Finite Group Databasebeta

← #6435562

Subgroups — #6435562

Order 512.

2150 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3141C211yesmaximal, characteristic, complemented
2C2×#2922C211yesmaximal, characteristic, complemented
3#35279C211yesmaximal, characteristic, complemented
4#35273C211yesmaximal, characteristic
5#29857C218yesmaximal, complemented
6#29845C212yesmaximal, complemented
7#29844C212yesmaximal, complemented
8#29843C212yesmaximal, complemented
9#29842C212yesmaximal, complemented
10#23616C214yesmaximal, complemented
11#23485C214yesmaximal, complemented
12C2×#1990C211yesmaximal, characteristic, complemented
13#21195C211yesmaximal, characteristic, complemented
14#20800C211yesmaximal, characteristic, complemented
15C2×#537C2212yes—
16C2×#532C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19#3141—28no—
20C2×#489C2214yes—
21C2×#487C2212yes—
22C2×#486C2212yes—
23#2922—28no—
24C2×#433C2214yes—
25C2×#431C2212yes—
26C2×#431C2212yes—
27C2×#428C2214yes—
28C2×#424C2212yes—
29C2×#417C2214yes—
30C2×#416C2214yes—
31C2×#408C2212yes—
32C2×C4×M16C2211yescharacteristic
33C22×C4×C8C2211yescharacteristic
34#2857C2212yes—
35#2857C2212yes—
36#2857C2212yes—
37#2856C2212yes—
38#2628C2214yes—
39#2620C2214yes—
40#2568—21no—
41#2567—21no—
42#2563—21no—
43#2542—21no—
44#2538—21no—
45#2538—21no—
46#2537—21no—
47#2537—21no—
48#2487—22no—
49#2416—22no—
50#2416—22no—
51#2409—22no—
52#2317C2212yes—
53#2311C2212yes—
54#2311C2212yes—
55#2285C2211yescharacteristic
56C2×#384C2211yescharacteristic
57#2068C2218yes—
58#2067C2218yes—
59#2066C2218yes—
60#2065C2218yes—
61#2036C2218yes—
62#2035C2218yes—
63#2034C2218yes—
64#2033C2218yes—
65#1990C2212yes—
66#1990C2212yes—
67#1990C2214yes—
68#1958C2214yes—
69#1957C2211yescharacteristic
70#1957C2211yescharacteristic
71#1956C2211yescharacteristic
72#1956C2211yescharacteristic
73#1847C2214yes—
74#1845C2212yes—
75#1844C2212yes—
76C2×#340C2214yes—
77C22×#41C2312yes—
78C22×#41C2311yescharacteristic
79C23×Q8—21no—
80C22×M16C2312yes—
81C23×C8C2312yes—
82#537—48no—
83#532—24no—
84#532—24no—
85C2×C4⋊Q8—21no—
86C2×C4⋊Q8—21no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×#127C2311yescharacteristic
93C2×#124C2312yes—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8C2314yes—
98C2×#121—24no—
99C2×#121—22no—
100C2×#121—21no—
101C2×#121—21no—
102C2×#121—22no—
103C2×#121—22no—
104C2×#121C2314yes—
105C4×#41—24no—
106C4×#41—24no—
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8D411yescharacteristic
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4D411yescharacteristic
112C2×C4×D4D412yes—
113C2×C4×D4D412yes—
114C2×C4×D4D411yescharacteristic
115C2×#117C2312yes—
116C2×#117C2311yescharacteristic
117C2×#117C2312yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125#489—216no—
126#487—28no—
127#486—28no—
128#433—216no—
129#431—28no—
130#431—28no—
131#428—48no—
132C2×#107C2318yes—
133C2×#106C2318yes—
134#424—28no—
135C2×C4⋊C8C2314yes—
136C2×C4⋊C8C2314yes—
137#417—216no—
138#416—48no—
139C2×#103C2318yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2318yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145#408—28no—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148C4×M16D418yescomplemented
149C2×#97C2312yes—
150C2×C4×C8—24no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2312yes—
154C2×C4×C8C2312yes—
155#400—24no—
156#399—24no—
157#396—24no—
158#396—24no—
159#395—24no—
160#394—24no—
161#394—24no—
162#393—24no—
163#391—22no—
164#391—22no—
165#391—22no—
166#391C2314yes—
167#390—22no—
168#390—22no—
169#390—22no—
170#390C2314yes—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2314yes—
176#389C2314yes—
177#386—24no—
178#386—24no—
179#385—24no—
180#384—22no—
181#384—21no—
182#384—21no—
183#384C2312yes—
184#384C2311yescharacteristic
185#384C2311yescharacteristic
186#384C2312yes—
187#382—22no—
188#382C2312yes—
189#380—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—21no—
194#380—21no—
195#380—22no—
196#380—22no—
197#380C2314yes—
198#380C2314yes—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201#340C2318yes—
202#340C2318yes—
203C2×#41—22no—
204C2×#41—22no—
205C2×#41—24no—
206C2×#41—24no—
207C2×#41—24no—
208C2×#41—24no—
209C2×#41—28no—
210C2×#41—28no—
211C22×Q8—22no—
212C22×Q8—22no—
213C22×Q8—44no—
214C22×Q8—82no—
215C22×Q8—82no—
216C22×Q8C2×D411yescharacteristic
217C22×Q8C2412yes—
218C22×D4—22no—
219C22×D4—24no—
220C22×D4—21no—
221C22×D4—21no—
222C22×D4C2×D412yes—
