Finite Group Databasebeta

← #196733

Subgroups — #196733

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2988C211yesmaximal, characteristic, complemented
2#35273C211yesmaximal, characteristic
3#20629C211yesmaximal, characteristic, complemented
4C2×#1547C211yesmaximal, characteristic, complemented
5C2×#1540C211yesmaximal, characteristic, complemented
6#9134C212yesmaximal, complemented
7#9133C212yesmaximal, complemented
8#9130C212yesmaximal, complemented
9#9129C212yesmaximal, complemented
10#8043C211yesmaximal, characteristic, complemented
11C2×#1296C211yesmaximal, characteristic, complemented
12C2×#537—21no—
13C2×#532C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15#2988—24no—
16#2988—24no—
17C2×C8×Q8—21no—
18C2×C8×D4—22no—
19C2×C8×D4—21no—
20C2×#431C2212yes—
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4⋊C8C2212yes—
25C22×C4⋊C8C2211yescharacteristic
26C2×#408—21no—
27#2857C2212yes—
28#2856C2212yes—
29#2620—22no—
30#2538—21no—
31#2538C412yes—
32#2537—21no—
33#2537C412yes—
34#2416—22no—
35#2416C414yes—
36#2311—21no—
37C2×#384C2211yescharacteristic
38#1798—22no—
39#1797C2212yes—
40#1797C2212yes—
41#1729—21no—
42#1547—24no—
43#1540—24no—
44C2×#332C2211yescharacteristic
45C2×#332C2211yescharacteristic
46C2×#332C2212yes—
47C2×#331C2211yescharacteristic
48C2×#331C2211yescharacteristic
49C2×#331C2212yes—
50#1296C2212yes—
51#1296C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54C22×#41—21no—
55C22×#41D411yescharacteristic
56C22×M16—21no—
57C23×C8—22no—
58C23×C8—21no—
59C23×C8—21no—
60#537—44no—
61#537—44no—
62#532—22no—
63#532—22no—
64#532—22no—
65#532—22no—
66C2×C4⋊Q8C2311yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#124—21no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×#121—24no—
72C2×#121—22no—
73C2×#121—22no—
74C2×#121—21no—
75C2×#121—22no—
76C2×#121C2×C412yes—
77C2×#121C2×C414yes—
78C4×#41—22no—
79C4×#41—22no—
80C4×#41—22no—
81C4×#41—22no—
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×Q8D411yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4C2×C412yes—
86C2×C4×D4C2×C412yes—
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4D412yes—
89C2×C4×D4D411yescharacteristic
90C2×#117D412yes—
91C2×#117D411yescharacteristic
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4C2×C412yes—
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97C8×Q8—28no—
98C8×D4—216no—
99C8×D4—28no—
100#431—22no—
101#431—24no—
102#431—24no—
103#431D414yescomplemented
104#423—22no—
105#423—22no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—24no—
110C2×C4⋊C8—24no—
111C2×C4⋊C8—22no—
112C2×C4⋊C8—22no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8D414yescomplemented
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2311yescharacteristic
122#408—44no—
123C2×C22⋊C8—22no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—22no—
126C2×C22⋊C8—22no—
127C2×C4×C8—22no—
128C2×C4×C8—21no—
129#396—24no—
130#396—24no—
131#394—24no—
132#394—24no—
133#391—22no—
134#390—22no—
135#390—22no—
136#390C2×C414yes—
137#389—22no—
138#389C2×C414yes—
139#386—24no—
140#384—21no—
141#384—21no—
142#384C2×C412yes—
143#384C2312yes—
144#380—22no—
145#380—22no—
146#380—22no—
147#380C2×C414yes—
148#336—22no—
149#336—22no—
150#332—22no—
151#332—22no—
152#332—24no—
153#331—22no—
154#331—22no—
155#331—24no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—42no—
161C2×#41—42no—
162C2×#41—42no—
163C2×#41—42no—
164C2×#41—42no—
165C2×#41—42no—
166C2×#41D814yes—
167C2×#41SD1614yes—
168C22×Q8—21no—
169C22×Q8C22⋊C411yescharacteristic
170C22×D4—22no—
171C22×D4—24no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4C22⋊C412yes—
175C22×D4C22⋊C411yescharacteristic
176C22×D4C22×C412yes—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4C22×C412yes—
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C2×M16—42no—
186C2×M16—42no—
187C2×M16—42no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—28no—
192C22×C8—24no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—42no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8—81no—
200C22×C8—82no—
201C22×C8—81no—
202C22×C8—82no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207#127—21no—
208#127—21no—
209#127—21no—
210#127—21no—
211#124—42no—
212#124—42no—
213C22⋊Q8—48no—
214C22⋊Q8—44no—
215C22⋊Q8—44no—
216#121—24no—
217#121—24no—
218#121—48no—
219#121—416no—
220#121—42no—
221#121—42no—
222#121—44no—
223#121—44no—
224#121—82no—
225#121—82no—
226#121—84no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—24no—
238C4×D4—24no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—42no—
242C4×D4—42no—
243#117—22no—
244#117—22no—
245#117—24no—
246#117—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—81no—
257C2×C4⋊C4—81no—
258C2×C4⋊C4—82no—
259C2×C4⋊C4C22⋊C411yescharacteristic
260C2×C4⋊C4C22⋊C412yes—
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22⋊C412yes—
263C2×C4⋊C4C22×C412yes—
264C2×C4⋊C4C22×C414yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4C22⋊C412yes—
272C2×C22⋊C4C22⋊C412yes—
