Finite Group Databasebeta

← #223152

Subgroups — #223152

Order 512.

806 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36202C211yesmaximal, characteristic
2#21271C211yesmaximal, characteristic, complemented
3#21270C211yesmaximal, characteristic, complemented
4#9892C211yesmaximal, characteristic, complemented
5#9892C211yesmaximal, characteristic, complemented
6#9124C211yesmaximal, characteristic, complemented
7#9123C211yesmaximal, characteristic, complemented
8#9123C211yesmaximal, characteristic, complemented
9#9123C211yesmaximal, characteristic, complemented
10#8722C211yesmaximal, characteristic, complemented
11#8720C211yesmaximal, characteristic, complemented
12#8581C211yesmaximal, characteristic, complemented
13C4×#329C211yesmaximal, characteristic, complemented
14#8032C211yesmaximal, characteristic, complemented
15#8031C211yesmaximal, characteristic, complemented
16C2×#432C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20#2864C2211yescharacteristic
21#2850C2211yescharacteristic
22#2850C2211yescharacteristic
23#2578—21no—
24#2531—21no—
25#2525—21no—
26#2516—21no—
27#2512C411yescharacteristic
28#2511C411yescharacteristic
29#2467—21no—
30#2457—21no—
31#2423C411yescharacteristic
32#2422C411yescharacteristic
33#2415C411yescharacteristic
34#2414C411yescharacteristic
35#2374—21no—
36#2373—21no—
37#2363C411yescharacteristic
38#2362C411yescharacteristic
39C4×#124C2211yescharacteristic
40#2285C2211yescharacteristic
41#2283C2211yescharacteristic
42C42×D4C2211yescharacteristic
43#1955—21no—
44#1954—21no—
45#1940—21no—
46#1939—21no—
47#1924—21no—
48#1924—21no—
49#1923—21no—
50#1923—21no—
51#1767C2211yescharacteristic
52#1767C2211yescharacteristic
53#1749C2211yescharacteristic
54#1749C2211yescharacteristic
55#1749C2211yescharacteristic
56#1749C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C4×#97C2211yescharacteristic
59C2×#329C2211yescharacteristic
60C2×#329C2211yescharacteristic
61C2×#329C2211yescharacteristic
62C2×#329C2211yescharacteristic
63#1295C2211yescharacteristic
64#1295C2211yescharacteristic
65#1295C2211yescharacteristic
66#1295C2211yescharacteristic
67#1295C2211yescharacteristic
68#1295C2211yescharacteristic
69#1295C2211yescharacteristic
70#1295C2211yescharacteristic
71#1295C2211yescharacteristic
72#1295C2211yescharacteristic
73#1295C2211yescharacteristic
74#1295C2211yescharacteristic
75C2×#126—21no—
76C2×#124C2311yescharacteristic
77C2×#123—21no—
78C2×C22⋊Q8C2×C411yescharacteristic
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×#121C2×C411yescharacteristic
81C2×#121C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×D4C2×C411yescharacteristic
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×C4×D4D411yescharacteristic
87C2×#117C2×C411yescharacteristic
88C2×#117C2×C411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91#432—22no—
92#432—22no—
93#432—22no—
94#432—22no—
95#432—41no—
96#432—41no—
97#432—41no—
98#432—41no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×#97C2311yescharacteristic
116C2×#97C2311yescharacteristic
117C2×#97C2311yescharacteristic
118C2×#97C2311yescharacteristic
119C2×#97C2311yescharacteristic
120C2×#97C2311yescharacteristic
121C2×#97C2311yescharacteristic
122C2×#97C2311yescharacteristic
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125#400—21no—
126#400—21no—
127#399—21no—
128#398—21no—
129#397—21no—
130#397—21no—
131#397—21no—
132#396—21no—
133#394—21no—
134#394—21no—
135#394—21no—
136#392—21no—
137#391C2×C411yescharacteristic
