Finite Group Databasebeta

← #193635

Subgroups — #193635

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2984C211yesmaximal, characteristic, complemented
2C2×C4×#137C211yesmaximal, characteristic
3#21849C211yesmaximal, characteristic
4C2×#1488C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic, complemented
6#8952C212yesmaximal, complemented
7#8950C212yesmaximal, complemented
8#8948C212yesmaximal, complemented
9#8946C212yesmaximal, complemented
10#7876C211yesmaximal, characteristic, complemented
11#7850C211yesmaximal, characteristic, complemented
12C22×#137C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15#2984—24no—
16#2984—24no—
17C2×#446—22no—
18C2×#446C2212yes—
19C2×C8×Q8C2212yes—
20C4×#137—24no—
21C4×#137—24no—
22C2×#441C411yescharacteristic
23C2×#441C411yescharacteristic
24C2×#440C411yescharacteristic
25C2×#440C411yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27C2×C4×Q16C2211yescharacteristic
28C2×C4×SD16C2211yescharacteristic
29C2×C4×SD16C2211yescharacteristic
30C2×#432—21no—
31C2×#432C2211yescharacteristic
32C2×#431—21no—
33C2×#431C2211yescharacteristic
34C2×#431C2211yescharacteristic
35C2×#428C411yescharacteristic
36C2×#423C2211yescharacteristic
37C2×#417C411yescharacteristic
38C2×#419C411yescharacteristic
39C2×C4×M16C2211yescharacteristic
40#1984C2212yes—
41#1954C2212yes—
42#1870—21no—
43#1870—21no—
44#1797—22no—
45#1488—24no—
46#1487—24no—
47C2×Q8⋊C8C2212yes—
48C2×Q8⋊C8C2212yes—
49C2×Q8⋊C8C2212yes—
50C2×D4⋊C8C2212yes—
51#1277C2212yes—
52#1277C2212yes—
53#1273C2212yes—
54#1273C2212yes—
55C22×#41C2×C411yescharacteristic
56C23×Q8C2×C411yescharacteristic
57C2×#137—24no—
58C2×#137—24no—
59C2×#137—24no—
60C22×Q16C2×C411yescharacteristic
61C22×Q16C2×C411yescharacteristic
62C22×SD16C2×C411yescharacteristic
63C22×SD16C2×C411yescharacteristic
64C22×M16—21no—
65C22×M16C2×C411yescharacteristic
66C22×M16C2×C411yescharacteristic
67C4×#41—22no—
68C4×#41—22no—
69C4×#41—22no—
70C4×#41—22no—
71C2×C4×Q8—21no—
72C2×C4×Q8—22no—
73C2×C4×Q8—21no—
74C2×C4×Q8—22no—
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8C2×C411yescharacteristic
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×Q8D414yes—
79C2×C4×Q8D414yes—
80C2×C4×Q8C2311yescharacteristic
81C2×C4×Q8C2311yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2311yescharacteristic
87C2×#117—21no—
88C2×#117C2×C411yescharacteristic
89C22×C4⋊C4—21no—
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C42—21no—
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94#446—216no—
95#446—24no—
96#446—24no—
97C8×Q8—24no—
98C8×Q8—24no—
99#441—24no—
100#441—24no—
101#440—24no—
102#440—24no—
103C4×Q16—22no—
104C4×Q16—22no—
105C4×Q16—22no—
106C4×Q16—22no—
107C4×SD16—22no—
108C4×SD16—22no—
109C4×SD16—22no—
110C4×SD16—22no—
111#432—28no—
112#432—22no—
113#432D414yes—
114#431—28no—
115#431—22no—
116#431—22no—
117#431—22no—
118#431—22no—
119#428—24no—
120C2×#107C2×C411yescharacteristic
121C2×#107C2×C411yescharacteristic
122C2×#106C2×C411yescharacteristic
123C2×#106C2×C411yescharacteristic
124#423—22no—
125#423—22no—
126C2×C4⋊C8—22no—
127C2×C4⋊C8—22no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×C4⋊C8D412yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2312yes—
133#419—24no—
134#417—24no—
135C2×#103C2×C411yescharacteristic
136C2×#103C2×C411yescharacteristic
137C2×#103C2×C411yescharacteristic
138C2×#103C2×C411yescharacteristic
