Finite Group Databasebeta

← #214716

Subgroups — #214716

Order 512.

1056 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35836C211yesmaximal, characteristic
2#20633C211yesmaximal, characteristic
3#17350C211yesmaximal, characteristic
4#9469C212yesmaximal, complemented
5#9383C212yesmaximal, complemented
6#9199C212yesmaximal, complemented
7#9021C212yesmaximal, complemented
8#8761C212yesmaximal, complemented
9#8380C212yesmaximal, complemented
10C2×#554C2211yescharacteristic
11C2×#518C2211yescharacteristic
12#2972C2212yes—
13C22×#99—21no—
14#2628C411yescharacteristic, complemented
15#2626C411yescharacteristic, complemented
16#2626C2211yescharacteristic
17#2580C411yescharacteristic, complemented
18#2564C411yescharacteristic, complemented
19#2564C2211yescharacteristic
20#2542C411yescharacteristic, complemented
21#2487C411yescharacteristic, complemented
22#2487C2211yescharacteristic
23#2454—21no—
24#2449—21no—
25#2449—21no—
26#2441—21no—
27#2437—21no—
28#2437—21no—
29#2396C411yescharacteristic, complemented
30C2×#399C411yescharacteristic, complemented
31C2×#399C2211yescharacteristic
32#2349C2211yescharacteristic
33D4×C4⋊C4—21no—
34D4×C22⋊C4—21no—
35#2315C2212yes—
36#1824C2214yescomplemented
37#1799—21no—
38#1799—21no—
39#1799C2212yes—
40#1799C2212yes—
41C2×#342—21no—
42#1520C2214yescomplemented
43#1460—21no—
44#1459—21no—
45#1459C2212yes—
46#1458—21no—
47C2×#323—21no—
48C2×#323C2212yes—
49#1294C2214yescomplemented
50#1289C2214yescomplemented
51C23×D4D411yescharacteristic
52#554—44no—
53C2×C4⋊Q8C2×C411yescharacteristic, complemented
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C411yescharacteristic, complemented
60C2×C22⋊Q8C2311yescharacteristic
61#518—44no—
62C2×C4×Q8C2×C411yescharacteristic, complemented
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2×C411yescharacteristic, complemented
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—22no—
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4C2311yescharacteristic
86C8×D4—44no—
87#431—22no—
88C2×#99—22no—
89C2×#99—22no—
90C2×#99—44no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8C2314yes—
93#400C2×C411yescharacteristic, complemented
94#400C2×C411yescharacteristic, complemented
95#400C2×C411yescharacteristic, complemented
96#399—21no—
97#399—21no—
98#399—21no—
99#399—21no—
100#399C2×C411yescharacteristic
101#399C2×C411yescharacteristic
102#399C2×C411yescharacteristic
103#399C2×C411yescharacteristic
104#399C2×C411yescharacteristic
105#399C2×C411yescharacteristic
106#399C2×C411yescharacteristic, complemented
107#399C2×C411yescharacteristic, complemented
108#399C2×C411yescharacteristic, complemented
109#398—21no—
110#396—21no—
111#396—21no—
112#396—21no—
113#396—21no—
114#396—21no—
115#395—21no—
116#395C2×C411yescharacteristic, complemented
117#393—21no—
118#393C2×C411yescharacteristic
119#393C2×C411yescharacteristic
120#393C2×C411yescharacteristic, complemented
121#390—21no—
122#390—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#387—21no—
129#387—21no—
130#386—21no—
131#386—21no—
132#386—21no—
133#386—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#385—44no—
145#384—21no—
146#382C2×C411yescharacteristic, complemented
147#380—21no—
148#380—21no—
149#380—21no—
150#380—21no—
151#380—21no—
152#380—22no—
153#380C2×C411yescharacteristic, complemented
154#380C2314yes—
155#380C2311yescharacteristic
156#379—21no—
157#379—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4C2×C411yescharacteristic, complemented
160C4×C22⋊C4—21no—
161C2×#95C2×C411yescharacteristic, complemented
162C2×#95C2311yescharacteristic
163#342—22no—
164#342—22no—
165#340—44no—
166#328—44no—
167#324—22no—
168#323—22no—
169#323—22no—
170#323—22no—
171C25C22⋊C411yescharacteristic
172C25C2×D411yescharacteristic
173C22×Q8—21no—
174C22×Q8C22×C411yescharacteristic
175C22×D4—42no—
176C22×D4—42no—
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22⋊C411yescharacteristic
179C22×D4SD1614yes—
180C22×D4Q1614yes—
181C22×D4C2×D411yescharacteristic
182C22×D4C2×D411yescharacteristic
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—42no—
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22⋊C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2411yescharacteristic
197C22×C8—24no—
198C22×C8—42no—
199C4⋊Q8—42no—
200#125—44no—
201C22⋊Q8—24no—
202C22⋊Q8—44no—
203C22⋊Q8—44no—
204C22⋊Q8—42no—
205C22⋊Q8—82no—
206C22⋊Q8—82no—
207C4×Q8—42no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—44no—
217C4×D4—44no—
218#117—42no—
219#117—42no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C2×D411yescharacteristic
253C2×C22⋊C4—24no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—82no—
278C2×C22⋊C4—82no—
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C22⋊C4C22×C411yescharacteristic
