Finite Group Databasebeta

← #6343659

Subgroups — #6343659

Order 512.

2833 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3507C211yesmaximal, characteristic
2C2×#2983C211yesmaximal, characteristic, complemented
3#34187C211yesmaximal, characteristic, complemented
4#17624C2116yesmaximal, complemented
5#17596C212yesmaximal, complemented
6#17595C212yesmaximal
7#17581C214yesmaximal, complemented
8#17567C214yesmaximal, complemented
9#3507—24no—
10#3507—24no—
11#3507C418yescomplemented
12#3507C2218yes—
13C2×#559C412yescomplemented
14C2×#557C412yescomplemented
15C2×#556—22no—
16C2×#556C412yescomplemented
17C2×#554C412yescomplemented
18C2×#552—22no—
19C2×#539—21no—
20C2×#539—21no—
21C2×#536—21no—
22C2×#533C2211yescharacteristic
23C2×#532C2211yescharacteristic
24C2×#529C2212yes—
25C2×#529C2212yes—
26C2×#520—21no—
27C22×C4×Q8C2211yescharacteristic
28#2983—22no—
29#2983—22no—
30#2983C2218yescomplemented
31C2×#446—24no—
32C2×#432C2212yes—
33C2×#432C2212yes—
34C2×#431C2212yes—
35#2356C2218yescomplemented
36#2355—24no—
37#2313—21no—
38#2313—21no—
39#2310—21no—
40#2285C2212yes—
41#2283C2212yes—
42#2283C2212yes—
43#2274—21no—
44#1521C22116yescomplemented
45#1520C2218yes—
46#1519C2218yes—
47#1515C2218yescomplemented
48#1514C2218yes—
49#1512C22116yescomplemented
50#1509C22116yescomplemented
51#1506C2218yescomplemented
52#1505C2218yes—
53#1500C22116yescomplemented
54C22×#41—21no—
55C23×Q8D411yescharacteristic
56#559—28no—
57#559C2×C4116yes—
58#557—28no—
59#557C2×C4116yes—
60#556—216no—
61#556—28no—
62#556—216no—
63#556C2×C4116yes—
64#554—28no—
65#554C2×C4116yes—
66#552—216no—
67#552—216no—
68#539—28no—
69#539—28no—
70#539—28no—
71#539—28no—
72#536—28no—
73#536—28no—
74#533—22no—
75#533—22no—
76#533C2×C414yes—
77#533C2314yes—
78#532—22no—
79#532—22no—
80#532C2×C414yes—
81#532C2314yes—
82C2×C4⋊Q8—22no—
83C2×C4⋊Q8C2×C411yescharacteristic
84C2×C4⋊Q8C2×C411yescharacteristic
85C2×C4⋊Q8C2×C411yescharacteristic
86C2×#127C2×C411yescharacteristic
87#529—24no—
88#529—24no—
89#529—24no—
90#529—24no—
91#529C2×C418yes—
92#529C2×C418yes—
93#529C2318yes—
94#529C2318yes—
95C2×#126—22no—
96C2×#126—22no—
97C2×#126—22no—
98C2×#126—22no—
99C2×#125—22no—
100C2×#125—22no—
101C2×#125—21no—
102C2×#125C2×C412yes—
103C2×#125C2×C412yes—
104C2×#124—22no—
105C2×#124—21no—
106C2×#124C2×C412yes—
107C2×#124C2×C412yes—
108C2×#123—22no—
109C2×#123—22no—
110C2×#123—22no—
111C2×#123—22no—
112C2×C22⋊Q8—22no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8C2312yes—
118C2×#121—21no—
119C2×#121C2312yes—
120#520—28no—
121#520—28no—
122C2×C4×Q8—22no—
123C2×C4×Q8—21no—
124C2×C4×Q8C2×C412yes—
125C2×C4×Q8C2×C412yes—
126C2×C4×Q8C2×C412yes—
127C2×C4×Q8C2×C412yes—
128C2×C4×Q8C2×C412yes—
129C2×C4×Q8C2×C414yes—
130C2×C4×Q8D412yes—
131C2×C4×Q8D414yes—
132C2×C4×Q8D412yes—