139C2×#103C2×C411yescharacteristic
140C2×#103C2×C411yescharacteristic
141C2×#102C2×C411yescharacteristic
142C2×#102C2×C411yescharacteristic
143C2×C22⋊C8—21no—
144C2×C22⋊C8—21no—
145C2×C22⋊C8—22no—
146C2×C22⋊C8—22no—
147C2×C22⋊C8D411yescharacteristic
148C2×C22⋊C8D411yescharacteristic
149C4×M16—22no—
150C4×M16—22no—
151C2×#97—22no—
152C2×#97—21no—
153C2×#97C2×C411yescharacteristic
154C2×#97C2×C411yescharacteristic
155C2×C4×C8—22no—
156C2×C4×C8C2312yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160#336—22no—
161#336—22no—
162Q8⋊C8—24no—
163Q8⋊C8—24no—
164Q8⋊C8—24no—
165D4⋊C8—24no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—22no—
169C2×#41—22no—
170C2×#41—41no—
171C2×#41—41no—
172C22×Q8—21no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—41no—
177C22×Q8—41no—
178C22×Q8C22⋊C414yes—
179C22×Q8C22×C411yescharacteristic
180C22×Q8C22×C411yescharacteristic
181C22×Q8C22×C411yescharacteristic
182C22×D4—21no—
183C22×D4C22×C411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4#4111yescharacteristic
192#137—232no—
193C2×Q16—22no—
194C2×Q16—22no—
195C2×Q16—22no—
196C2×Q16—22no—
197C2×Q16—22no—
198C2×Q16M1614yescomplemented
199C2×SD16—22no—
200C2×SD16—22no—
201C2×SD16—22no—
202C2×SD16—22no—
203C2×SD16—22no—
204C2×SD16M1614yescomplemented
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—22no—
208C2×M16—22no—
209C2×M16—41no—
210C2×M16—41no—
211C2×M16—42no—
212C2×M16—82no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8C22×C411yescharacteristic
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C2×D412yes—
221C22×C8C2×D412yes—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8C22⋊C414yes—
239C4×Q8C22⋊C414yes—
240C4×Q8C2×D412yes—
241C4×Q8C2×D412yes—
242C4×Q8C2×D412yes—
243C4×Q8C2×D412yes—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—21no—
247C4×D4—21no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—42no—
253C4×D4—42no—
254#117—22no—
255#117—22no—
256#117—42no—
257#117—42no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4C22⋊C414yes—
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—42no—
286C2×C42—42no—
287C2×C42C22×C411yescharacteristic
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2×D411yescharacteristic
292C2×C42#4112yes—
293C2×C42#4111yescharacteristic
294C2×C42C2411yescharacteristic
295#107—22no—
296#107—22no—
297#106—22no—
298#106—22no—
299C4⋊C8—28no—
300C4⋊C8—28no—
301C4⋊C8—24no—
302C4⋊C8—22no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—42no—
306C4⋊C8—42no—
307C4⋊C8C2×D414yes—
308#103—22no—
309#103—22no—
310#103—22no—
311#103—22no—
312#103—22no—
313#103—22no—
314#102—22no—
315#102—22no—
316C22⋊C8—24no—
317C22⋊C8—22no—
318C22⋊C8—22no—
319C22⋊C8—42no—
320C22⋊C8—82no—
321C22⋊C8—82no—
322#97—28no—
323#97—22no—
324#97—22no—
325#97—24no—
326C4×C8—28no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—22no—
332C4×C8—22no—
333C4×C8—22no—
334C4×C8—22no—
335C24—21no—
336C24C4×D411yescharacteristic
337#41—28no—
338#41—44no—
339#41—44no—
340#41—44no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—24no—
347C2×Q8—28no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8C2×C22⋊C412yes—
362C2×Q8C2×C22⋊C412yes—
363C2×Q8C2×M1612yes—
364C2×Q8C2×M1612yes—
365C2×Q8C2×M1612yes—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4C2×M1612yes—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—42no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4C2×C22⋊C412yes—
