Finite Group Databasebeta

← #355706

Subgroups — #355706

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1923C211yesmaximal, characteristic
2C2×#1850C211yesmaximal, characteristic
3C2×C4×C4⋊C8C211yesmaximal, characteristic, complemented
4C2×C8×C22⋊C4C211yesmaximal, characteristic
5C4×#388C211yesmaximal, characteristic
6#10921C211yesmaximal, characteristic
7#10921C211yesmaximal, characteristic
8#10876C211yesmaximal, characteristic, complemented
9#10876C211yesmaximal, characteristic, complemented
10#10876C211yesmaximal, characteristic, complemented
11#10874C211yesmaximal, characteristic, complemented
12#10834C211yesmaximal, characteristic, complemented
13#10795C211yesmaximal, characteristic, complemented
14#10714C211yesmaximal, characteristic
15C8×#95C211yesmaximal, characteristic, complemented
16C24×C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C2×#388C412yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C43C2211yescharacteristic
34#1923C418yescomplemented
35#1850C418yescomplemented
36C4×C4⋊C8—24no—
37C8×C22⋊C4—24no—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
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
54#1437C412yes—
55#1436C412yes—
56#1434C412yes—
57C4×#95C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C24×C4C2311yescharacteristic
60C23×C8—22no—
61C23×C8—22no—
62C23×C8—41no—
63C23×C8—41no—
64C23×C8—41no—
65C23×C8—41no—
66C23×C8C2×C414yes—
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C23×C8C2311yescharacteristic
70C23×C8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C22×C4⋊C4C2×C412yes—
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C411yescharacteristic
80C22×C42C2×C412yes—
81C22×C42C2×C412yes—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—41no—
93C2×C4⋊C8—41no—
94C2×C4⋊C8C2×C414yes—
95C2×C4⋊C8C2×C414yes—
96C2×C4⋊C8C2×C414yes—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—41no—
100C2×C22⋊C8—41no—
101C2×C22⋊C8—41no—
102C2×C22⋊C8—41no—
103C2×C22⋊C8C2×C414yes—
104C2×C22⋊C8C2×C414yes—
105C2×C22⋊C8C2×C414yes—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—41no—
110C2×C4×C8—41no—
111C2×C4×C8—41no—
112C2×C4×C8—41no—
113C2×C4×C8—41no—
114C2×C4×C8—41no—
115C2×C4×C8—41no—
116C2×C4×C8—41no—
117C2×C4×C8—41no—
118C2×C4×C8—41no—
119C2×C4×C8C2×C414yes—
120C2×C4×C8D414yes—
121#388C8116yescomplemented
122C4×C4⋊C4—41no—
123C4×C4⋊C4—41no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—41no—
127C4×C22⋊C4—41no—
128C4×C22⋊C4C2×C414yes—
129C4×C22⋊C4C2×C414yes—
130C2×#95C2×C412yes—
131C2×#95C2×C412yes—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C411yescharacteristic
136C2×#95C2×C411yescharacteristic
137C2×#95C2×C412yes—
138C2×#95C2×C412yes—
139C43—22no—
140C43—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145#336—22no—
146#336—22no—
147#336—41no—
148#336—41no—
149#336—41no—
150#336—41no—
151#336—41no—
152#336—41no—
153#336C2×C414yes—
154#336C2×C414yes—
155#336C2×C414yes—
156#336C2×C414yes—
157C25C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4C22×C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C412yes—
174C23×C4C22×C412yes—
175C23×C4C22×C412yes—
176C23×C4C22×C412yes—
177C23×C4C22×C412yes—
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C412yes—
181C23×C4C2411yescharacteristic
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×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—28no—
227C22×C8—44no—
228C22×C8—44no—
229C22×C8—44no—
230C22×C8—44no—
231C22×C8—44no—
232C22×C8—44no—
233C22×C8C22×C412yes—
234C22×C8C22×C412yes—
235C22×C8C22×C412yes—
236C22×C8C22×C412yes—
237C22×C8C22×C412yes—
238C22×C8C22×C412yes—
239C22×C8C22×C412yes—
240C22×C8C22×C412yes—
241C22×C8C2×D412yes—
242C22×C8C2×D412yes—
243C22×C8#4112yes—
244C22×C8#4112yes—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C2×C818yescomplemented
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4C4214yes—
254C2×C22⋊C4C4214yes—
255C2×C22⋊C4C2×C818yes—
256C2×C22⋊C4C2×C818yes—
257C2×C22⋊C4C2×C818yes—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—24no—
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—42no—
292C2×C42—42no—
293C2×C42C2×C818yescomplemented
294C2×C42C22×C412yes—
295C2×C42C22×C412yes—
296C2×C42C22×C412yes—
297C2×C42C22×C412yes—
298C2×C42C2×D412yes—
299C2×C42#4112yes—
300C4⋊C8—28no—
301C4⋊C8—28no—
302C4⋊C8—44no—
303C22⋊C8—28no—
304C22⋊C8—44no—
305C22⋊C8—44no—
306C4×C8—44no—
307C4×C8—44no—
308C4×C8—44no—
309C4×C8—44no—
310C4×C8—44no—
311#95—24no—
312#95—24no—
313#95—42no—
314#95—42no—
315#95—42no—
316#95C4214yes—
317#95C4214yes—
318#95C2×C818yescomplemented
319#95C2×C818yescomplemented
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24C2×C4212yes—
330C24C22×C814yes—
331C24C23×C411yescharacteristic
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
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—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×C4C2×C4212yes—
