Finite Group Databasebeta

← #188133

Subgroups — #188133

Order 512.

2428 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#542C211yesmaximal, characteristic
2#21753C212yesmaximal, complemented
3#17392C212yesmaximal, complemented
4#17362C212yesmaximal, complemented
5#8556C212yesmaximal, complemented
6#8555C212yesmaximal, complemented
7#8552C212yesmaximal, complemented
8#8551C212yesmaximal, complemented
9C2×#542—28no—
10C2×#542—28no—
11C2×#542—24no—
12C2×#542—24no—
13C22×C4⋊Q8C2211yescharacteristic
14C22×#126C411yescharacteristic
15C22×#126C2211yescharacteristic
16C22×#124C411yescharacteristic
17C22×#124C411yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#123C411yescharacteristic
20C22×#123C2211yescharacteristic
21C22×C22⋊Q8C411yescharacteristic
22C22×C22⋊Q8C2211yescharacteristic
23C22×C22⋊Q8C2211yescharacteristic
24C22×#121C411yescharacteristic
25C22×C4×Q8C411yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C22×#117C411yescharacteristic
28C2×#407C2214yes—
29#1901—28no—
30#1821C2214yes—
31#1784C2214yes—
32#1468—28no—
33#1465—24no—
34#1463C2214yes—
35#1463C2214yes—
36#1462C2214yes—
37#1462C2214yes—
38#1459—24no—
39C23×Q8C2×C411yescharacteristic
40C23×Q8C2×C411yescharacteristic
41C23×Q8C2311yescharacteristic
42C23×D4C2×C411yescharacteristic
43C23×D4C2×C411yescharacteristic
44C23×D4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C22×M16—24no—
49#542—232no—
50#542—232no—
51#542—232no—
52#542—232no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×#126—24no—
57C2×#126—24no—
58C2×#126—24no—
59C2×#126—24no—
60C2×#126—24no—
61C2×#126—24no—
62C2×#124—24no—
63C2×#124—24no—
64C2×#124—24no—
65C2×#124—24no—
66C2×#124—24no—
67C2×#124—24no—
68C2×#124—24no—
69C2×#124—24no—
70C2×#124—24no—
71C2×#123—22no—
72C2×#123—22no—
73C2×#123—24no—
74C2×#123—22no—
75C2×#123—22no—
76C2×#123—24no—
77C2×#123—24no—
78C2×#123D418yes—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8D418yes—
90C2×#121—24no—
91C2×#121—24no—
92C2×#121—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—22no—
96C2×C4×Q8—22no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4D414yes—
100C2×C4×D4D414yes—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117—22no—
104C2×#117—22no—
105C22×C4⋊C4C2×C411yescharacteristic
106C22×C4⋊C4C2×C411yescharacteristic
107C22×C4⋊C4C2×C411yescharacteristic
108C22×C4⋊C4C2×C411yescharacteristic
109C22×C4⋊C4C2×C411yescharacteristic
110C22×C4⋊C4C2×C411yescharacteristic
111C22×C4⋊C4C2×C411yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4C2×C411yescharacteristic
116C22×C22⋊C4C2×C411yescharacteristic
117C22×C22⋊C4C2×C411yescharacteristic
118C22×C22⋊C4C2×C411yescharacteristic
119C22×C22⋊C4C2×C411yescharacteristic
120C22×C22⋊C4C2×C411yescharacteristic
121C22×C22⋊C4C2×C411yescharacteristic
122C22×C22⋊C4C2×C411yescharacteristic
123C22×C22⋊C4C2×C411yescharacteristic
124C22×C22⋊C4C2×C411yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C42C2×C411yescharacteristic
128C22×C42C2×C411yescharacteristic
129C22×C42C2×C411yescharacteristic
130C22×C42C2×C411yescharacteristic
131C2×#101—28no—
132#407—24no—
133#407—24no—
134#407—24no—
135#407C2318yes—
136C2×C22⋊C8—24no—
137C2×C22⋊C8—24no—
138#343—28no—
139#324—28no—
140#324—28no—
141#323—28no—
142#323—28no—
143C25C22×C411yescharacteristic
144C25C22×C411yescharacteristic
145C25C22×C411yescharacteristic
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—44no—
155C22×Q8—82no—
156C22×Q8—82no—
157C22×Q8C2×D414yes—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—44no—
166C22×D4—82no—
167C22×D4—82no—
168C22×D4C22⋊C414yes—
169C22×D4C22⋊C414yes—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—44no—
176C23×C4—82no—
177C23×C4—82no—
178C23×C4C22⋊C412yes—
179C23×C4C22⋊C412yes—
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×C4C2×D414yes—
192C23×C4C2×D414yes—
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2411yescharacteristic
196C2×M16—28no—
197C2×M16—44no—
198C2×M16—88no—
199C22×C8—44no—
200C4⋊Q8—232no—
201#126—232no—
202#126—232no—
203#124—232no—
204#124—232no—
205#124—232no—
206#123—216no—
207#123—216no—
208#123—28no—
209#123—28no—
