Finite Group Databasebeta

← #354507

Subgroups — #354507

Order 512.

1224 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1919C211yesmaximal, characteristic
2C2×#1920C211yesmaximal, characteristic
3C2×#1776C211yesmaximal, characteristic
4#13783C211yesmaximal, characteristic
5#13750C211yesmaximal, characteristic
6#10907C211yesmaximal, characteristic
7#10907C211yesmaximal, characteristic
8#10885C211yesmaximal, characteristic
9#10884C211yesmaximal, characteristic
10#10807C211yesmaximal, characteristic
11#10806C211yesmaximal, characteristic
12#10703C211yesmaximal, characteristic
13C4×#340C211yesmaximal, characteristic
14C4×#340C211yesmaximal, characteristic
15#10694C211yesmaximal, characteristic
16C22×#107C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×#97C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1920C418yescomplemented
32#1919C418yescomplemented
33#1776C418yescomplemented
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52#1436—21no—
53#1436—21no—
54C4⋊#95—21no—
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57#1434—21no—
58C4×#95—21no—
59C4×#95—21no—
60C4×#95—21no—
61C4×#95—21no—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C42—21no—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42—21no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×#107—22no—
88C2×#107—22no—
89C2×#107C2×C414yes—
90C2×#106C2×C414yes—
91C2×#106C2×C414yes—
92C2×#106C2×C414yes—
93C2×#97—22no—
94C2×#97—22no—
95C2×#97C2×C414yes—
96C2×C4×C8—22no—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99#389—22no—
100#389—22no—
101#389C2×C414yes—
102#389C2×C414yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2×C414yes—
115C4×C4⋊C4C2×C414yes—
116C4×C4⋊C4C2×C414yes—
117C4×C4⋊C4C2×C414yes—
118C4×C4⋊C4Q814yes—
119#376—22no—
120#376—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95D411yescharacteristic
136C2×#95D411yescharacteristic
137C2×#95D411yescharacteristic
138C2×#95D411yescharacteristic
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340C2×C414yes—
146#340C2×C414yes—
147#340C2×C414yes—
148#340C2×C414yes—
149#340C2×C414yes—
150#340C2×C414yes—
151#340C2×C414yes—
152#340C2×C414yes—
153#336—22no—
154#336—22no—
155#336C2×C414yes—
156#336C2×C414yes—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8C22×C412yes—
195C22×C8C22×C412yes—
196C22×C8C22×C412yes—
197C22×C8C22×C412yes—
198C22×C8C22×C412yes—
199C22×C8C22×C412yes—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
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—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—81no—
233C2×C4⋊C4—81no—
234C2×C4⋊C4—81no—
235C2×C4⋊C4—81no—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C22×C412yes—
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C22×C412yes—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C412yes—
248C2×C4⋊C4C2×Q812yes—
249C2×C4⋊C4C2×Q812yes—
250C2×C4⋊C4#4112yes—
251C2×C4⋊C4#4112yes—
252C2×C4⋊C4#4112yes—
253C2×C4⋊C4#4112yes—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—42no—
270C2×C42—81no—
271C2×C42—81no—
272C2×C42—81no—
273C2×C42—81no—
274C2×C42—81no—
275C2×C42—81no—
276C2×C42—81no—
277C2×C42—81no—
278C2×C42—81no—
279C2×C42—81no—
280C2×C42—81no—
281C2×C42—81no—
282C2×C42—81no—
283C2×C42—81no—
284C2×C42—81no—
285C2×C42—81no—
286C2×C42C22×C412yes—
287C2×C42C22×C412yes—
288C2×C42C22×C412yes—
289C2×C42C2×Q812yes—
290C2×C42#4112yes—
291C2×C42#4112yes—
292C2×C42#4112yes—
293C2×C42#4112yes—
294C2×C42#4112yes—
295C2×C42#4112yes—
296#107—28no—
297#106C42116yescomplemented
298#97—28no—
299C4×C8—44no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—42no—
314#95—81no—
315#95—81no—
316#95—81no—
317#95—81no—
318C24C22×D411yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4C4×D412yes—
408C22×C4C4×D412yes—
409C22×C4C4×D412yes—
410C22×C4C4×D412yes—
411C22×C4C4×D412yes—
412C22×C4C4×D412yes—
413C22×C4C22⋊Q812yes—
414C22×C4C22⋊Q812yes—
415C22×C4#12312yes—
416C22×C4#12312yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C23×C411yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C22×Q811yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4C2×#4111yescharacteristic
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
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—
457C2×C8—41no—
458C2×C8C2×C4214yes—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4C2×C4214yes—
489C4⋊C4C2×C4214yes—
490C4⋊C4#11714yes—