403C22×C4C2×C4212yes—
404C22×C4C2×C4212yes—
405C22×C4C2×C4212yes—
406C22×C4C2×C4212yes—
407C22×C4C4×D414yes—
408C22×C4C22×C814yes—
409C22×C4C22×C814yes—
410C22×C4C22×C814yes—
411C22×C4C22×C814yes—
412C22×C4C22×C814yes—
413C22×C4C22×C814yes—
414C22×C4#13114yes—
415C22×C4C23×C411yescharacteristic
416C22×C4C23×C411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4H(5,𝔽2)11yescharacteristic
422C22×C4H(5,𝔽2)11yescharacteristic
423C22×C4H(5,𝔽2)11yescharacteristic
424C22×C4H(5,𝔽2)11yescharacteristic
425C22×C4H(5,𝔽2)11yescharacteristic
426C22×C42−1+411yescharacteristic
427C22×C42−1+411yescharacteristic
428C22×C42−1+411yescharacteristic
429C2×C8—22no—
430C2×C8—22no—
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—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—416no—
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—
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×C8C4×D414yes—
485C2×C8C4×D414yes—
486C4⋊C4—48no—
487C22⋊C4—216no—
488C22⋊C4—44no—
489C22⋊C4—48no—
490C22⋊C4C4×C8116yescomplemented
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—22no—
496C42—22no—
497C42—22no—
498C42—22no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—216no—
504C42—48no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23C2×C4×C814yes—
540C23C22×C4211yescharacteristic
541C23C2×C4×D411yescharacteristic
542C23C4×#4111yescharacteristic
543C23#51811yescharacteristic
544C23#52011yescharacteristic
545C23#52011yescharacteristic
546C23#52011yescharacteristic
547C23C23×C812yes—
548C23C2×#13112yes—
549C23#56812yes—
550C23#56812yes—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—41no—
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×C4C2×C4×C814yes—
612C2×C4C2×C4×C814yes—
613C2×C4C8×D418yes—
614C2×C4C2×C4×D411yescharacteristic
615C2×C4C2×C4×D411yescharacteristic
616C2×C4C4×#4111yescharacteristic
617C2×C4C4×#4111yescharacteristic
618C2×C4#51811yescharacteristic
619C2×C4#51811yescharacteristic
620C2×C4#51811yescharacteristic
621C2×C4#51911yescharacteristic
622C2×C4#52011yescharacteristic
623C2×C4#52011yescharacteristic
624C2×C4#52011yescharacteristic
625C2×C4#52011yescharacteristic
626C2×C4#53411yescharacteristic
627C2×C4#53511yescharacteristic
628C2×C4#53611yescharacteristic
629C2×C4#53711yescharacteristic
630C2×C4#53811yescharacteristic
631C2×C4#53811yescharacteristic
632C2×C4#53811yescharacteristic
633C2×C4#53911yescharacteristic
634C2×C4#53911yescharacteristic
635C2×C4#53911yescharacteristic
636C2×C4#54011yescharacteristic
637C2×C4#54111yescharacteristic
638C2×C4#54211yescharacteristic
639C2×C4#54211yescharacteristic
640C2×C4#54211yescharacteristic
641C2×C4#54211yescharacteristic
642C8—24no—
643C8—24no—
644C8—42no—
645C8—42no—
646C8—42no—
647C8—42no—
648C8—42no—
649C8—42no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—42no—
654C22—44no—
655C22—44no—
656C22—42no—
657C22—42no—
658C22—44no—
659C22—44no—
660C22—42no—
661C22C42×D411yescharacteristic
662C22#227411yescharacteristic
663C22#227511yescharacteristic
664C22#232211yescharacteristic
665C22#232411yescharacteristic
666C22#233411yescharacteristic
667C22#233711yescharacteristic
668C22C22×C4×C811yescharacteristic
669C22#287311yescharacteristic
670C22C4×#13111yescharacteristic
671C22#287511yescharacteristic
672C22C2×C8×D412yes—
673C22#293012yes—
674C22C8×#4112yes—
675C22#296712yes—
676C22#297212yes—
677C22#298112yes—
678C22#298812yes—
679C22#298812yes—
680C22#298812yes—
681C22#299212yes—
682C22#299212yes—
683C22#299312yes—
684C4—24no—
685C4—28no—
686C4—24no—
687C4—42no—
688C4—42no—
689C4—44no—
690C4—44no—
691C4—44no—
692C4—42no—
693C4#232111yescharacteristic
694C4#232311yescharacteristic
695C4#232511yescharacteristic
696C4#232611yescharacteristic
697C4#233211yescharacteristic
698C4#233411yescharacteristic
699C4#233511yescharacteristic
700C4#233511yescharacteristic
701C2—44no—
702C2#1378911yescharacteristic
703C2C4×C8×D411yescharacteristic
704C2#2048711yescharacteristic
705C2#2054311yescharacteristic
706C2#2055311yescharacteristic
707C2#2055511yescharacteristic
708C2#2056611yescharacteristic
709C2#2206312yes—
710C2#2213312yes—
711C2#2234112yes—
712C2#2238512yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC22C22×C4×C8
socleC24C23×C4
prime radicalC25C22×C4
LCSG ▷ C22 ▷ 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×C4×C4⋊C8C2semidirect
8. #10876C2semidirect
9. #10876C2semidirect
10. #10876C2semidirect
11. #10874C2semidirect
12. #10834C2semidirect
13. #10795C2semidirect
15. C8×#95C2semidirect
34. #1923C4semidirect
35. #1850C4semidirect
121. #388C8semidirect
247. C2×C4⋊C4C2×C8semidirect
293. C2×C42C2×C8semidirect
318. #95C2×C8semidirect
319. #95C2×C8semidirect
490. C22⋊C4C4×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#19232
C2×#18502
C2×C4×C4⋊C82
C2×C8×C22⋊C42
C4×#3882
#109212
#108762
#108742
#108342
#107952
#107142
C8×#952