Finite Group Databasebeta

← #188104

Subgroups — #188104

Order 512.

2358 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#556C211yesmaximal, characteristic
2#20727C212yesmaximal, complemented
3#17398C212yesmaximal, complemented
4#17357C212yesmaximal, complemented
5#8556C212yesmaximal, complemented
6#8555C212yesmaximal, complemented
7#8552C212yesmaximal
8#8551C212yesmaximal, complemented
9C2×#556—28no—
10C2×#556—24no—
11C2×#556—24no—
12C2×#556—24no—
13C2×#556—24no—
14C22×C4⋊Q8C2211yescharacteristic
15C22×#126C411yescharacteristic
16C22×#126C411yescharacteristic
17C22×#126C2211yescharacteristic
18C22×#126C2211yescharacteristic
19C22×#124C411yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C22⋊Q8C411yescharacteristic
22C22×C22⋊Q8C2211yescharacteristic
23C22×C4×Q8C411yescharacteristic
24C22×C4×Q8C411yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26C22×C4×D4C411yescharacteristic
27C22×#117C411yescharacteristic
28C22×#117C2211yescharacteristic
29#2920C2214yes—
30#1821C2214yes—
31#1801—24no—
32#1784C2214yes—
33C4×#101—24no—
34#1473—28no—
35#1464—24no—
36#1463C2214yes—
37#1463C2214yes—
38#1462C2214yes—
39#1462C2214yes—
40#1458—24no—
41C23×Q8C2×C411yescharacteristic
42C23×Q8C2×C411yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47#556—232no—
48#556—232no—
49#556—232no—
50#556—232no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×#126—24no—
55C2×#126—22no—
56C2×#126—22no—
57C2×#126—24no—
58C2×#126—24no—
59C2×#126—24no—
60C2×#126—22no—
61C2×#126—22no—
62C2×#126—24no—
63C2×#126—24no—
64C2×#126—24no—
65C2×#126—22no—
66C2×#126—22no—
67C2×#126—22no—
68C2×#126—22no—
69C2×#126D418yes—
70C2×#124—24no—
71C2×#124—24no—
72C2×#124—22no—
73C2×#124—22no—
74C2×#124—24no—
75C2×#124—24no—
76C2×#124—24no—
77C2×C22⋊Q8—24no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C4×Q8—24no—
85C2×C4×Q8—22no—
86C2×C4×Q8—22no—
87C2×C4×Q8—24no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8D414yes—
96C2×C4×Q8D414yes—
97C2×C4×D4—24no—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4—24no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—24no—
104C2×#117—24no—
105C2×#117—24no—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117D414yes—
109C2×#117D414yes—
110C22×C4⋊C4C2×C411yescharacteristic
111C22×C4⋊C4C2×C411yescharacteristic
112C22×C4⋊C4C2×C411yescharacteristic
113C22×C4⋊C4C2×C411yescharacteristic
114C22×C4⋊C4C2×C411yescharacteristic
115C22×C4⋊C4C2×C411yescharacteristic
116C22×C4⋊C4C2×C411yescharacteristic
117C22×C4⋊C4C2×C411yescharacteristic
118C22×C4⋊C4C2×C411yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4C2×C411yescharacteristic
123C22×C22⋊C4C2×C411yescharacteristic
124C22×C22⋊C4C2×C411yescharacteristic
125C22×C22⋊C4C2×C411yescharacteristic
126C22×C22⋊C4C2×C411yescharacteristic
127C22×C22⋊C4C2×C411yescharacteristic
128C22×C22⋊C4C2×C411yescharacteristic
129C22×C22⋊C4C2×C411yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C42C2×C411yescharacteristic
133C22×C42C2×C411yescharacteristic
134C22×C42C2×C411yescharacteristic
135C22×C42C2×C411yescharacteristic
136C22×C42C2×C411yescharacteristic
137C22×C42C2×C411yescharacteristic
138C22×C42C2311yescharacteristic
139#432—24no—
140#432—24no—
141#431—24no—
142#431—24no—
143#423—24no—
144C2×#101—28no—
145#407C2318yes—
146C4×M16—24no—
147#343—28no—
148#324—28no—
149#324—28no—
150#323—28no—
151#323—28no—
152C25C22×C411yescharacteristic
153C25C22×C411yescharacteristic
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—22no—
160C22×Q8—22no—
161C22×Q8—22no—
162C22×Q8—44no—
163C22×Q8—44no—
164C22×Q8—44no—
165C22×Q8C2×D414yes—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—44no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4C22⋊C414yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C2411yescharacteristic
192C2×M16—28no—
193C4⋊Q8—232no—
194#126—232no—
195#126—216no—
196#126—216no—
197#126—216no—
