Finite Group Databasebeta

← #6374540

Subgroups — #6374540

Order 512.

3516 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3444C211yesmaximal, characteristic
2C22×#432C211yesmaximal, characteristic, complemented
3C22×#423C211yesmaximal, characteristic, complemented
4#21854C214yesmaximal, complemented
5#21853C214yesmaximal, complemented
6#21852C214yesmaximal, complemented
7#21851C214yesmaximal, complemented
8#21301C212yesmaximal, complemented
9#21188C212yesmaximal, complemented
10#20897C214yesmaximal, complemented
11#20629C214yesmaximal, complemented
12C23×M16C2212yes—
13#3444—28no—
14#3444—28no—
15C2×#533C411yescharacteristic
16C2×#533C411yescharacteristic
17C2×#533C2211yescharacteristic
18C2×#532C411yescharacteristic
19C2×#532C411yescharacteristic
20C2×#532C2211yescharacteristic
21C2×#529C412yes—
22C2×#529C412yes—
23C2×#529C412yes—
24C2×#529C412yes—
25C2×#529C2212yes—
26C2×#529C2212yes—
27C2×#525C412yes—
28C2×#525C412yes—
29C2×#525C2212yes—
30C2×C4×#41C2212yes—
31C2×C4×#41C2212yes—
32C2×C4×#41C2212yes—
33C23×C42C2211yescharacteristic
34C2×#432—28no—
35C2×#432C2212yes—
36C2×#432C2212yes—
37C2×#432C2212yes—
38C2×#432C2212yes—
39C2×#423—28no—
40C2×#423C2212yes—
41C2×#423C2212yes—
42C2×#423C2212yes—
43C2×#423C2212yes—
44C22×C4⋊C8C2212yes—
45C22×C4⋊C8C2212yes—
46C22×C4⋊C8C2212yes—
47C2×#408C2218yes—
48C22×C22⋊C8C2212yes—
49C22×#97C2212yes—
50#1985C2218yescomplemented
51#1985C2218yescomplemented
52#1955C2218yescomplemented
53#1955C2218yescomplemented
54#1870C22116yes—
55#1869C22116yes—
56#1798C22116yescomplemented
57#1797C22116yescomplemented
58#1729C2218yes—
59C22×#41C2×C412yes—
60C22×#41C2×C412yes—
61C22×#41C2312yes—
62C24×C4C2×C411yescharacteristic
63C24×C4C2×C411yescharacteristic
64C24×C4C2311yescharacteristic
65C22×M16—28no—
66C22×M16—48no—
67C22×M16C2314yes—
68C22×M16C2314yes—
69C23×C8C2314yes—
70C23×C8C2314yes—
71#533—28no—
72#533—28no—
73#533—24no—
74#533—24no—
75#532—28no—
76#532—28no—
77#532—24no—
78#532—24no—
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×C4⋊Q8C2×C411yescharacteristic
82C2×#127C2×C411yescharacteristic
83#529—216no—
84#529—216no—
85#529—216no—
86#529—216no—
87#529—28no—
88#529—28no—
89#529—28no—
90#529—28no—
91C2×#126C2×C414yes—
92C2×#126C2×C414yes—
93C2×#125C2×C412yes—
94C2×#125C2×C412yes—
95C2×#125C2×C412yes—
96C2×#124C2×C412yes—
97C2×#124C2×C412yes—
98C2×#124C2×C412yes—
99#525—216no—
100#525—216no—
101#525—28no—
102#525—28no—
103C2×#123C2×C414yes—
104C2×#123C2×C414yes—
105C2×#123C2×C414yes—
106C2×C22⋊Q8C2×C412yes—
107C2×C22⋊Q8C2×C412yes—
108C2×C22⋊Q8C2×C412yes—
109C2×C22⋊Q8C2×C412yes—
110C2×C22⋊Q8C2×C412yes—
111C2×C22⋊Q8C2×C412yes—
112C2×#121C2×C412yes—
113C2×#121C2×C412yes—
114C2×#121C2×C412yes—
115C2×#121C2×C412yes—
116C2×#121C2×C412yes—
117C2×#121C2×C412yes—
118C2×#120C2×C414yes—
119C4×#41—28no—
120C4×#41—28no—
121C4×#41—28no—
122C4×#41—28no—
123C4×#41—28no—
124C4×#41—28no—
125C2×C4×Q8C2×C412yes—
126C2×C4×Q8C2×C412yes—
127C2×C4×Q8C2×C412yes—
128C2×C4×Q8C2×C412yes—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×D4C2×C414yes—
132C2×C4×D4C2×C414yes—
133C2×C4×D4C2×C414yes—
134C2×C4×D4C2×C414yes—
135C2×C4×D4C2×C412yes—
136C2×C4×D4C2×C412yes—
137C2×C4×D4C2×C412yes—
138C2×C4×D4C2×C412yes—
139C2×C4×D4C2314yes—
140C2×C4×D4C2314yes—
141C2×C4×D4C2312yes—
