Finite Group Databasebeta

← #357133

Subgroups — #357133

Order 512.

1356 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1924C211yesmaximal, characteristic
2C2×#1923C211yesmaximal, characteristic
3C2×#1770C211yesmaximal, characteristic, complemented
4C2×#1771C211yesmaximal, characteristic, complemented
5C2×#1750C211yesmaximal, characteristic
6C2×#1750C211yesmaximal, characteristic
7#13844C211yesmaximal, characteristic
8#10922C211yesmaximal, characteristic
9#10922C211yesmaximal, characteristic
10#10877C211yesmaximal, characteristic, complemented
11#10876C211yesmaximal, characteristic, complemented
12#10875C211yesmaximal, characteristic, complemented
13#10874C211yesmaximal, characteristic, complemented
14#10696C211yesmaximal, characteristic, complemented
15#10696C211yesmaximal, characteristic, complemented
16C22×#117C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×#97C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35#1924C418yescomplemented
36#1923C418yescomplemented
37#1771—24no—
38#1770—24no—
39#1750—24no—
40#1750—24no—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C2×#336C2211yescharacteristic
44C2×#336C2211yescharacteristic
45C2×#336C2211yescharacteristic
46C2×#336C2211yescharacteristic
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55#1444C412yes—
56#1442C412yes—
57#1439C412yes—
58#1439C412yes—
59C4×#95C2211yescharacteristic
60C4×#95C2211yescharacteristic
61C24×C4C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—22no—
74C2×#117—41no—
75C2×#117—41no—
76C22×C4⋊C4C2×C412yes—
77C22×C4⋊C4C2×C412yes—
78C22×C4⋊C4C2×C412yes—
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C411yescharacteristic
81C22×C22⋊C4C2×C412yes—
82C22×C22⋊C4C2×C411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8C2×C414yes—
94C2×C4⋊C8C2×C414yes—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8C2×C414yes—
104C2×C22⋊C8C2×C414yes—
105C2×C22⋊C8C2×C414yes—
106C2×C22⋊C8C2×C414yes—
107C2×#97—22no—
108C2×#97—22no—
109C2×#97—22no—
110C2×#97—22no—
111C2×#97—22no—
112C2×#97—22no—
113C2×#97—22no—
114C2×#97—22no—
115C2×#97—22no—
116C2×#97—22no—
117C2×#97—41no—
118C2×#97—41no—
119C2×#97—41no—
120C2×#97—41no—
121C2×#97C2×C414yes—
122C2×C4×C8—22no—
123C2×C4×C8—22no—
124C2×C4×C8—22no—
125C2×C4×C8—22no—
126C2×C4×C8—22no—
127C2×C4×C8—22no—
128C2×C4×C8—41no—
129C2×C4×C8—41no—
130C2×C4×C8C2×C414yes—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C22⋊C4—41no—
136C4×C22⋊C4—41no—
137C4×C22⋊C4C2×C414yes—
138C4×C22⋊C4C2×C414yes—
139C2×#95C2×C412yes—
140C2×#95C2×C412yes—
141C2×#95C2×C412yes—
142C2×#95C2×C412yes—
143C2×#95C2×C412yes—
144C2×#95C2×C412yes—
145C2×#95C2×C412yes—
146C2×#95C2×C411yescharacteristic
147C2×#95C2×C411yescharacteristic
148#336—22no—
149#336—22no—
150#336—22no—
151#336—22no—
152#336—22no—
153#336—22no—
154#336—41no—
155#336—41no—
156#336C2×C414yes—
157#336C2×C414yes—
158#336C2×C414yes—
159#336C2×C414yes—
160#336D414yes—
161#336D414yes—
162C25C22×C411yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4C22×C412yes—
176C23×C4C22×C412yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C412yes—
182C23×C4C22×C412yes—
183C23×C4C22×C412yes—
184C23×C4C22×C412yes—
185C23×C4C22×C411yescharacteristic
186C23×C4C2411yescharacteristic
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8C22×C412yes—
228C22×C8C22×C412yes—
229C22×C8C22×C412yes—
230C22×C8C22×C412yes—
231C22×C8C22×C412yes—
232C22×C8C22×C412yes—
233C22×C8C22×C412yes—
234C22×C8C22×C412yes—
235C22×C8C2×D412yes—
236C22×C8C2×D412yes—
237C22×C8C2×D412yes—
238C22×C8C2×D412yes—
239C22×C8#4112yes—
240C22×C8#4112yes—
241C22×C8#4112yes—
242C22×C8#4112yes—
243#117—44no—
244#117—44no—
245#117—44no—
246#117—44no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4C4214yes—
265C2×C22⋊C4C4214yes—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42C22×C412yes—
295C2×C42C22×C412yes—
296C2×C42C22×C412yes—
297C2×C42C22×C412yes—
298C2×C42C2×D412yes—
299C2×C42C2×D412yes—
300C2×C42#4112yes—
301C2×C42#4112yes—
302C4⋊C8—28no—
303C4⋊C8—28no—
304C22⋊C8—28no—
305C22⋊C8—28no—
306C22⋊C8—28no—
307C22⋊C8—28no—
308#97—28no—
309#97—44no—
310#97—44no—
311#97—44no—
312#97—44no—
313C4×C8—28no—
314C4×C8—44no—
315C4×C8—44no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—42no—
320#95—42no—