198#126—216no—
199#126—28no—
200#126—28no—
201#126—28no—
202#126—28no—
203#124—232no—
204#124—216no—
205#124—216no—
206C22⋊Q8—216no—
207C22⋊Q8—216no—
208C22⋊Q8—232no—
209C4×Q8—216no—
210C4×Q8—216no—
211C4×Q8—216no—
212C4×Q8—216no—
213C4×Q8—28no—
214C4×Q8—28no—
215C4×Q8—28no—
216C4×Q8—28no—
217C4×D4—216no—
218C4×D4—216no—
219#117—216no—
220#117—216no—
221#117—28no—
222#117—28no—
223#117—28no—
224#117—28no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4C22⋊C412yes—
256C2×C4⋊C4C22⋊C412yes—
257C2×C4⋊C4C22⋊C412yes—
258C2×C4⋊C4C22⋊C412yes—
259C2×C4⋊C4C22⋊C414yes—
260C2×C4⋊C4C22⋊C414yes—
261C2×C4⋊C4C22⋊C412yes—
262C2×C4⋊C4C22⋊C412yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D414yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4C22⋊C414yes—
294C2×C22⋊C4C22⋊C412yes—
295C2×C22⋊C4C22⋊C412yes—
296C2×C22⋊C4C22⋊C414yes—
297C2×C22⋊C4C2×D414yes—
298C2×C22⋊C4C2×D414yes—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42C22⋊C414yes—
318C2×C42C22⋊C414yes—
319C2×C42C22⋊C414yes—
320C2×C42C2×D414yes—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C4⋊C8—44no—
326C4⋊C8—44no—
327#101—416no—
328C22⋊C8—28no—
329C22⋊C8—28no—
330#97—44no—
331C4×C8—44no—
332C24—21no—
333C24—21no—
334C24—21no—
335C24—21no—
336C24—42no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—42no—
341C24—42no—
342C24C2×C22⋊C412yes—
343C24C2×C22⋊C412yes—
344C24C23×C411yescharacteristic
345C2×Q8—28no—
346C2×Q8—28no—
347C2×Q8—28no—
348C2×Q8—28no—
349C2×Q8—44no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×D4—28no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—44no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—28no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4C2×C22⋊C412yes—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C2×C22⋊C412yes—
415C22×C4C2×C22⋊C412yes—
416C22×C4C2×C22⋊C412yes—
417C22×C4C2×C22⋊C412yes—
418C22×C4C2×C22⋊C412yes—
419C22×C4C2×C22⋊C412yes—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C412yes—
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C412yes—
427C22×C4C2×C22⋊C412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431M16—416no—
432C2×C8—48no—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—28no—
436C4⋊C4—28no—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—28no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C22⋊C4—28no—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—28no—
455C22⋊C4—28no—
456C22⋊C4—28no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C42—28no—
464C42—28no—
465C42—28no—
466C42—24no—
467C42—24no—
468C42—24no—
469C42—24no—
470C42—24no—
471C42—24no—
472C42#12014yes—
473C42#12014yes—
474C42#12014yes—
475C42#12014yes—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—24no—
482C23—24no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—88no—
500C23—88no—
501C23C22×C22⋊C411yescharacteristic
502C23C22×C22⋊C411yescharacteristic
503C23C22×C22⋊C411yescharacteristic
504Q8—416no—
505Q8—416no—
506Q8—416no—
507D4—416no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—416no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4#37912yes—
563C2×C4#37912yes—
564C2×C4#37912yes—
565C2×C4#37912yes—
566C2×C4#41014yes—
567C2×C4#41314yes—
568C2×C4C2×#12014yes—
569C8—88no—
570C22—28no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22C2×#37911yescharacteristic
590C22C2×#41012yes—
591C22C2×#41312yes—
592C22#288912yes—
593C4—24no—
594C4—24no—
595C4—24no—
596C4—22no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—42no—
608C2—22no—
609C2—22no—
610C2—22no—
611C2—82no—
612C2—82no—
613C2#2058411yescharacteristic
614C2#2059111yescharacteristic
615C2#2059311yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. #17398C2semidirect
4. #17357C2semidirect
5. #8556C2semidirect
6. #8555C2semidirect
8. #8551C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5562
#207272
#173982
#173572
#85562
#85552
#85522
#85512