142C2×C4×D4C2312yes—
143C2×#117C2×C412yes—
144C2×#117C2×C412yes—
145C2×#117C2×C414yes—
146C2×#117C2×C414yes—
147C2×#117C2314yes—
148C2×#117C2312yes—
149C22×C42C2×C411yescharacteristic
150C22×C42C2×C411yescharacteristic
151C22×C42C2×C411yescharacteristic
152C22×C42C2×C411yescharacteristic
153C22×C42C2×C411yescharacteristic
154C22×C42C2×C411yescharacteristic
155C22×C42D418yes—
156C22×C42D418yes—
157C22×C42C2314yes—
158C22×C42C2311yescharacteristic
159C22×C42C2311yescharacteristic
160C22×C42C2311yescharacteristic
161C22×C42C2311yescharacteristic
162C22×C42C2311yescharacteristic
163C22×C42C2311yescharacteristic
164#432—28no—
165#432—28no—
166#432—28no—
167#432—28no—
168#423—28no—
169#423—28no—
170#423—28no—
171#423—28no—
172C2×C4⋊C8—28no—
173C2×C4⋊C8—28no—
174C2×C4⋊C8—28no—
175C2×C4⋊C8C2314yes—
176C2×C4⋊C8C2314yes—
177C2×C4⋊C8C2314yes—
178C2×C4⋊C8C2314yes—
179C2×C4⋊C8C2314yes—
180C2×C4⋊C8C2314yes—
181#408—416no—
182C2×C22⋊C8—28no—
183C2×C22⋊C8C2314yes—
184C2×C22⋊C8C2314yes—
185C2×C22⋊C8C23132yes—
186C2×#97—28no—
187C2×#97C2314yes—
188C2×#97C2314yes—
189#336C23132yes—
190C25C22×C411yescharacteristic
191C2×#41—28no—
192C2×#41—28no—
193C2×#41—24no—
194C2×#41—24no—
195C2×#41—28no—
196C2×#41—28no—
197C2×#41—48no—
198C2×#41—48no—
199C22×Q8C22×C412yes—
200C22×Q8C22×C412yes—
201C22×Q8C22×C412yes—
202C22×D4C22×C414yes—
203C22×D4C22×C412yes—
204C22×D4C22×C414yes—
205C22×D4C22×C414yes—
206C22×D4C22×C412yes—
207C22×D4C22×C412yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4C22⋊C418yes—
213C23×C4C22⋊C418yes—
214C23×C4C22×C414yes—
215C23×C4C22×C411yescharacteristic
216C23×C4C22×C411yescharacteristic
217C23×C4C22×C411yescharacteristic
218C23×C4C22×C411yescharacteristic
219C23×C4C22×C411yescharacteristic
220C23×C4C22×C411yescharacteristic
221C23×C4C22×C411yescharacteristic
222C23×C4C22×C411yescharacteristic
223C23×C4C22×C414yes—
224C23×C4C22×C411yescharacteristic
225C23×C4C22×C411yescharacteristic
226C23×C4C22×C411yescharacteristic
227C23×C4C22×C411yescharacteristic
228C23×C4C22×C411yescharacteristic
229C23×C4C22×C411yescharacteristic
230C23×C4C22×C411yescharacteristic
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2414yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C2×M16—28no—
238C2×M16—28no—
239C2×M16—216no—
240C2×M16—48no—
241C2×M16—432no—
242C22×C8—28no—
243C22×C8—28no—
244C22×C8—28no—
245C22×C8—28no—
246C22×C8—48no—
247C22×C8C2418yes—
248C22×C8C2418yes—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252#127—24no—
253#126—216no—
254#126—216no—
255#125—28no—
256#125—28no—
257#125—28no—
258#124—28no—
259#124—28no—
260#124—28no—
261#123—216no—
262#123—216no—
263#123—216no—
264C22⋊Q8—28no—
265C22⋊Q8—28no—
266C22⋊Q8—28no—
267C22⋊Q8—28no—
268C22⋊Q8—28no—
269C22⋊Q8—28no—
270#121—28no—
271#121—28no—
272#121—28no—
273#121—28no—
274#121—28no—
275#121—28no—
276#120—216no—
277C4×Q8—28no—
278C4×Q8—28no—
279C4×Q8—28no—
280C4×Q8—28no—
281C4×Q8—24no—
282C4×Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×D4—216no—
286C4×D4—216no—
287C4×D4—216no—
288C4×D4—216no—
289C4×D4—28no—
290C4×D4—28no—
291C4×D4—28no—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—28no—
295C4×D4—28no—
296C4×D4—28no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4—24no—
301#117—28no—
302#117—28no—
303#117—24no—