321#95—42no—
322#95—42no—
323#95C4214yes—
324#95C4214yes—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24C2×C4212yes—
336C24C23×C411yescharacteristic
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
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×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4C2×C4212yes—
413C22×C4C2×C4212yes—
414C22×C4C2×C4212yes—
415C22×C4C2×C4212yes—
416C22×C4C2×C4212yes—
417C22×C4C4×D414yes—
418C22×C4C4×D414yes—
419C22×C4#13114yes—
420C22×C4C23×C411yescharacteristic
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4H(5,𝔽2)11yescharacteristic
431C22×C4H(5,𝔽2)11yescharacteristic
432C22×C42−1+411yescharacteristic
433C22×C42−1+411yescharacteristic
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—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8C4×D414yes—
487C2×C8C4×D414yes—
488C2×C8C4×D414yes—
489C2×C8C4×D414yes—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C22⋊C4—28no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23C22×C4211yescharacteristic
545C23C2×C4×D411yescharacteristic
546C23C2×C4×D411yescharacteristic
547C23C4×#4111yescharacteristic
548C23C4×#4111yescharacteristic
549C23#51811yescharacteristic
550C23#51911yescharacteristic
551C23C2×#13112yes—
552C23#56812yes—
553C23#56812yes—
554C23#56812yes—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
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—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4#40514yes—
618C2×C4C2×C4×D411yescharacteristic
619C2×C4C2×C4×D411yescharacteristic
620C2×C4C2×C4×D411yescharacteristic
621C2×C4C2×C4×D411yescharacteristic
622C2×C4C4×#4111yescharacteristic
623C2×C4C4×#4111yescharacteristic
624C2×C4C4×#4111yescharacteristic
625C2×C4C4×#4111yescharacteristic
626C2×C4#52011yescharacteristic
627C2×C4#52011yescharacteristic
628C2×C4#52011yescharacteristic
629C2×C4#52011yescharacteristic
630C2×C4D4211yescharacteristic
631C2×C4#54611yescharacteristic
632C2×C4#54611yescharacteristic
633C2×C4#54711yescharacteristic
634C2×C4#54711yescharacteristic
635C2×C4#54811yescharacteristic
636C2×C4#54811yescharacteristic
637C2×C4#55111yescharacteristic
638C2×C4#55111yescharacteristic
639C2×C4#55211yescharacteristic
640C2×C4#55211yescharacteristic
641C2×C4#55311yescharacteristic
642C2×C4#55311yescharacteristic
643C2×C4#55611yescharacteristic
644C2×C4#55611yescharacteristic
645C2×C4#55911yescharacteristic
646C8—24no—
647C8—24no—
648C8—24no—
649C8—24no—
650C8—42no—
651C8—42no—
652C8—42no—
653C8—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22C42×D411yescharacteristic
666C22C42×D411yescharacteristic
667C22#227411yescharacteristic
668C22#235811yescharacteristic
669C22#236211yescharacteristic
670C22#236211yescharacteristic
671C22#237411yescharacteristic
672C22C2×#40511yescharacteristic
673C22#287311yescharacteristic
674C22#287511yescharacteristic
675C22#287511yescharacteristic
676C22#293012yes—
677C22#293212yes—
678C22#293312yes—
679C22#293512yes—
680C22#296712yes—
681C22#296712yes—
682C22#296812yes—
683C22#296912yes—
684C22#297812yes—
685C22#298112yes—
686C22#298512yes—
687C22#298712yes—
688C4—24no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—44no—
693C4—44no—
694C4—42no—
695C4—82no—
696C4—82no—
697C4#235911yescharacteristic
698C4#235911yescharacteristic
699C4#236111yescharacteristic
700C4#236111yescharacteristic
701C4#236711yescharacteristic
702C4#236711yescharacteristic
703C4#237311yescharacteristic
704C4#237311yescharacteristic
705C2—82no—
706C2#1382311yescharacteristic
707C2#2048711yescharacteristic
708C2#2048811yescharacteristic
709C2#2049311yescharacteristic
710C2#2049511yescharacteristic
711C2#2053711yescharacteristic
712C2#2054211yescharacteristic
713C2#2284912yes—
714C2#2305812yes—
715C2#2305912yes—
716C2#2330112yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C22×C42
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1770C2semidirect
4. C2×#1771C2semidirect
10. #10877C2semidirect
11. #10876C2semidirect
12. #10875C2semidirect
13. #10874C2semidirect
14. #10696C2semidirect
15. #10696C2semidirect
35. #1924C4semidirect
36. #1923C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19242
C2×#19232
C2×#17702
C2×#17712
C2×#17502
#138442
#109222
#108772
#108762
#108752
#108742
#106962