404C22×C4C4×D412yes—
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4#12012yes—
409C22×C4#12112yes—
410C22×C4#12112yes—
411C22×C4#12312yes—
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C2×#4111yescharacteristic
415Q16—28no—
416Q16—28no—
417SD16—28no—
418SD16—28no—
419M16—28no—
420M16—44no—
421M16—84no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—44no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8C2×M1612yes—
447C2×C8C2×M1612yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—42no—
464C4⋊C4C2×C22⋊C412yes—
465C4⋊C4C2×C22⋊C412yes—
466C22⋊C4—42no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—42no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—24no—
475C42—24no—
476C42—41no—
477C42—41no—
478C42—42no—
479C42—42no—
480C42C2×C22⋊C412yes—
481C42C4×D412yes—
482C42#12012yes—
483C42#12112yes—
484C42#12112yes—
485C42#12312yes—
486C42C22×D411yescharacteristic
487C42C22×D411yescharacteristic
488C42C22×D411yescharacteristic
489C42C2×#4111yescharacteristic
490C23—21no—
491C23—21no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23#38612yes—
500C23C2×C4×D411yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—22no—
504Q8—22no—
505Q8—22no—
506Q8—22no—
507Q8—22no—
508Q8—48no—
509Q8—41no—
510Q8—41no—
511Q8—41no—
512Q8—41no—
513Q8#40714yes—
514D4—22no—
515D4—22no—
516D4—48no—
517D4—41no—
518D4—41no—
519C2×C4—21no—
520C2×C4—21no—
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—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—48no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—84no—
554C2×C4—82no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4#38612yes—
558C2×C4#38612yes—
559C2×C4#38612yes—
560C2×C4C22×C22⋊C411yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C2×#12011yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×#12311yescharacteristic
566C2×C4C22×M1611yescharacteristic
567C2×C4C22×M1611yescharacteristic
568C2×C4#56811yescharacteristic
569C2×C4#56811yescharacteristic
570C2×C4#57712yes—
571C2×C4#57812yes—
572C8—22no—
573C8—22no—
574C8—41no—
575C8—41no—
576C8—42no—
577C8—42no—
578C8—44no—
579C8—82no—
580C8—82no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—42no—
585C22—42no—
586C22—44no—
587C22—42no—
588C22—82no—
589C22—81no—
590C22—81no—
591C22C2×#38611yescharacteristic
592C22D4×M1611yescharacteristic
593C22D4×M1611yescharacteristic
594C22#294712yes—
595C22#295512yes—
596C4—22no—
597C4—24no—
598C4—21no—
599C4—22no—
600C4—21no—
601C4—22no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—81no—
607C4—81no—
608C4C2×#40711yescharacteristic
609C4#287911yescharacteristic
610C4D4×M1611yescharacteristic
611C4D4×M1611yescharacteristic
612C4#301312yes—
613C4#304311yescharacteristic
614C4#304411yescharacteristic
615C2—42no—
616C2—82no—
617C2#1805112yes—
618C2#1805312yes—
619C2#2089811yescharacteristic
620C2#2091711yescharacteristic
621C2#2093011yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2984C2semidirect
5. C2×#1487C2semidirect
6. #8952C2semidirect
7. #8950C2semidirect
8. #8948C2semidirect
9. #8946C2semidirect
10. #7876C2semidirect
11. #7850C2semidirect
198. C2×Q16M16semidirect
204. C2×SD16M16semidirect

Maximal subgroups

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

subgroupindex
C2×#29842
C2×C4×#1372
#218492
C2×#14882
C2×#14872
#89522
#89502
#89482
#89462
#78762
#78502