138#391C2×C411yescharacteristic
139#391C2×C411yescharacteristic
140#390—21no—
141#390—21no—
142#390C2×C411yescharacteristic
143#390C2×C411yescharacteristic
144#390C2×C411yescharacteristic
145#389—21no—
146#389C2×C411yescharacteristic
147#389C2×C411yescharacteristic
148#389C2×C411yescharacteristic
149#389C2×C411yescharacteristic
150#388—21no—
151#388—21no—
152#388—21no—
153#388—21no—
154#388—21no—
155#388—21no—
156#388—21no—
157#388—21no—
158#388—21no—
159#388—21no—
160#388—21no—
161#388—21no—
162#388—21no—
163#387—21no—
164#387—21no—
165#387—21no—
166#387—21no—
167#387—21no—
168#387—21no—
169#387—21no—
170#387—21no—
171#386—21no—
172#386—21no—
173#386—21no—
174#386—21no—
175#386—21no—
176#385—21no—
177#384C2×C411yescharacteristic
178#384C2×C411yescharacteristic
179#383—21no—
180#382C2311yescharacteristic
181#382C2311yescharacteristic
182#382C2311yescharacteristic
183#381—21no—
184#380—21no—
185#380—21no—
186#380C2×C411yescharacteristic
187#380C2×C411yescharacteristic
188C4×C4⋊C4—21no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4C2×C411yescharacteristic
192C4×C4⋊C4C2×C411yescharacteristic
193C4×C4⋊C4C2×C411yescharacteristic
194C4×C4⋊C4D411yescharacteristic
195C4×C4⋊C4D411yescharacteristic
196C4×C22⋊C4—21no—
197C4×C22⋊C4—21no—
198C4×C22⋊C4—21no—
199C4×C22⋊C4—21no—
200C4×C22⋊C4C2×C411yescharacteristic
201C4×C22⋊C4C2×C411yescharacteristic
202C4×C22⋊C4D411yescharacteristic
203C4×C22⋊C4D411yescharacteristic
204C4×C22⋊C4D411yescharacteristic
205C4×C22⋊C4D411yescharacteristic
206#376—21no—
207C43C2311yescharacteristic
208#336—21no—
209#336—21no—
210#336—21no—
211#336—21no—
212#336—21no—
213#336—21no—
214#329—22no—
215#329—22no—
216#329—41no—
217#329—41no—
218C22×Q8C22×C411yescharacteristic
219C22×D4—21no—
220C22×D4—21no—
221C22×D4C22⋊C411yescharacteristic
222C22×D4C22×C411yescharacteristic
223C23×C4—21no—
224C23×C4C22×C411yescharacteristic
225C23×C4C22×C411yescharacteristic
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238#126—44no—
239#124—21no—
240#124—21no—
241#124—21no—
242#124—21no—
243#123—44no—
244C22⋊Q8—42no—
245C22⋊Q8—81no—
246#121—42no—
247#121—81no—
248C4×Q8—81no—
249C4×D4—42no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—81no—
253#117—81no—
254#117—81no—
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⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22⋊C411yescharacteristic
268C2×C4⋊C4C22⋊C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
293C2×C22⋊C4C22⋊C411yescharacteristic
294C2×C22⋊C4C22⋊C411yescharacteristic
295C2×C22⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C42—21no—
299C2×C42—21no—
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×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
313C2×C42C22×C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2411yescharacteristic
324C4⋊C8—42no—
325C4⋊C8—42no—
326C4⋊C8—42no—
327C4⋊C8—42no—
328C4⋊C8—42no—
329C4⋊C8—42no—
330C22⋊C8—42no—
331C22⋊C8—42no—
332C22⋊C8—42no—
333C22⋊C8—42no—
334C22⋊C8—42no—
335C22⋊C8—42no—
336C22⋊C8—42no—
337C22⋊C8—42no—
338#97—21no—
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—42no—
353#97—42no—
354#97#4112yes—
355#97#4112yes—
356C4×C8—42no—
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#95—21no—
377#95—21no—
378#95C22⋊C411yescharacteristic
379#95C22⋊C411yescharacteristic
380#95C22⋊C411yescharacteristic
381#95C22⋊C411yescharacteristic
382#95C22⋊C411yescharacteristic
383#95C22⋊C411yescharacteristic
384C24—21no—
385C24C2×C22⋊C411yescharacteristic
