Finite Group Databasebeta

← #355674

Subgroups — #355674

Order 512.

1514 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
1C2×#1923C211yesmaximal, characteristic
2C2×#1923C211yesmaximal, characteristic
3C2×C8×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊C8C211yesmaximal, characteristic
5C4×#385C211yesmaximal, characteristic
6#10921C211yesmaximal, characteristic
7#10921C211yesmaximal, characteristic
8#10876C211yesmaximal, characteristic, complemented
9#10876C211yesmaximal, characteristic, complemented
10#10874C211yesmaximal, characteristic, complemented
11#10874C211yesmaximal, characteristic, complemented
12#10834C211yesmaximal, characteristic, complemented
13#10797C211yesmaximal, characteristic, complemented
14#10712C211yesmaximal, characteristic
15C8×#95C211yesmaximal, characteristic, complemented
16C23×C42C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C22⋊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
28C22×C4×C8C2211yescharacteristic
29C2×#385C412yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1923C418yescomplemented
35#1923C418yescomplemented
36C8×C4⋊C4—24no—
37C4×C22⋊C8—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#1436C412yes—
55#1436C412yes—
56C22⋊#95C412yes—
57C4×#95C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C24×C4C2×C412yes—
60C24×C4C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2×C412yes—
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C412yes—
73C22×C22⋊C4C2×C411yescharacteristic
74C22×C42—22no—
75C22×C42—22no—
76C22×C42—41no—
77C22×C42—41no—
78C22×C42—41no—
79C22×C42—41no—
80C22×C42C2×C412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—41no—
90C2×C4⋊C8—41no—
91C2×C4⋊C8C2×C414yescomplemented
92C2×C4⋊C8C2×C414yescomplemented
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—41no—
98C2×C22⋊C8—41no—
99C2×C22⋊C8—41no—
100C2×C22⋊C8—41no—
101C2×C22⋊C8C2×C414yes—
102C2×C22⋊C8C2×C414yes—
103C2×C22⋊C8C2×C414yes—
104C2×C22⋊C8C2×C414yes—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
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×C8—41no—
120C2×C4×C8C2×C414yescomplemented
121C2×C4×C8C2×C414yescomplemented
122C2×C4×C8D414yes—
123#385C8116yescomplemented
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—41no—
127C4×C4⋊C4—41no—
128C4×C22⋊C4—41no—
129C4×C22⋊C4—41no—
130C4×C22⋊C4C2×C414yes—
131C4×C22⋊C4C2×C414yes—
132C2×#95C2×C412yes—
133C2×#95C2×C412yes—
134C2×#95C2×C412yes—
135C2×#95C2×C412yes—
136C2×#95C2×C412yes—
137C2×#95C2×C411yescharacteristic
138C2×#95C2×C411yescharacteristic
139C2×#95C2×C412yes—
140C2×#95C2×C412yes—
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×C414yescomplemented
154#336C2×C414yescomplemented
155#336C2×C414yescomplemented
156#336C2×C414yescomplemented
157C25C22×C411yescharacteristic
158C23×C4—24no—
159C23×C4—24no—
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×C4—21no—
171C23×C4—21no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4C2×C818yes—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C412yes—
181C23×C4C22×C412yes—
182C23×C4C22×C412yes—
183C23×C4C22×C412yes—
184C23×C4C22×C412yes—
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C412yes—
188C23×C4C2411yescharacteristic
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×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
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—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C2×C818yes—
250C2×C4⋊C4C2×C818yes—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4C4214yes—
255C2×C22⋊C4C4214yes—
256C2×C22⋊C4C2×C818yes—
257C2×C22⋊C4C2×C818yes—
258C2×C42—21no—
259C2×C42—21no—
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—24no—
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—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42C22×C412yes—
296C2×C42C22×C412yes—
297C2×C42C22×C412yes—
298C2×C42C22×C412yes—
299C2×C42C2×D412yes—
300C2×C42#4112yes—
301C4⋊C8—28no—
302C4⋊C8—44no—
303C22⋊C8—28no—
304C22⋊C8—28no—
305C22⋊C8—44no—
306C22⋊C8—44no—
307C4×C8—28no—
308C4×C8—44no—
309C4×C8—44no—
310C4×C8—44no—
311C4×C8—44no—
312C4×C8—44no—
313#95—24no—
314#95—24no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95C4214yes—
319#95C4214yes—
320#95C2×C818yes—
321#95C2×C818yes—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24C2×C4212yes—
332C24C22×C814yes—
333C24C23×C411yescharacteristic
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
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—216no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
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×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4C2×C4212yes—
412C22×C4C2×C4212yes—
