Finite Group Databasebeta

← #188294

Subgroups — #188294

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#117C211yesmaximal, characteristic
2#20810C212yesmaximal, complemented
3C2×#1464C212yesmaximal
4#8560C212yesmaximal, complemented
5#8559C212yesmaximal
6#8558C212yesmaximal, complemented
7#8557C212yesmaximal
8#7910C212yesmaximal, complemented
9C22×#117C411yescharacteristic
10C22×#117C2211yescharacteristic
11C23×C42C411yescharacteristic
12C2×#432C2214yes—
13C4×#117—22no—
14C4×#117—22no—
15C4×#117—22no—
16C4×#117—22no—
17C2×C4×C4⋊C4C411yescharacteristic
18C2×C4×C4⋊C4C411yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C22⋊C4C411yescharacteristic
22C2×C4×C22⋊C4C411yescharacteristic
23C2×C4×C22⋊C4C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25C2×#376C411yescharacteristic
26C2×#376C2211yescharacteristic
27C2×C43C411yescharacteristic
28C2×C43C2211yescharacteristic
29#1842—22no—
30#1831—22no—
31#1797C2214yes—
32#1797C2214yes—
33#1767—22no—
34#1767—22no—
35#1464—28no—
36C2×#324C2214yes—
37C2×#323C2214yes—
38#1280C2214yes—
39#1280C2214yes—
40C24×C4C2×C411yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2×C411yescharacteristic
43C2×#117—24no—
44C2×#117—24no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—24no—
50C2×#117D414yes—
51C2×#117Q814yes—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4C2×C411yescharacteristic
57C22×C22⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C411yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C42—24no—
61C22×C42—21no—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42—21no—
65C22×C42—24no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C411yescharacteristic
76C22×C42C2×C411yescharacteristic
77C22×C42C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#432—28no—
83#432—28no—
84C2×C4⋊C8—24no—
85C2×C4⋊C8—24no—
86C2×C4⋊C8—24no—
87C2×C22⋊C8C2318yes—
88C2×#97—24no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105#376—22no—
106#376—22no—
107#376—22no—
108#376—22no—
109C2×#95C2×C411yescharacteristic
110C2×#95C2×C411yescharacteristic
111C2×#95C2×C411yescharacteristic
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C411yescharacteristic
114C2×#95C2×C411yescharacteristic
115C2×#95C2311yescharacteristic
116C2×#95C2311yescharacteristic
117C43—21no—
118C43—21no—
119C43—21no—
120C43—21no—
121C43D412yes—
122C43D412yes—
123C43D412yes—
124C43D412yes—
125#336—24no—
126#336—24no—
127#324—28no—
128#323—28no—
129C25C22×C411yescharacteristic
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4C4⋊C414yes—
149C23×C4M1614yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C2411yescharacteristic
164C22×C8—28no—
165#117—216no—
166#117—216no—
167#117—28no—
168#117—28no—
169#117—28no—
170#117—28no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C4⋊C414yes—
181C2×C4⋊C4M1614yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×Q812yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4C4⋊C414yes—
194C2×C22⋊C4M1614yes—
195C2×C22⋊C4C2×D412yes—
196C2×C22⋊C4C2×Q812yes—
197C2×C42—28no—
198C2×C42—28no—
199C2×C42—28no—
200C2×C42—28no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42C22⋊C412yes—
240C2×C42C22⋊C412yes—
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C412yes—
246C2×C42C22⋊C412yes—
247C2×C42C22⋊C412yes—
248C2×C42C22⋊C412yes—
249C2×C42C4⋊C414yes—
250C2×C42M1614yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×Q812yes—
259C2×C42#4114yes—
260C4⋊C8—48no—
261C4⋊C8—48no—
262C4⋊C8—48no—
263C22⋊C8—28no—
264C22⋊C8—28no—
265#97—48no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24C2×C4⋊C412yes—
283C24C2×M1612yes—
284C24C2×M1612yes—
285C24C23×C411yescharacteristic
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—28no—
291C22×C4—28no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—28no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—28no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—48no—
337C22×C4—48no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C4⋊C412yes—
370C22×C4C2×C4⋊C412yes—
371C22×C4C2×C4⋊C412yes—
372C22×C4C2×C4⋊C412yes—
373C22×C4C2×C4⋊C412yes—
374C22×C4#11712yes—
375C22×C4#11712yes—
376C22×C4C2×M1612yes—
377C22×C4C2×M1612yes—
378C22×C4C2×M1612yes—
379C22×C4C2×M1612yes—
380C22×C4C2×M1612yes—
381C22×C4C2×M1612yes—
382C22×C4C2×M1612yes—
383C22×C4C2×M1612yes—
384C22×C4C2×M1612yes—
385C22×C4C2×M1612yes—
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C22×Q811yescharacteristic
389C22×C4C2×#4112yes—
390C2×C8—48no—
391C2×C8—48no—
392C2×C8—48no—
393C4⋊C4—28no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—28no—
397C4⋊C4—28no—
398C22⋊C4—28no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—28no—
402C22⋊C4—28no—
403C42—24no—
404C42—24no—
405C42—24no—
406C42—24no—
407C42—24no—
408C42—24no—
409C42—24no—
410C42—24no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—24no—
416C42—24no—
417C42—24no—
418C42—24no—
419C42—24no—
420C42—24no—
421C42—24no—
422C42—24no—
423C42—42no—
424C42—42no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42#12114yes—
436C42#12114yes—
437C42C22⋊Q814yes—
438C42C22⋊Q814yes—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23#42314yes—
466C23C22×C22⋊C411yescharacteristic
467C23C22×C4⋊C411yescharacteristic
468C23C2×#11711yescharacteristic
469C23C22×M1611yescharacteristic
470C23C22×M1611yescharacteristic
471C23C22×M1611yescharacteristic
472C23C22×M1611yescharacteristic
473C2×C4—216no—
474C2×C4—216no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—48no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4#38012yes—
552C2×C4#38012yes—
553C2×C4#38014yes—
554C2×C4#40712yes—
555C2×C4#40714yes—
556C2×C4#40712yes—
557C2×C4#41012yes—
558C2×C4#41012yes—
559C2×C4#41312yes—
560C2×C4#41312yes—
561C2×C4#42314yes—
562C2×C4C2×#12112yes—
563C2×C4C2×C22⋊Q812yes—
564C8—88no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—24no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22C2×#38011yescharacteristic
589C22C2×#40711yescharacteristic
590C22C2×#40711yescharacteristic
591C22C2×#41011yescharacteristic
592C22C2×#41011yescharacteristic
593C22C2×#41311yescharacteristic
594C22C2×#41311yescharacteristic
595C22C2×#42311yescharacteristic
596C22C2×#42311yescharacteristic
597C22#292011yescharacteristic
598C22#292011yescharacteristic
599C4—22no—
600C4—22no—
601C4—24no—
602C4—22no—
603C4—22no—
604C4—24no—
605C4—22no—
606C4—22no—
607C4—22no—
608C4—24no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—42no—
616C4#146414yescomplemented
617C2—21no—
618C2—21no—
619C2—21no—
620C2—21no—
621C2—44no—
622C2C2×#146411yescharacteristic
623C2#1735711yescharacteristic
624C2#1736311yescharacteristic
625C2#1736611yescharacteristic
626C2#2062611yescharacteristic
627C2#2063211yescharacteristic
628C2#2063711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20810C2semidirect
4. #8560C2semidirect
6. #8558C2semidirect
8. #7910C2semidirect
616. C4#1464semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1172
#208102
C2×#14642
#85602
#85592
#85582
#85572
#79102