304#117—24no—
305#117—216no—
306#117—216no—
307#117—28no—
308#117—28no—
309C2×C4⋊C4C22×C412yes—
310C2×C4⋊C4C22×C412yes—
311C2×C4⋊C4C22×C414yes—
312C2×C4⋊C4C22×C414yes—
313C2×C4⋊C4C22×C412yes—
314C2×C4⋊C4C22×C412yes—
315C2×C4⋊C4C22×C414yes—
316C2×C4⋊C4C22×C414yes—
317C2×C4⋊C4C22×C412yes—
318C2×C4⋊C4C22×C412yes—
319C2×C4⋊C4C22×C414yes—
320C2×C4⋊C4C22×C414yes—
321C2×C22⋊C4C22×C414yes—
322C2×C22⋊C4C22×C414yes—
323C2×C22⋊C4C22×C414yes—
324C2×C22⋊C4C22×C414yes—
325C2×C22⋊C4C22×C414yes—
326C2×C22⋊C4C22×C414yes—
327C2×C22⋊C4C22×C414yes—
328C2×C22⋊C4C22×C414yes—
329C2×C22⋊C4C22×C414yes—
330C2×C42—28no—
331C2×C42—28no—
332C2×C42C22⋊C418yes—
333C2×C42C22⋊C418yes—
334C2×C42C22⋊C418yes—
335C2×C42C22⋊C418yes—
336C2×C42C22⋊C418yes—
337C2×C42C22⋊C418yes—
338C2×C42C22×C414yes—
339C2×C42C22×C414yes—
340C2×C42C22×C411yescharacteristic
341C2×C42C22×C411yescharacteristic
342C2×C42C22×C411yescharacteristic
343C2×C42C22×C411yescharacteristic
344C2×C42C22×C411yescharacteristic
345C2×C42C22×C411yescharacteristic
346C2×C42C22×C411yescharacteristic
347C2×C42C22×C411yescharacteristic
348C2×C42C22×C411yescharacteristic
349C2×C42C22×C411yescharacteristic
350C2×C42C22×C411yescharacteristic
351C2×C42C22×C411yescharacteristic
352C2×C42C2×D414yes—
353C2×C42C2×D414yes—
354C2×C42C2×D414yes—
355C2×C42C2×D414yes—
356C2×C42C2×D414yes—
357C2×C42C2×D414yes—
358C2×C42C2×D414yes—
359C2×C42C2×D414yes—
360C2×C42C2×D414yes—
361C2×C42C2×D414yes—
362C2×C42C2×D414yes—
363C2×C42C2×D414yes—
364C2×C42C2414yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C4⋊C8—28no—
370C4⋊C8—28no—
371C4⋊C8—28no—
372C4⋊C8—28no—
373C4⋊C8—28no—
374C4⋊C8—28no—
375C22⋊C8—28no—
376C22⋊C8—28no—
377C22⋊C8—432no—
378#97—28no—
379#97—28no—
380C24—24no—
381C24—24no—
382C24—42no—
383C24C2×C22⋊C414yes—
384C24C23×C414yes—
385C24C23×C411yescharacteristic
386C24C23×C411yescharacteristic
387C24C23×C411yescharacteristic
388#41—432no—
389#41—432no—
390#41—48no—
391#41—48no—
392#41—416no—
393C2×Q8—24no—
394C2×Q8—24no—
395C2×Q8—24no—
396C2×Q8—28no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×D4—28no—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—216no—
403C2×D4—216no—
404C2×D4—24no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—28no—
408C2×D4—44no—
409C2×D4—44no—
410C2×D4—48no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—216no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—216no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—48no—
443C22×C4C2×C22⋊C414yes—
444C22×C4C2×C22⋊C414yes—
445C22×C4C2×C22⋊C414yes—
446C22×C4C2×C22⋊C414yes—
447C22×C4C2×C22⋊C414yes—
448C22×C4C2×C22⋊C414yes—
449C22×C4C2×C22⋊C414yes—
450C22×C4C2×C22⋊C414yes—
451C22×C4C2×C22⋊C414yes—
452C22×C4C2×C22⋊C414yes—
453C22×C4C2×C22⋊C414yes—
454C22×C4C2×C22⋊C414yes—
455C22×C4C2×C22⋊C414yes—
456C22×C4C2×C22⋊C414yes—
457C22×C4C2×C22⋊C414yes—
458C22×C4C23×C414yes—
459C22×C4C23×C414yes—
460C22×C4C23×C414yes—
461C22×C4C23×C411yescharacteristic
462C22×C4C23×C411yescharacteristic
463C22×C4C23×C411yescharacteristic
464C22×C4C23×C411yescharacteristic
465C22×C4C23×C411yescharacteristic
466C22×C4C23×C411yescharacteristic
467C22×C4C23×C411yescharacteristic
468C22×C4C23×C411yescharacteristic
469C22×C4C23×C411yescharacteristic
470C22×C4C23×C411yescharacteristic
471C22×C4C23×C411yescharacteristic