133C2×C4×Q8D414yes—
134C2×C4×Q8C2314yes—
135C2×C4×D4—22no—
136C2×C4×D4—22no—
137C2×C4×D4—22no—
138C2×C4×D4—21no—
139C2×C4×D4C2×C412yes—
140C2×C4×D4C2×C412yes—
141C2×C4×D4C2×C412yes—
142C2×#117—21no—
143C2×#117—22no—
144C2×#117—22no—
145C2×#117—22no—
146C2×#117—22no—
147C22×C4⋊C4—22no—
148C22×C4⋊C4D411yescharacteristic
149C22×C4⋊C4D411yescharacteristic
150C22×C4⋊C4D411yescharacteristic
151C22×C42C2311yescharacteristic
152C22×C42C2311yescharacteristic
153C22×C42C2311yescharacteristic
154#446—216no—
155#446—216no—
156#432—24no—
157#432—22no—
158#432—22no—
159#432C2314yes—
160#432C2314yes—
161#432C2318yes—
162#431—24no—
163#431C2318yes—
164C2×C4⋊C8—22no—
165C2×C4⋊C8—22no—
166C2×C4⋊C8—22no—
167C2×C22⋊C8C2314yes—
168C2×#97—22no—
169C2×#97—22no—
170C2×C4×C8—22no—
171#391—28no—
172#387—28no—
173#387—28no—
174#384—22no—
175#383C2314yes—
176#383C2314yes—
177#382—22no—
178#382—22no—
179#382C2314yes—
180#381—22no—
181#381—22no—
182#380—22no—
183#380C2314yes—
184C4×C4⋊C4—28no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4C2314yes—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191#376C2314yes—
192#376C2314yes—
193#328C23132yes—
194#327C23132yes—
195C2×#41—24no—
196C2×#41—24no—
197C2×#41—48no—
198C22×Q8—21no—
199C22×Q8—21no—
200C22×Q8—44no—
201C22×Q8C22⋊C412yes—
202C22×Q8C22⋊C412yes—
203C22×Q8C22⋊C414yes—
204C22×Q8C22×C412yes—
205C22×Q8C2×D414yes—
206C22×D4—22no—
207C22×D4—21no—
208C22×D4—21no—
209C22×D4C22×C412yes—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D411yescharacteristic
219C23×C4C2411yescharacteristic
220C22×C8—24no—
221C4⋊Q8—216no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225C4⋊Q8C22×C414yes—
226C4⋊Q8C22×C414yes—
227C4⋊Q8C22×C414yes—
228#127—22no—
229#127C22×C414yes—
230#126—216no—
231#126—216no—
232#126—28no—
233#126—28no—
234#126—28no—
235#126—28no—
236#125—216no—
237#125—216no—
238#125—24no—
239#125—24no—
240#125—24no—
241#125—24no—
242#125C22×C418yes—
243#125C22×C418yes—
244#124—216no—
245#124—24no—
246#124—24no—
247#124—24no—
248#124—24no—
249#124C22×C418yes—
250#124C22×C418yes—
251#123—28no—
252#123—28no—
253#123—28no—
254#123—28no—
255#123—216no—
256#123—216no—
257C22⋊Q8—216no—
258C22⋊Q8—216no—
259C22⋊Q8—24no—
260C22⋊Q8—24no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263C22⋊Q8—216no—
264C22⋊Q8—216no—
265C22⋊Q8C22×C414yes—
266C22⋊Q8C2414yes—
267#121—24no—
268#121—24no—
269#121—22no—
270#121—22no—
271#121C22×C414yes—
272#121C2414yes—
273C4×Q8—216no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—24no—
279C4×Q8C22⋊C418yes—
280C4×Q8C22⋊C418yes—
281C4×Q8C22⋊C418yes—
282C4×Q8C22⋊C418yes—
283C4×Q8C22×C418yes—
284C4×Q8C22×C418yes—
285C4×Q8C22×C418yes—
286C4×Q8C22×C418yes—
287C4×Q8C22×C418yes—
288C4×Q8C2×D418yes—
289C4×Q8C2×D418yes—
290C4×D4—28no—