386C24C2×C22⋊C411yescharacteristic
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—41no—
390C2×Q8—41no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—81no—
401C2×D4—81no—
402C2×D4—81no—
403C2×D4—81no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—41no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—41no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—42no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C2×C22⋊C411yescharacteristic
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C411yescharacteristic
442C22×C4C2×C22⋊C411yescharacteristic
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C411yescharacteristic
445C22×C4C2×C22⋊C411yescharacteristic
446C22×C4C2×C22⋊C411yescharacteristic
447C22×C4C2×C22⋊C411yescharacteristic
448C22×C4C23×C411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—42no—
502C22⋊C4—42no—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C22⋊C4—81no—
515C22⋊C4—81no—
516C22⋊C4—81no—
517C22⋊C4—81no—
518C22⋊C4—81no—
519C22⋊C4—81no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—22no—
533C42—22no—
534C42—41no—
535C42—41no—
536C42—41no—
537C42—41no—
538C42—42no—
539C42—81no—
540C42—81no—
541C42—81no—
542C42—81no—
543C42C2×#4111yescharacteristic
544C42C2×#4111yescharacteristic
545C42H(5,𝔽2)11yescharacteristic
546C422−1+411yescharacteristic
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23C22×C22⋊C411yescharacteristic
569Q8—81no—
570Q8—81no—
571D4—44no—
572D4—81no—
573D4—81no—
574D4—162no—
575D4—162no—
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—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—81no—
633C2×C4—81no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—81no—
639C2×C4—81no—
640C2×C4—81no—
641C2×C4—81no—
642C2×C4—162no—
643C2×C4#41312yes—
644C2×C4#42112yes—
645C2×C4#42112yes—
646C2×C4C4×#4111yescharacteristic
647C2×C4#52011yescharacteristic
648C2×C4#54611yescharacteristic
649C2×C4#54611yescharacteristic
650C2×C4#54711yescharacteristic
651C2×C4#54711yescharacteristic
652C8—42no—
653C8—42no—
654C8—42no—
655C8—42no—
656C8—81no—
657C8—81no—
658C8—81no—
659C8—81no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22#233911yescharacteristic
682C22C2×#41311yescharacteristic
683C22#288811yescharacteristic
684C22C2×#42111yescharacteristic
685C22C2×#42111yescharacteristic
686C22#290111yescharacteristic
687C22#290111yescharacteristic
688C4—22no—
689C4—21no—
690C4—21no—
691C4—22no—
692C4—21no—
693C4—21no—
694C4—21no—
695C4—21no—
696C4—22no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—41no—
703C4—41no—
704C4—81no—
705C4—81no—
706C4—81no—
707C4—81no—
708C4—81no—
709C4—81no—
710C4—81no—
711C2—81no—
712C2—81no—
713C2—161no—
714C2#1766211yescharacteristic
715C2#1767911yescharacteristic
716C2#1768411yescharacteristic
717C2#1769111yescharacteristic
718C2#2248211yescharacteristic
719C2#2250711yescharacteristic
720C2#2250711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2414C4
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
2. #21271C2semidirect
3. #21270C2semidirect
4. #9892C2semidirect
5. #9892C2semidirect
6. #9124C2semidirect
7. #9123C2semidirect
8. #9123C2semidirect
9. #9123C2semidirect
10. #8722C2semidirect
11. #8720C2semidirect
12. #8581C2semidirect
13. C4×#329C2semidirect
14. #8032C2semidirect
15. #8031C2semidirect

Maximal subgroups

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

subgroupindex
#362022
#212712
#212702
#98922
#91242
#91232
#87222
#87202
#85812
C4×#3292
#80322
#80312