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4H(5,𝔽2)14yes—
479C22×C42−1+414yes—
480C22×C4C2511yescharacteristic
481M16—416no—
482M16—416no—
483C2×C8—28no—
484C2×C8—28no—
485C2×C8—28no—
486C2×C8—216no—
487C2×C8—28no—
488C2×C8—416no—
489C2×C8—416no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—28no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—28no—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—28no—
510C22⋊C4—28no—
511C42—28no—
512C42—28no—
513C42—24no—
514C42—24no—
515C42C2×C22⋊C414yes—
516C42C2×C22⋊C414yes—
517C42C2×C22⋊C414yes—
518C42C2×C22⋊C414yes—
519C42C2×C22⋊C414yes—
520C42C2×C22⋊C414yes—
521C42C2×C22⋊C414yes—
522C42C2×C22⋊C414yes—
523C42C2×C22⋊C414yes—
524C42C2×C22⋊C414yes—
525C42C2×C22⋊C414yes—
526C42C2×C22⋊C414yes—
527C42C22×D412yes—
528C42C22×D412yes—
529C42C22×D412yes—
530C42C22×D412yes—
531C42C22×D412yes—
532C42C22×D412yes—
533C42C22×D412yes—
534C42C22×D412yes—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—28no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—44no—
549C23—416no—
550C23C22×C22⋊C412yes—
551C23C22×C22⋊C412yes—
552C23C22×C22⋊C412yes—
553C23#52014yes—
554C23#56814yes—
555C23C24×C411yescharacteristic
556Q8—48no—
557Q8—48no—
558Q8—48no—
559D4—48no—
560D4—48no—
561D4—416no—
562D4—416no—
563D4—416no—
564D4—48no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—28no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—48no—
606C2×C4C22×C22⋊C412yes—
607C2×C4C22×C22⋊C412yes—
608C2×C4C22×C22⋊C412yes—
609C2×C4C22×C22⋊C412yes—
610C2×C4C22×C22⋊C412yes—
611C2×C4C22×C22⋊C412yes—
612C2×C4C22×C22⋊C412yes—
613C2×C4C22×C22⋊C412yes—
614C2×C4C22×C22⋊C412yes—
615C2×C4C22×C22⋊C412yes—
616C2×C4C22×C22⋊C412yes—
617C2×C4#52014yes—
618C2×C4#53514yes—
619C2×C4#53614yes—
620C2×C4#53718yes—
621C2×C4#56814yes—
622C2×C4C23×D411yescharacteristic
623C2×C4C23×D411yescharacteristic
624C2×C4C2×H(5,𝔽2)12yes—
625C2×C4C2×2−1+412yes—
626C8—48no—
627C8—48no—
628C22—28no—
629C22—28no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—416no—
638C22—44no—
639C22—416no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22#231114yes—
644C22#287914yes—
645C22C23×C22⋊C411yescharacteristic
646C22C2×#52012yes—
647C22C2×#56812yes—
648C22C2×#56812yes—
649C4—22no—
650C4—22no—
651C4—22no—
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4#231114yes—
662C4#287914yes—
663C4C2×#53511yescharacteristic
664C4C2×#53611yescharacteristic
665C4C2×#53712yes—
666C2—24no—
667C2—24no—
668C2—42no—
669C2—84no—
670C2C2×#231111yescharacteristic
671C2C2×#287911yescharacteristic
672C2C2×#287911yescharacteristic
673C2#6026914yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC2×#3444C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C22×#432C2semidirect
3. C22×#423C2semidirect
4. #21854C2semidirect
5. #21853C2semidirect
6. #21852C2semidirect
7. #21851C2semidirect
8. #21301C2semidirect
9. #21188C2semidirect
10. #20897C2semidirect
11. #20629C2semidirect
50. #1985C22semidirect
51. #1985C22semidirect
52. #1955C22semidirect
53. #1955C22semidirect
56. #1798C22semidirect
57. #1797C22semidirect

Maximal subgroups

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

subgroupindex
C2×#34442
C22×#4322
C22×#4232
#218542
#218532
#218522
#218512
#213012
#211882
#208972
#206292