊Q8—48no—
338C22⋊Q8—42no—
339C22⋊Q8—42no—
340C22⋊Q8—44no—
341C22⋊Q8—88no—
342C4×Q8—24no—
343C4×Q8—24no—
344C4×Q8—42no—
345C4×Q8—42no—
346C4×Q8—44no—
347C4×Q8—44no—
348C4×Q8—44no—
349C4×D4—44no—
350C4×D4—48no—
351#117—22no—
352#117—22no—
353#117—22no—
354#117—22no—
355#117—22no—
356#117—22no—
357#117—22no—
358#117—22no—
359#117—24no—
360#117—48no—
361#117—42no—
362#117C2×D414yes—
363#117C2×D414yes—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—24no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—24no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—24no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—22no—
386C2×C4⋊C4—22no—
387C2×C4⋊C4—24no—
388C2×C4⋊C4—24no—
389C2×C4⋊C4—42no—
390C2×C4⋊C4—48no—
391C2×C4⋊C4—84no—
392C2×C4⋊C4C2×D411yescharacteristic
393C2×C4⋊C4C2×D411yescharacteristic
394C2×C4⋊C4C2412yes—
395C2×C4⋊C4C2412yes—
396C2×C4⋊C4C2412yes—
397C2×C4⋊C4C2412yes—
398C2×C4⋊C4C2411yescharacteristic
399C2×C4⋊C4C2411yescharacteristic
400C2×C4⋊C4C2411yescharacteristic
401C2×C4⋊C4C2411yescharacteristic
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—22no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—22no—
409C2×C22⋊C4—24no—
410C2×C22⋊C4—24no—
411C2×C22⋊C4C2×D412yes—
412C2×C22⋊C4C2×D414yes—
413C2×C22⋊C4C2×D414yes—
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—22no—
417C2×C42—22no—
418C2×C42—22no—
419C2×C42—22no—
420C2×C42—22no—
421C2×C42—21no—
422C2×C42—21no—
423C2×C42C2×D411yescharacteristic
424C2×C42C2×D411yescharacteristic
425C2×C42C2412yes—
426C2×C42C2412yes—
427#107—24no—
428#107—24no—
429#107—42no—
430#107C2×D414yes—
431#107C2×D414yes—
432#106—24no—
433#106—24no—
434#106—42no—
435#106C2×D414yes—
436#106C2×D414yes—
437C4⋊C8—24no—
438C4⋊C8—24no—
439#103—28no—
440#103—24no—
441#103—24no—
442#103—24no—
443#103—24no—
444#103—24no—
445#103—24no—
446#103—24no—
447#103—24no—
448#103—44no—
449#103—44no—
450#103—44no—
451C22⋊C8—24no—
452C22⋊C8—24no—
453C22⋊C8—44no—
454C4×C8—42no—
455C4×C8—42no—
456#95—22no—
457#95—22no—
458#95—22no—
459#95—22no—
460#95—24no—
461#95—24no—
462#95—24no—
463#95—24no—
464#95C2×D414yes—
465#95C2×D414yes—
466C24C22×D412yes—
467C24C22×D411yescharacteristic
468#41—44no—
469#41—44no—
470#41—44no—
471#41—44no—
472C2×Q8—22no—
473C2×Q8—22no—
474C2×Q8—24no—
475C2×Q8—22no—
476C2×Q8—22no—
477C2×Q8—28no—
478C2×Q8—28no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8—22no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—22no—
485C2×Q8—22no—
486C2×Q8—22no—
487C2×Q8—22no—
488C2×Q8—22no—
489C2×Q8—44no—
490C2×Q8—42no—
491C2×Q8—42no—
492C2×Q8—44no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—42no—
496C2×Q8—44no—
497C2×Q8—44no—
498C2×Q8—44no—
499C2×Q8—44no—
500C2×Q8—44no—
501C2×Q8—84no—
502C2×Q8C2×#4114yes—
503C2×D4—24no—
504C2×D4—24no—
505C2×D4—24no—
506C2×D4—22no—
507C2×D4—22no—
508C2×D4—22no—
509C22×C4—24no—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—24no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—22no—
520C22×C4—22no—
521C22×C4—22no—
522C22×C4—22no—
523C22×C4—22no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—22no—
527C22×C4—22no—
528C22×C4—22no—
529C22×C4—44no—
530C22×C4—42no—
531C22×C4—42no—
532C22×C4—42no—
533C22×C4—42no—
534C22×C4—42no—
535C22×C4—42no—
536C22×C4—42no—
537C22×C4—82no—
538C22×C4—82no—
539C22×C4—84no—
540C22×C4C22×D412yes—
541C22×C4C22×D412yes—
542C22×C4C22×D412yes—
543C22×C4C22×D412yes—
544C22×C4C22×D412yes—
545C22×C4C22×D411yescharacteristic
546C22×C4C22×D412yes—
547C22×C4C2×#4112yes—
548C22×C4H(5,𝔽2)12yes—
549C22×C4H(5,𝔽2)12yes—
550C22×C42−1+412yes—
551C22×C42−1+412yes—
552C22×C4C2511yescharacteristic
553M16—44no—
554M16—44no—
555C2×C8—28no—
556C2×C8—21no—
557C2×C8—21no—