291C4×D4—28no—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—216no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—24no—
299C4×D4—24no—
300C4×D4C22×C418yes—
301C4×D4C22×C418yes—
302C4×D4C22×C418yes—
303#117—24no—
304#117—24no—
305#117—216no—
306#117—28no—
307#117—28no—
308#117—28no—
309#117—28no—
310#117—48no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—216no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4C22⋊C412yes—
328C2×C4⋊C4C22⋊C412yes—
329C2×C4⋊C4C22⋊C414yes—
330C2×C4⋊C4C22⋊C412yes—
331C2×C4⋊C4C22⋊C412yes—
332C2×C4⋊C4C22⋊C412yes—
333C2×C4⋊C4C22⋊C412yes—
334C2×C4⋊C4C22⋊C414yes—
335C2×C4⋊C4C22⋊C414yes—
336C2×C4⋊C4C22×C412yes—
337C2×C4⋊C4C22×C412yes—
338C2×C4⋊C4C2×D414yes—
339C2×C4⋊C4C2×D414yes—
340C2×C4⋊C4C2×D414yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—24no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42C22×C411yescharacteristic
358C2×C42C22×C412yes—
359C2×C42C22×C412yes—
360C2×C42C22×C411yescharacteristic
361C2×C42C22×C412yes—
362C2×C42C22×C412yes—
363C2×C42C22×C412yes—
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2×D412yes—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2×D412yes—
374C2×C42C2412yes—
375C2×C42C2412yes—
376C2×C42C2412yes—
377C4⋊C8—28no—
378C4⋊C8—28no—
379C4⋊C8—24no—
380C4⋊C8—24no—
381C22⋊C8—24no—
382C22⋊C8C2418yes—
383#97—28no—
384#97—24no—
385#97—24no—
386C4×C8—28no—
387#95—24no—
388#95—24no—
389#95—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95C2418yes—
395C24—22no—
396C24C22×D411yescharacteristic
397#41—416no—
398#41—416no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—48no—
405C2×Q8—44no—
406C2×Q8—44no—
407C2×Q8—44no—
408C2×Q8—44no—
409C2×Q8—82no—
410C2×Q8C2×C22⋊C418yes—
411C2×Q8C2×C22⋊C418yes—
412C2×Q8C23×C414yes—
413C2×D4—24no—
414C2×D4—24no—
415C2×D4—22no—
416C2×D4—24no—
417C2×D4—24no—
418C2×D4—44no—
419C2×D4—44no—
420C2×D4—48no—
421C2×D4—82no—
422C2×D4C23×C414yes—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—42no—
434C22×C4—48no—
435C22×C4—81no—
436C22×C4—81no—
437C22×C4—81no—
438C22×C4—84no—
439C22×C4—84no—
440C22×C4C2×C22⋊C411yescharacteristic
441C22×C4C2×C22⋊C412yes—
442C22×C4C2×C22⋊C412yes—
443C22×C4C2×C22⋊C411yescharacteristic
444C22×C4C2×C22⋊C412yes—
445C22×C4C2×C22⋊C412yes—
446C22×C4C2×C22⋊C412yes—
447C22×C4C2×C22⋊C412yes—
448C22×C4C2×C22⋊C412yes—
449C22×C4C2×C22⋊C411yescharacteristic
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C412yes—
452C22×C4C2×C22⋊C412yes—
453C22×C4C2×C22⋊C412yes—
454C22×C4C23×C411yescharacteristic
455C22×C4C23×C411yescharacteristic
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C2511yescharacteristic
466C2×C8—28no—
467C2×C8—48no—
468C4⋊C4—24no—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—28no—
474C4⋊C4—28no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—28no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—84no—
