Finite Group Databasebeta

← #223858

Subgroups — #223858

Order 512.

796 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36192C211yesmaximal, characteristic
2#21271C211yesmaximal, characteristic
3#21107C211yesmaximal, characteristic
4#9894C211yesmaximal, characteristic, complemented
5#9891C211yesmaximal, characteristic, complemented
6#9126C211yesmaximal, characteristic, complemented
7#9125C211yesmaximal, characteristic, complemented
8#9125C211yesmaximal, characteristic, complemented
9#9121C211yesmaximal, characteristic, complemented
10#8722C211yesmaximal, characteristic, complemented
11#8721C211yesmaximal, characteristic, complemented
12#8582C211yesmaximal, characteristic, complemented
13#8227C211yesmaximal, characteristic, complemented
14#8034C211yesmaximal, characteristic, complemented
15#8031C211yesmaximal, characteristic, complemented
16C2×#432C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2859C2211yescharacteristic
21#2851C2211yescharacteristic
22#2851C2211yescharacteristic
23#2552C411yescharacteristic
24#2552C411yescharacteristic
25#2522—21no—
26#2522—21no—
27#2522—21no—
28#2522—21no—
29#2454—21no—
30#2454—21no—
31#2441C411yescharacteristic
32#2441C411yescharacteristic
33#2441C411yescharacteristic
34#2441C411yescharacteristic
35#2370—21no—
36#2370—21no—
37#2354C411yescharacteristic
38#2354C411yescharacteristic
39C4×#125C2211yescharacteristic
40#2285C2211yescharacteristic
41#2285C2211yescharacteristic
42C42×D4C2211yescharacteristic
43#1955—21no—
44#1940—21no—
45#1924—21no—
46#1923—21no—
47#1923—21no—
48#1917—21no—
49#1869—21no—
50#1850—21no—
51#1767C2211yescharacteristic
52#1767C2211yescharacteristic
53#1749C2211yescharacteristic
54#1749C2211yescharacteristic
55#1749C2211yescharacteristic
56#1749C2211yescharacteristic
57#1745C2211yescharacteristic
58C4×#97C2211yescharacteristic
59C2×#330C2211yescharacteristic
60C2×#330C2211yescharacteristic
61C2×#330C2211yescharacteristic
62C2×#330C2211yescharacteristic
63#1295C2211yescharacteristic
64#1295C2211yescharacteristic
65#1295C2211yescharacteristic
66#1295C2211yescharacteristic
67#1295C2211yescharacteristic
68#1295C2211yescharacteristic
69#1295C2211yescharacteristic
70#1295C2211yescharacteristic
71#1295C2211yescharacteristic
72#1295C2211yescharacteristic
73#1295C2211yescharacteristic
74#1295C2211yescharacteristic
75C23×C8—21no—
76C2×#125—21no—
77C2×#125—21no—
78C2×#125C2×C411yescharacteristic
79C2×#125C2×C411yescharacteristic
80C2×#125C2311yescharacteristic
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4D411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×#117C2×C411yescharacteristic
85C2×#117C2×C411yescharacteristic
86C2×#117C2×C411yescharacteristic
87C2×#117C2×C411yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#432—22no—
91#432—22no—
92#432—22no—
93#432—41no—
94#432—41no—
95#432—41no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×#97—22no—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—41no—
112C2×#97C2311yescharacteristic
113C2×#97C2311yescharacteristic
114C2×#97C2311yescharacteristic
115C2×#97C2311yescharacteristic
116C2×#97C2311yescharacteristic
117C2×#97C2311yescharacteristic
118C2×#97C2311yescharacteristic
119C2×#97C2311yescharacteristic
120C2×C4×C8—21no—
121#399—21no—
122#399—21no—
123#399—21no—
124#399—21no—
125#398—21no—
126#398—21no—
127#398—21no—
128#398—21no—
129#396—21no—
130#396—21no—
131#396—21no—
132#396—21no—
133#390—21no—
134#390—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389—21no—
143#389—21no—
144#389—21no—
145#389C2×C411yescharacteristic
146#389C2×C411yescharacteristic
147#389C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#389C2×C411yescharacteristic
150#389C2×C411yescharacteristic
151#388—21no—
152#388—21no—
153#388—21no—
154#388—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#387—21no—
160#387—21no—
161#387—21no—
162#387—21no—
163#387—21no—
164#387—21no—
165#385—21no—
166#385—21no—
167#385—21no—
168#385—21no—
169#384C2×C411yescharacteristic
170#384C2×C411yescharacteristic
171#384C2×C411yescharacteristic
172#384C2×C411yescharacteristic
173#384C2×C411yescharacteristic
174#384C2×C411yescharacteristic
175#382C2311yescharacteristic
176#382C2311yescharacteristic
177#382C2311yescharacteristic
178#380—21no—
179#380—21no—
180#380—21no—
181#380—21no—
182#380C2×C411yescharacteristic
183#380C2×C411yescharacteristic
184#380C2×C411yescharacteristic
185#380C2×C411yescharacteristic
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4C2×C411yescharacteristic
195C4×C4⋊C4C2×C411yescharacteristic
196C4×C4⋊C4C2×C411yescharacteristic
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D411yescharacteristic
199C4×C22⋊C4C2×C411yescharacteristic
200C4×C22⋊C4C2×C411yescharacteristic