558C2×C8—21no—
559C2×C8—21no—
560C2×C8—22no—
561C2×C8—22no—
562C2×C8—22no—
563C2×C8—22no—
564C2×C8—22no—
565C2×C8—22no—
566C2×C8—22no—
567C2×C8—22no—
568C2×C8—22no—
569C2×C8—44no—
570C2×C8C22×D412yes—
571C2×C8C22×D412yes—
572C4⋊C4—22no—
573C4⋊C4—22no—
574C4⋊C4—21no—
575C4⋊C4—21no—
576C4⋊C4—21no—
577C4⋊C4—21no—
578C4⋊C4—22no—
579C4⋊C4—22no—
580C4⋊C4—22no—
581C4⋊C4—22no—
582C4⋊C4—22no—
583C4⋊C4—22no—
584C4⋊C4—22no—
585C4⋊C4—22no—
586C4⋊C4—44no—
587C4⋊C4—48no—
588C4⋊C4—48no—
589C4⋊C4—44no—
590C4⋊C4—44no—
591C4⋊C4—44no—
592C4⋊C4—44no—
593C4⋊C4—48no—
594C4⋊C4—44no—
595C4⋊C4—44no—
596C4⋊C4—44no—
597C4⋊C4—42no—
598C4⋊C4—42no—
599C4⋊C4—44no—
600C4⋊C4—48no—
601C4⋊C4—44no—
602C4⋊C4—44no—
603C4⋊C4—44no—
604C4⋊C4—84no—
605C4⋊C4—84no—
606C4⋊C4C22×D412yes—
607C4⋊C4C22×D412yes—
608C4⋊C4C22×D412yes—
609C4⋊C4C22×D412yes—
610C22⋊C4—24no—
611C22⋊C4—24no—
612C22⋊C4—24no—
613C22⋊C4—28no—
614C22⋊C4—28no—
615C22⋊C4—44no—
616C22⋊C4—44no—
617C22⋊C4—44no—
618C22⋊C4—48no—
619C22⋊C4—42no—
620C22⋊C4—42no—
621C22⋊C4—84no—
622C42—24no—
623C42—24no—
624C42—22no—
625C42—22no—
626C42—22no—
627C42—22no—
628C42—22no—
629C42—22no—
630C42—44no—
631C42—44no—
632C42—44no—
633C42—42no—
634C42—42no—
635C23—22no—
636C23—22no—
637C23—22no—
638C23—22no—
639C23—22no—
640C23—24no—
641C23—24no—
642C23#53612yes—
643C23D4212yes—
644C23#54612yes—
645C23C23×D411yescharacteristic
646Q8—24no—
647Q8—24no—
648Q8—42no—
649Q8—48no—
650Q8—44no—
651Q8—42no—
652Q8—42no—
653Q8—42no—
654Q8—42no—
655Q8—42no—
656Q8—84no—
657Q8—164no—
658D4—48no—
659D4—44no—
660C2×C4—216no—
661C2×C4—22no—
662C2×C4—22no—
663C2×C4—22no—
664C2×C4—22no—
665C2×C4—22no—
666C2×C4—22no—
667C2×C4—22no—
668C2×C4—22no—
669C2×C4—22no—
670C2×C4—22no—
671C2×C4—22no—
672C2×C4—22no—
673C2×C4—24no—
674C2×C4—22no—
675C2×C4—24no—
676C2×C4—24no—
677C2×C4—42no—
678C2×C4—48no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—42no—
701C2×C4—84no—
702C2×C4—164no—
703C2×C4—162no—
704C2×C4—162no—
705C2×C4#53612yes—
706C2×C4#53714yes—
707C2×C4#53914yes—
708C2×C4D4212yes—
709C2×C4#54612yes—
710C2×C4#54814yes—
711C2×C4#56214yes—
712C2×C4#57714yes—
713C2×C4#57814yes—
714C2×C4C23×D411yescharacteristic
715C2×C4C22×#4111yescharacteristic
716C2×C4C2×H(5,𝔽2)11yescharacteristic
717C2×C4C2×H(5,𝔽2)11yescharacteristic
718C2×C4C2×2−1+411yescharacteristic
719C2×C4C2×2−1+411yescharacteristic
720C8—22no—
721C8—22no—
722C8—42no—
723C8—42no—
724C8—42no—
725C22—22no—
726C22—22no—
727C22—24no—
728C22—24no—
729C22—42no—
730C22—42no—
731C22—42no—
732C22—44no—
733C22—44no—
734C22#268114yes—
735C22C2×#53611yescharacteristic
736C22C2×D4211yescharacteristic
737C22C2×#54611yescharacteristic
738C22C2×#57712yes—
739C22C2×#57812yes—
740C4—24no—
741C4—24no—
742C4—42no—
743C4—44no—
744C4—42no—
745C4—42no—
746C4—42no—
747C4—42no—
748C4—42no—
749C4—42no—
750C4—82no—
751C4—82no—
752C4—82no—
753C4—82no—
754C4—82no—
755C4#268114yes—
756C4C2×#53711yescharacteristic
757C4C2×#53911yescharacteristic
758C4C2×#54811yescharacteristic
759C4C2×#56211yescharacteristic
760C2—24no—
761C2—82no—
762C2C2×#268111yescharacteristic
763C2#6087812yes—
764C2#6108711yescharacteristic
765C2#6109111yescharacteristic
766C2#6116812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3082C2semidirect
5. #31122C2semidirect
6. #31120C2semidirect
7. #31119C2semidirect
8. #31117C2semidirect
9. #23658C2semidirect
10. #23613C2semidirect
11. #23507C2semidirect
14. #23457C2semidirect
15. C2×#1956C2semidirect
16. #21151C2semidirect
19. #20877C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30972
C2×#30822
#356672
#355322
#311222
#311202
#311192
#311172
#236582
#236132
#235072
#234872
#234842
#234572
C2×#19562
#211512
#210462
#209132
#208772