486C4⋊C4C2×C22⋊C418yes—
487C4⋊C4C2×C22⋊C418yes—
488C4⋊C4C2×C22⋊C418yes—
489C4⋊C4C2×C22⋊C418yes—
490C4⋊C4C2×C22⋊C418yes—
491C4⋊C4C2×C22⋊C418yes—
492C4⋊C4C23×C414yes—
493C4⋊C4C23×C414yes—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—24no—
497C22⋊C4—28no—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—24no—
505C22⋊C4—48no—
506C42—28no—
507C42—28no—
508C42—28no—
509C42—48no—
510C42—84no—
511C42—84no—
512C42C2×C22⋊C414yes—
513C42C2×C22⋊C414yes—
514C42C2×C22⋊C412yes—
515C42C2×C22⋊C412yes—
516C42C2×C22⋊C414yes—
517C42C2×C22⋊C414yes—
518C42C2×C22⋊C412yes—
519C42C2×C22⋊C412yes—
520C42C23×C412yes—
521C42C23×C414yes—
522C42C23×C414yes—
523C42C23×C412yes—
524C42C22×D414yes—
525C42C22×D414yes—
526C42C22×D412yes—
527C42C22×D412yes—
528C23—24no—
529C23—41no—
530C23—41no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23C22×C22⋊C411yescharacteristic
535C23C22×C22⋊C411yescharacteristic
536C23#57514yes—
537C23C23×D411yescharacteristic
538Q8—48no—
539Q8—48no—
540Q8—44no—
541Q8—48no—
542Q8—44no—
543Q8—84no—
544D4—44no—
545D4—48no—
546D4—48no—
547D4—48no—
548D4—44no—
549D4—84no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—416no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—48no—
585C2×C4—48no—
586C2×C4—48no—
587C2×C4—88no—
588C2×C4—88no—
589C2×C4C22×C22⋊C412yes—
590C2×C4C22×C22⋊C414yes—
591C2×C4C22×C22⋊C414yes—
592C2×C4C22×C22⋊C411yescharacteristic
593C2×C4C22×C22⋊C414yes—
594C2×C4C22×C22⋊C414yes—
595C2×C4C22×C22⋊C412yes—
596C2×C4C22×C22⋊C412yes—
597C2×C4C22×C22⋊C412yes—
598C2×C4C22×C22⋊C411yescharacteristic
599C2×C4C24×C411yescharacteristic
600C2×C4C23×D411yescharacteristic
601C8—48no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—88no—
613C22#289612yes—
614C22#289612yes—
615C22C23×C22⋊C411yescharacteristic
616C22C2×#57512yes—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—22no—
621C4—22no—
622C4—24no—
623C4—24no—
624C4—24no—
625C4—24no—
626C4—22no—
627C4—22no—
628C4—44no—
629C4—44no—
630C4—44no—
631C4—88no—
632C4#288412yes—
633C2—22no—
634C2—22no—
635C2—22no—
636C2—84no—
637C2C2×#288411yescharacteristic
638C2C2×#289611yescharacteristic
639C2#6009311yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2983C2semidirect
3. #34187C2semidirect
4. #17624C2semidirect
5. #17596C2semidirect
7. #17581C2semidirect
8. #17567C2semidirect
11. #3507C4semidirect
13. C2×#559C4semidirect
14. C2×#557C4semidirect
16. C2×#556C4semidirect
17. C2×#554C4semidirect
30. #2983C22semidirect
35. #2356C22semidirect
44. #1521C22semidirect
47. #1515C22semidirect
49. #1512C22semidirect
50. #1509C22semidirect
51. #1506C22semidirect
53. #1500C22semidirect

Maximal subgroups

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

subgroupindex
C2×#35072
C2×#29832
#341872
#176242
#175962
#175952
#175812
#175672