491C4⋊C4C4×Q814yes—
492C4⋊C4C4×Q814yes—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—84no—
502C42—84no—
503C42—84no—
504C42—84no—
505C42—84no—
506C42—84no—
507C42—84no—
508C42—84no—
509C42#11714yes—
510C42C4×Q814yes—
511C42#12514yes—
512C42#12614yes—
513C23C2×C4×D411yescharacteristic
514C23C2×C4×D411yescharacteristic
515C23C2×C4×D411yescharacteristic
516C23C2×C22⋊Q811yescharacteristic
517C23C2×#12311yescharacteristic
518C23#52511yescharacteristic
519C23#52511yescharacteristic
520C23#57611yescharacteristic
521C23#57611yescharacteristic
522C23#57711yescharacteristic
523C23#57711yescharacteristic
524C23#57711yescharacteristic
525C23#57711yescharacteristic
526C23#57811yescharacteristic
527C23#57811yescharacteristic
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
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—42no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—81no—
620C2×C4—81no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—81no—
624C2×C4#38714yes—
625C2×C4#38714yes—
626C2×C4C22×C4211yescharacteristic
627C2×C4C2×#11711yescharacteristic
628C2×C4C2×#11711yescharacteristic
629C2×C4C2×C4×Q811yescharacteristic
630C2×C4C2×C4×Q811yescharacteristic
631C2×C4C2×C4×Q811yescharacteristic
632C2×C4C4×#4111yescharacteristic
633C2×C4C4×#4111yescharacteristic
634C2×C4C4×#4111yescharacteristic
635C2×C4C4×#4111yescharacteristic
636C2×C4C4×#4111yescharacteristic
637C2×C4C4×#4111yescharacteristic
638C2×C4C4×#4111yescharacteristic
639C2×C4C4×#4111yescharacteristic
640C2×C4C4×#4111yescharacteristic
641C2×C4C4×#4111yescharacteristic
642C2×C4C2×#12511yescharacteristic
643C2×C4C2×#12611yescharacteristic
644C2×C4#52911yescharacteristic
645C2×C4#52911yescharacteristic
646C2×C4#52911yescharacteristic
647C2×C4#52911yescharacteristic
648C2×C4#52911yescharacteristic
649C2×C4#52911yescharacteristic
650C2×C4#52911yescharacteristic
651C2×C4#52911yescharacteristic
652C2×C4#53311yescharacteristic
653C2×C4#53311yescharacteristic
654C8—24no—
655C8—42no—
656C8—42no—
657C8—42no—
658C22C42×D411yescharacteristic
659C22C2×#38711yescharacteristic
660C22#229011yescharacteristic
661C22C4×#12311yescharacteristic
662C22C4×#12311yescharacteristic
663C22C4×C22⋊Q811yescharacteristic
664C22C4×C22⋊Q811yescharacteristic
665C22#294611yescharacteristic
666C22#294611yescharacteristic
667C22#294611yescharacteristic
668C22#294611yescharacteristic
669C22#294711yescharacteristic
670C22#294711yescharacteristic
671C22#294711yescharacteristic
672C22#294711yescharacteristic
673C22#294911yescharacteristic
674C22#294911yescharacteristic
675C22#295111yescharacteristic
676C22#295111yescharacteristic
677C22#308111yescharacteristic
678C22#308211yescharacteristic
679C22#308311yescharacteristic
680C22#308411yescharacteristic
681C22#309611yescharacteristic
682C22#309711yescharacteristic
683C22#309811yescharacteristic
684C22#309911yescharacteristic
685C22#311811yescharacteristic
686C22#311911yescharacteristic
687C22#312011yescharacteristic
688C22#312111yescharacteristic
689C22#312211yescharacteristic
690C22#312511yescharacteristic
691C22#312511yescharacteristic
692C22#312711yescharacteristic
693C4—24no—
694C4—24no—
695C4—44no—
696C4—44no—
697C4—42no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—44no—
702C4—44no—
703C4—44no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4C4×#11711yescharacteristic
708C4C42×Q811yescharacteristic
709C4#229511yescharacteristic
710C4#229811yescharacteristic
711C4C4×#12611yescharacteristic
712C4C4×#12611yescharacteristic
713C4C4×#12511yescharacteristic
714C4C4×#12511yescharacteristic
715C2C4×#38711yescharacteristic
716C2#2050411yescharacteristic
717C2#2050411yescharacteristic
718C2#2103511yescharacteristic
719C2#2103611yescharacteristic
720C2#2112611yescharacteristic
721C2#2112711yescharacteristic
722C2#2158411yescharacteristic
723C2#2158711yescharacteristic
724C2#2158911yescharacteristic
725C2#2159011yescharacteristic
726C2#2162011yescharacteristic
727C2#2162111yescharacteristic
728C2#2162311yescharacteristic
729C2#2162611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
31. #1920C4semidirect
32. #1919C4semidirect
33. #1776C4semidirect
297. #106C42semidirect

Maximal subgroups

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

subgroupindex
C2×#19192
C2×#19202
C2×#17762
#137832
#137502
#109072
#108852
#108842
#108072
#108062
#107032
C4×#3402
#106942