201C4×C22⋊C4D411yescharacteristic
202C4×C22⋊C4D411yescharacteristic
203C4×C22⋊C4D411yescharacteristic
204C4×C22⋊C4D411yescharacteristic
205C43C2311yescharacteristic
206#336—21no—
207#336—21no—
208#336—21no—
209#336—21no—
210#336—21no—
211#336—21no—
212#336—21no—
213#336—21no—
214#330—22no—
215#330—22no—
216#330—41no—
217#330—41no—
218C22×D4C22⋊C411yescharacteristic
219C23×C4C22×C411yescharacteristic
220C23×C4C22×C411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—42no—
234C22×C8—42no—
235C22×C8—42no—
236#125—21no—
237#125—21no—
238#125—21no—
239#125—21no—
240#125—44no—
241#125—44no—
242#125—81no—
243#125—81no—
244C4×D4—24no—
245C4×D4—42no—
246C4×D4—42no—
247#117—81no—
248#117—81no—
249#117—81no—
250#117—81no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4C22⋊C411yescharacteristic
276C2×C4⋊C4C22⋊C411yescharacteristic
277C2×C4⋊C4C22⋊C411yescharacteristic
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C4⋊C4C22×C411yescharacteristic
283C2×C4⋊C4C22×C411yescharacteristic
284C2×C4⋊C4C22×C411yescharacteristic
285C2×C4⋊C4C22×C411yescharacteristic
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4C22⋊C411yescharacteristic
295C2×C22⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42C22×C411yescharacteristic
315C2×C42C22×C411yescharacteristic
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2411yescharacteristic
327C4⋊C8—42no—
328C4⋊C8—42no—
329C4⋊C8—42no—
330C4⋊C8—42no—
331C4⋊C8—42no—
332C22⋊C8—42no—
333C22⋊C8—42no—
334C22⋊C8—42no—
335C22⋊C8—42no—
336C22⋊C8—42no—
337C22⋊C8—42no—
338C22⋊C8—42no—
339#97—21no—
340#97—21no—
341#97—21no—
342#97—21no—
343#97—21no—
344#97—21no—
345#97—21no—
346#97—21no—
347#97—21no—
348#97—21no—
349#97—21no—
350#97—21no—
351#97—21no—
352#97—21no—
353#97—42no—
354#97—42no—
355#97#4112yes—
356#97#4112yes—
357C4×C8—42no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95C22⋊C411yescharacteristic
377#95C22⋊C411yescharacteristic
378#95C22⋊C411yescharacteristic
379#95C22⋊C411yescharacteristic
380#95C22⋊C411yescharacteristic
381#95C22⋊C411yescharacteristic
382C24C2×C22⋊C411yescharacteristic
383C24C2×C22⋊C411yescharacteristic
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—41no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—41no—
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C23×C411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—48no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—81no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C22⋊C4—81no—
504C22⋊C4—81no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C22⋊C4—81no—
509C22⋊C4#10114yes—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—22no—
523C42—22no—
524C42—41no—
525C42—41no—
526C42—41no—
527C42—41no—
528C42—42no—
529C42—42no—
530C42—81no—
531C42—81no—
532C42—81no—
533C42—81no—
534C42C2×#4111yescharacteristic
535C42C2×#4111yescharacteristic
536C42H(5,𝔽2)11yescharacteristic
537C422−1+411yescharacteristic
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23C2×#10112yes—
552C23C22×C22⋊C411yescharacteristic
553D4—44no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4C2×#10112yes—
623C2×C4C2×#10112yes—
624C2×C4#42112yes—
625C2×C4#42112yes—
626C2×C4C4×#4111yescharacteristic
627C2×C4#52011yescharacteristic
628C2×C4#54611yescharacteristic
629C2×C4#54611yescharacteristic
630C2×C4#54711yescharacteristic
631C2×C4#54711yescharacteristic
632C8—42no—
633C8—42no—
634C8—42no—
635C8—42no—
636C8—81no—
637C8—81no—
638C8—81no—
639C8—81no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22#233911yescharacteristic
655C22C22×#10111yescharacteristic
656C22#288811yescharacteristic
657C22C2×#42111yescharacteristic
658C22C2×#42111yescharacteristic
659C22#290111yescharacteristic
660C22#290111yescharacteristic
661C4—22no—
662C4—22no—
663C4—21no—
664C4—21no—
665C4—22no—
666C4—21no—
667C4—21no—
668C4—22no—
669C4—21no—
670C4—21no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—81no—
678C4—81no—
679C4—81no—
680C4—81no—
681C4—81no—
682C4—81no—
683C4—81no—
684C4—81no—
685C2—42no—
686C2—81no—
687C2#1767211yescharacteristic
688C2#1768411yescharacteristic
689C2#1768411yescharacteristic
690C2#1768911yescharacteristic
691C2#2248111yescharacteristic
692C2#2250611yescharacteristic
693C2#2250711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9894C2semidirect
5. #9891C2semidirect
6. #9126C2semidirect
7. #9125C2semidirect
8. #9125C2semidirect
9. #9121C2semidirect
10. #8722C2semidirect
11. #8721C2semidirect
12. #8582C2semidirect
13. #8227C2semidirect
14. #8034C2semidirect
15. #8031C2semidirect

Maximal subgroups

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

subgroupindex
#361922
#212712
#211072
#98942
#98912
#91262
#91252
#91212
#87222
#87212
#85822
#82272
#80342
#80312