413C22×C4C2×C4212yes—
414C22×C4C2×C4212yes—
415C22×C4C2×C4212yes—
416C22×C4C4×D414yes—
417C22×C4C22×C814yes—
418C22×C4C22×C814yes—
419C22×C4C22×C814yes—
420C22×C4C22×C814yes—
421C22×C4C22×C814yes—
422C22×C4C22×C814yes—
423C22×C4#13114yes—
424C22×C4C23×C411yescharacteristic
425C22×C4C23×C411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4H(5,𝔽2)11yescharacteristic
431C22×C4H(5,𝔽2)11yescharacteristic
432C22×C4H(5,𝔽2)11yescharacteristic
433C22×C4H(5,𝔽2)11yescharacteristic
434C22×C4H(5,𝔽2)11yescharacteristic
435C22×C42−1+411yescharacteristic
436C22×C42−1+411yescharacteristic
437C22×C42−1+411yescharacteristic
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—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×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8C4×D414yes—
493C2×C8C4×D414yes—
494C4⋊C4—216no—
495C4⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—48no—
498C22⋊C4C4×C8116yescomplemented
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—48no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23C2×C4×C814yes—
547C23C22×C4211yescharacteristic
548C23C2×C4×D411yescharacteristic
549C23C4×#4111yescharacteristic
550C23#51811yescharacteristic
551C23#51811yescharacteristic
552C23#51911yescharacteristic
553C23#52011yescharacteristic
554C23C23×C812yes—
555C23C2×#13112yes—
556C23#56812yes—
557C23#56812yes—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—432no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
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—42no—
618C2×C4—42no—
619C2×C4C2×C4×C814yes—
620C2×C4C2×C4×C814yes—
621C2×C4C8×D418yes—
622C2×C4C2×C4×D411yescharacteristic
623C2×C4C2×C4×D411yescharacteristic
624C2×C4C4×#4111yescharacteristic
625C2×C4C4×#4111yescharacteristic
626C2×C4#51811yescharacteristic
627C2×C4#51811yescharacteristic
628C2×C4#52011yescharacteristic
629C2×C4#52011yescharacteristic
630C2×C4#52011yescharacteristic
631C2×C4#52011yescharacteristic
632C2×C4#52011yescharacteristic
633C2×C4#52011yescharacteristic
634C2×C4#53411yescharacteristic
635C2×C4#53511yescharacteristic
636C2×C4#53611yescharacteristic
637C2×C4#53711yescharacteristic
638C2×C4#53811yescharacteristic
639C2×C4#53811yescharacteristic
640C2×C4#53811yescharacteristic
641C2×C4#53911yescharacteristic
642C2×C4#53911yescharacteristic
643C2×C4#53911yescharacteristic
644C2×C4#54011yescharacteristic
645C2×C4#54111yescharacteristic
646C2×C4#54211yescharacteristic
647C2×C4#54211yescharacteristic
648C2×C4#54211yescharacteristic
649C2×C4#54211yescharacteristic
650C8—24no—
651C8—24no—
652C8—42no—
653C8—42no—
654C8—42no—
655C8—42no—
656C8—42no—
657C8—42no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—42no—
662C22—44no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—44no—
667C22—44no—
668C22—42no—
669C22C42×D411yescharacteristic
670C22#227411yescharacteristic
671C22#227511yescharacteristic
672C22#232111yescharacteristic
673C22#232311yescharacteristic
674C22#233211yescharacteristic
675C22#233511yescharacteristic
676C22C22×C4×C811yescharacteristic
677C22#287311yescharacteristic
678C22C4×#13111yescharacteristic
679C22#287511yescharacteristic
680C22C2×C8×D412yes—
681C22#293012yes—
682C22C8×#4112yes—
683C22#296712yes—
684C22#297212yes—
685C22#297212yes—
686C22#297612yes—
687C22#297612yes—
688C22#298212yes—
689C22#298512yes—
690C22#298812yes—
691C22#299312yes—
692C4—24no—
693C4—28no—
694C4—24no—
695C4—42no—
696C4—44no—
697C4—44no—
698C4—42no—
699C4—44no—
700C4—42no—
701C4#232211yescharacteristic
702C4#232411yescharacteristic
703C4#232511yescharacteristic
704C4#232611yescharacteristic
705C4#233411yescharacteristic
706C4#233411yescharacteristic
707C4#233511yescharacteristic
708C4#233711yescharacteristic
709C2—44no—
710C2#1378911yescharacteristic
711C2C4×C8×D411yescharacteristic
712C2#2048711yescharacteristic
713C2#2053311yescharacteristic
714C2#2054111yescharacteristic
715C2#2055511yescharacteristic
716C2#2056111yescharacteristic
717C2#2202812yes—
718C2#2209812yes—
719C2#2231712yes—
720C2#2235712yes—
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×C8×C4⋊C4C2semidirect
8. #10876C2semidirect
9. #10876C2semidirect
10. #10874C2semidirect
11. #10874C2semidirect
12. #10834C2semidirect
13. #10797C2semidirect
15. C8×#95C2semidirect
34. #1923C4semidirect
35. #1923C4semidirect
91. C2×C4⋊C8C2×C4semidirect
92. C2×C4⋊C8C2×C4semidirect
120. C2×C4×C8C2×C4semidirect
121. C2×C4×C8C2×C4semidirect
123. #385C8semidirect
153. #336C2×C4semidirect
154. #336C2×C4semidirect
155. #336C2×C4semidirect
156. #336C2×C4semidirect
498. 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×C8×C4⋊C42
C2×C4×C22⋊C82
C4×#3852
#109212
#108762
#108742
#108342
#107972
#107122
C8×#952