210#123—28no—
211#123—28no—
212C22⋊Q8—216no—
213C22⋊Q8—216no—
214C22⋊Q8—232no—
215C22⋊Q8—28no—
216C22⋊Q8—28no—
217C22⋊Q8—28no—
218C22⋊Q8—28no—
219#121—232no—
220C4×Q8—216no—
221C4×Q8—216no—
222C4×D4—28no—
223C4×D4—28no—
224C4×D4—28no—
225C4×D4—28no—
226#117—216no—
227#117—216no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
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—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
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⋊C414yes—
256C2×C4⋊C4C22⋊C414yes—
257C2×C4⋊C4C22⋊C414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D414yes—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
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—22no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
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⋊C4C22⋊C414yes—
292C2×C22⋊C4C22⋊C414yes—
293C2×C22⋊C4C22⋊C414yes—
294C2×C22⋊C4C22⋊C414yes—
295C2×C22⋊C4C22⋊C414yes—
296C2×C22⋊C4C2×D414yes—
297C2×C22⋊C4C2×D414yes—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42C22⋊C412yes—
310C2×C42C22⋊C412yes—
311#101—88no—
312C22⋊C8—28no—
313C22⋊C8—28no—
314C22⋊C8—44no—
315C22⋊C8—44no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24C2×C22⋊C412yes—
330C24C2×C22⋊C412yes—
331C24C2×C22⋊C412yes—
332C24C2×C22⋊C412yes—
333C24C2×C22⋊C412yes—
334C24C23×C411yescharacteristic
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—28no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—84no—
344C2×Q8—84no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×Q8—84no—
348C2×D4—28no—
349C2×D4—28no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—44no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—84no—
356C2×D4—84no—
357C2×D4—84no—
358C2×D4—84no—
359C2×D4—84no—
360C2×D4—84no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—24no—
364C22×C4—28no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
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—24no—
384C22×C4—24no—
385C22×C4—48no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—84no—
402C22×C4—82no—
403C22×C4—84no—
404C22×C4—82no—
405C22×C4—84no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C412yes—
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⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4#12014yes—
423C22×C4#12014yes—
424C22×C4#12014yes—
425C22×C4#12014yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429M16—416no—
430M16—88no—
431C2×C8—48no—
432C2×C8—88no—
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—
446C22⋊C4—28no—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C22⋊C4—28no—
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—
459C42—28no—
460C42—28no—
461C42—28no—
462C42—24no—
463C42—24no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
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—88no—
497C23—88no—
498C23—88no—
499C23#37914yes—
500C23C22×C22⋊C411yescharacteristic
501C23C22×C22⋊C411yescharacteristic
502C23C22×C22⋊C411yescharacteristic
503Q8—416no—
504Q8—88no—
505Q8—88no—
506D4—416no—
507D4—88no—
508D4—88no—
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—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—416no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
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—84no—
562C2×C4—84no—
563C2×C4#37912yes—
564C2×C4#37912yes—
565C2×C4#41314yes—
566C2×C4C2×#12014yes—
567C8—88no—
568C22—28no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—84no—
593C22C2×#37911yescharacteristic
594C22#288312yes—
595C22C2×#41312yes—
596C22#288912yes—
597C4—24no—
598C4—22no—
599C4—22no—
600C4—24no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—42no—
610C2—22no—
611C2—22no—
612C2—22no—
613C2—44no—
614C2—82no—
615C2—82no—
616C2#2058511yescharacteristic
617C2#2059111yescharacteristic
618C2#2059311yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21753C2semidirect
3. #17392C2semidirect
4. #17362C2semidirect
5. #8556C2semidirect
6. #8555C2semidirect
7. #8552C2semidirect
8. #8551C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5422
#217532
#173922
#173622
#85562
#85552
#85522
#85512