273C2×C22⋊C4C22⋊C412yes—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42C22×C412yes—
279C2×C42C22×C411yescharacteristic
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42#4112yes—
287C2×C42#4112yes—
288C2×C42C2411yescharacteristic
289C4⋊C8—28no—
290C4⋊C8—24no—
291C4⋊C8—22no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—22no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—22no—
300C4⋊C8—42no—
301C4⋊C8—42no—
302C4⋊C8C2×D412yes—
303C4⋊C8C2×D414yes—
304C4⋊C8C2×D412yes—
305C22⋊C8—28no—
306C22⋊C8—24no—
307C22⋊C8—44no—
308C22⋊C8—44no—
309C4×C8—28no—
310C4×C8—24no—
311#95—24no—
312#95—24no—
313C24—22no—
314C24C2×C22⋊C412yes—
315C24C2×C22⋊C411yescharacteristic
316#41—28no—
317#41—28no—
318#41—28no—
319#41—28no—
320#41—84no—
321#41—84no—
322#41—84no—
323#41—84no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—41no—
332C2×Q8#10214yes—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—84no—
358C2×D4—84no—
359C2×D4#10218yes—
360C2×D4#10214yes—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C412yes—
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C4×D412yes—
396C22×C4C4×D412yes—
397C22×C4#12012yes—
398C22×C4#12112yes—
399C22×C4#12112yes—
400C22×C4#12312yes—
401C22×C4C2×D814yes—
402C22×C4C2×D812yes—
403C22×C4C2×SD1614yes—
404C22×C4C2×SD1612yes—
405C22×C4#13512yes—
406C22×C4#13512yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410M16—84no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—42no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—42no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—42no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—42no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—42no—
429C2×C8—84no—
430C2×C8—88no—
431C2×C8—84no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—44no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C22⋊C4—24no—
450C22⋊C4—24no—
451C22⋊C4—24no—
452C22⋊C4—48no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—48no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42C4×D412yes—
465C42C4×D412yes—
466C42#12012yes—
467C42#12112yes—
468C42#12112yes—
469C42#12312yes—
470C42C22×D411yescharacteristic
471C42C22×D411yescharacteristic
472C42C2×#4111yescharacteristic
473C42C2×#4111yescharacteristic
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23#38612yes—
492C23C2×#10214yes—
493C23C2×#10212yes—
494C23#41612yes—
495C23C22×C22⋊C411yescharacteristic
496Q8—28no—
497Q8—81no—
498Q8—81no—
499Q8—81no—
500Q8—81no—
501D4—216no—
502D4—28no—
503D4—82no—
504D4—82no—
505D4—88no—
506D4—81no—
507D4—81no—
508D4—81no—
509D4—81no—
510D4—81no—
511D4—81no—
512D4—81no—
513D4—81no—
514D4—168no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—88no—
552C2×C4—164no—
553C2×C4#38612yes—
554C2×C4#38612yes—
555C2×C4#38612yes—
556C2×C4C2×#10214yes—
557C2×C4C2×#10212yes—
558C2×C4#41612yes—
559C2×C4#46514yes—
560C2×C4#46514yes—
561C2×C4#46614yes—
562C2×C4#46614yes—
563C2×C4C2×C4×D411yescharacteristic
564C2×C4C2×C4×D411yescharacteristic
565C2×C4C2×#12011yescharacteristic
566C2×C4C2×#12111yescharacteristic
567C2×C4C2×#12111yescharacteristic
568C2×C4C2×#12311yescharacteristic
569C2×C4C22×D811yescharacteristic
570C2×C4C22×SD1611yescharacteristic
571C2×C4C2×#13511yescharacteristic
572C2×C4C2×#13511yescharacteristic
573C2×C4#57611yescharacteristic
574C2×C4#57711yescharacteristic
575C2×C4#57711yescharacteristic
576C2×C4#57811yescharacteristic
577C8—44no—
578C8—42no—
579C8—42no—
580C8—81no—
581C8—81no—
582C8—82no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—24no—
590C22—24no—
591C22—24no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22#189314yes—
604C22C2×#38611yescharacteristic
605C22#288811yescharacteristic
606C22#288911yescharacteristic
607C22C22×#10211yescharacteristic
608C22C2×#41611yescharacteristic
609C22#289711yescharacteristic
610C22#289811yescharacteristic
611C4—21no—
612C4—21no—
613C4—22no—
614C4—22no—
615C4—24no—
616C4—42no—
617C4—42no—
618C4—84no—
619C4—82no—
620C4#189314yes—
621C4#301211yescharacteristic
622C4#301311yescharacteristic
623C4C2×#46511yescharacteristic
624C4C2×#46511yescharacteristic
625C4C2×#46611yescharacteristic
626C4C2×#46611yescharacteristic
627C4#309811yescharacteristic
628C4#309911yescharacteristic
629C2—24no—
630C2—44no—
631C2—162no—
632C2#1772011yescharacteristic
633C2#1772111yescharacteristic
634C2#1772611yescharacteristic
635C2#1772711yescharacteristic
636C2#2091511yescharacteristic
637C2C2×#189311yescharacteristic
638C2#2099211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35273C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2988C2semidirect
3. #20629C2semidirect
4. C2×#1547C2semidirect
5. C2×#1540C2semidirect
6. #9134C2semidirect
7. #9133C2semidirect
8. #9130C2semidirect
9. #9129C2semidirect
10. #8043C2semidirect
11. C2×#1296C2semidirect
103. #431D4semidirect
115. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29882
#352732
#206292
C2×#15472
C2×#15402
#91342
#91332
#91302
#91292
#80432
C2×#12962