284C2×C22⋊C4C22×C411yescharacteristic
285C2×C22⋊C4C22×C411yescharacteristic
286C2×C22⋊C4C2×D411yescharacteristic
287C2×C22⋊C4C2×D411yescharacteristic
288C2×C22⋊C4C2×D411yescharacteristic
289C2×C22⋊C4C2×D411yescharacteristic
290C2×C22⋊C4C2×D411yescharacteristic
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C22⋊C411yescharacteristic
294C2×C42C22⋊C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C4⋊C8—42no—
298#99—44no—
299#99—44no—
300C22⋊C8—24no—
301C22⋊C8—24no—
302C22⋊C8—42no—
303C4×C8—42no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—24no—
318#95—42no—
319#95C22×C411yescharacteristic
320#95C22×C411yescharacteristic
321#95C22×C411yescharacteristic
322#95C22×C411yescharacteristic
323C24—22no—
324C24—22no—
325C24—41no—
326C24—41no—
327C24—81no—
328C24—81no—
329C24—81no—
330C24—81no—
331C24#10314yes—
332C24C2×C22⋊C411yescharacteristic
333C24C2×C22⋊C411yescharacteristic
334C24C2×C22⋊C411yescharacteristic
335C24C2×C22⋊C411yescharacteristic
336C24C2×SD1612yes—
337C24C2×Q1612yes—
338C24C22×D411yescharacteristic
339C2×Q8—22no—
340C2×Q8—42no—
341C2×Q8—81no—
342C2×Q8—81no—
343C2×Q8—81no—
344C2×D4—22no—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4—22no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4#10314yes—
355C2×D4#10314yes—
356C2×D4C2×SD1612yes—
357C2×D4C2×SD1612yes—
358C2×D4C2×Q1612yes—
359C2×D4C2×Q1612yes—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—84no—
399C22×C4—162no—
400C22×C4—162no—
401C22×C4#10314yes—
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4H(5,𝔽2)14yes—
413C2×C8—44no—
414C2×C8—84no—
415C2×C8—162no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—81no—
435C4⋊C4—81no—
436C4⋊C4—81no—
437C4⋊C4—81no—
438C4⋊C4—81no—
439C4⋊C4—81no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4—81no—
443C4⋊C4—84no—
444C4⋊C4—162no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—41no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—81no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C22⋊C4—81no—
470C22⋊C4—81no—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C22⋊C4—84no—
476C42—22no—
477C42—22no—
478C42—41no—
479C42—41no—
480C42—42no—
481C42—42no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—24no—
491C23—24no—
492C23—41no—
493C23—41no—
494C23—44no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—82no—
504C23—82no—
505C23—82no—
506C23—82no—
507C23—84no—
508C23—161no—
509C23—161no—
510C23—162no—
511C23—162no—
512C23C2×#10312yes—
513C23C2×#10312yes—
514C23C2×#10312yes—
515C23C2×#10312yes—
516C23C22×C22⋊C411yescharacteristic
517C23#53412yes—
518C23C22×SD1611yescharacteristic
519C23C22×Q1611yescharacteristic
520C23#57512yes—
521Q8—82no—
522Q8—162no—
523D4—24no—
524D4—24no—
525D4—24no—
526D4—24no—
527D4—44no—
528D4—44no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—44no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—84no—
577C2×C4—84no—
578C2×C4—84no—
579C2×C4—84no—
580C2×C4—84no—
581C2×C4—161no—
582C2×C4—161no—
583C2×C4—161no—
584C2×C4—161no—
585C2×C4—162no—
586C2×C4—162no—
587C2×C4—168no—
588C2×C4C2×#10312yes—
589C2×C4C2×#10312yes—
590C2×C4#51812yes—
591C2×C4#53412yes—
592C8—84no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—42no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—84no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—84no—
622C22—164no—
623C22#230811yescharacteristic
624C22C22×#10311yescharacteristic
625C22#289611yescharacteristic
626C22#318712yes—
627C22#318912yes—
628C4—22no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4—164no—
647C2—21no—
648C2—22no—
649C2—21no—
650C2—22no—
651C2—42no—
652C2—42no—
653C2—42no—
654C2—42no—
655C2#2175911yescharacteristic
656C2#2177711yescharacteristic
657C2#2178211yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9469C2semidirect
5. #9383C2semidirect
6. #9199C2semidirect
7. #9021C2semidirect
8. #8761C2semidirect
9. #8380C2semidirect
14. #2628C4semidirect
15. #2626C4semidirect
17. #2580C4semidirect
18. #2564C4semidirect
20. #2542C4semidirect
21. #2487C4semidirect
29. #2396C4semidirect
30. C2×#399C4semidirect
36. #1824C22semidirect
42. #1520C22semidirect
49. #1294C22semidirect
50. #1289C22semidirect
53. C2×C4⋊Q8C2×C4semidirect
59. C2×C22⋊Q8C2×C4semidirect
62. C2×C4×Q8C2×C4semidirect
77. C22×C4⋊C4C2×C4semidirect
93. #400C2×C4semidirect
94. #400C2×C4semidirect
95. #400C2×C4semidirect
106. #399C2×C4semidirect
107. #399C2×C4semidirect
108. #399C2×C4semidirect
116. #395C2×C4semidirect
120. #393C2×C4semidirect
146. #382C2×C4semidirect
153. #380C2×C4semidirect
159. C4×C4⋊C4C2×C4semidirect
161. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#358362
#206332
#173502
#94692
#93832
#91992
#90212
#87612
#83802