223C22×D4C2×D411yescharacteristic
224C22×D4C2412yes—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4C2×D412yes—
229C23×C4C2×D412yes—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C2×M16—24no—
235C2×M16—24no—
236C2×M16C2×D418yes—
237C22×C8—24no—
238C22×C8—24no—
239C22×C8—24no—
240C22×C8C2412yes—
241C22×C8C2412yes—
242C22×C8C2414yes—
243C4⋊Q8—24no—
244C4⋊Q8—24no—
245C4⋊Q8—22no—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—22no—
249C4⋊Q8—44no—
250C4⋊Q8—44no—
251#127—22no—
252#127—22no—
253#124—24no—
254#124—24no—
255C22⋊Q8—48no—
256C22⋊Q8—48no—
257C22⋊Q8—84no—
258C22⋊Q8—84no—
259#121—48no—
260#121—416no—
261#121—48no—
262#121—44no—
263#121—44no—
264#121—84no—
265#121—84no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×D4—28no—
269C4×D4—28no—
270C4×D4—24no—
271C4×D4—24no—
272C4×D4—44no—
273C4×D4—44no—
274#117—22no—
275#117—22no—
276#117—24no—
277#117—24no—
278#117—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—22no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D414yes—
330C2×C42#4114yes—
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#107—48no—
335#106—48no—
336C4⋊C8—28no—
337C4⋊C8—28no—
338#103—216no—
339#103—44no—
340#103—44no—
341#102—216no—
342#102—44no—
343#102—44no—
344C22⋊C8—28no—
345C22⋊C8—28no—
346#97C2×D418yes—
347C4×C8—22no—
348C4×C8—22no—
349C4×C8—24no—
350C4×C8C2×D418yes—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—24no—
358#95—24no—
359C24—22no—
360C24C22×D412yes—
361C24C22×D411yescharacteristic
362#41—48no—
363#41—48no—
364#41—416no—
365#41—416no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—24no—
372C2×Q8—24no—
373C2×Q8—44no—
374C2×Q8—44no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×Q8—816no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—24no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—24no—
389C2×D4—24no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—48no—
394C2×D4—42no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—42no—
400C2×D4—84no—
401C2×D4—84no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—82no—
427C22×C4—82no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4#12114yes—
431C22×C4#12114yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C22×D412yes—
441C22×C4C2×#4112yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C42−1+414yes—
444C22×C4C2511yescharacteristic
445M16—216no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—48no—
455C2×C8C22×D418yes—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—24no—
461C4⋊C4—24no—
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—28no—
465C4⋊C4—48no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—42no—
475C4⋊C4—44no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—84no—
481C4⋊C4—84no—
482C22⋊C4—24no—
483C22⋊C4—24no—
484C22⋊C4—24no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4—84no—
493C22⋊C4—84no—
494C42—24no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—44no—
499C42#12114yes—
500C42#12114yes—
501C42C22×D412yes—
502C42C22×D412yes—
503C42C2×#4112yes—
504C42C2×#4112yes—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23C2×#12112yes—
524C23#53512yes—
525C23#53612yes—
526C23C23×D411yescharacteristic
527Q8—44no—
528Q8—44no—
529Q8—44no—
530Q8—88no—
531Q8—88no—
532D4—48no—
533D4—44no—
534D4—44no—
535D4—44no—
536D4—88no—
537D4—84no—
538D4—84no—
539D4—168no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—28no—
545C2×C4—28no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—88no—
575C2×C4—88no—
576C2×C4—164no—
577C2×C4C2×#12112yes—
578C2×C4C2×#12112yes—
579C2×C4C2×#12112yes—
580C2×C4C2×#12112yes—
581C2×C4C2×#12112yes—
582C2×C4#53512yes—
583C2×C4#53612yes—
584C2×C4#53718yes—
585C2×C4#54218yes—
586C2×C4#57612yes—
587C2×C4#57712yes—
588C2×C4#57712yes—
589C2×C4#57812yes—
590C2×C4C23×D411yescharacteristic
591C2×C4C22×#4111yescharacteristic
592C2×C4C2×H(5,𝔽2)12yes—
593C2×C4C2×2−1+412yes—
594C8—28no—
595C8—42no—
596C8—42no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—42no—
610C22—82no—
611C22—82no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22#262014yes—
618C22C22×#12111yescharacteristic
619C22C2×#53511yescharacteristic
620C22C2×#53611yescharacteristic
621C22C2×#57611yescharacteristic
622C22C2×#57711yescharacteristic
623C22C2×#57711yescharacteristic
624C22C2×#57811yescharacteristic
625C4—22no—
626C4—22no—
627C4—42no—
628C4—42no—
629C4—42no—
630C4—84no—
631C4—82no—
632C4—82no—
633C4—84no—
634C4—82no—
635C4#262014yes—
636C4#306214yes—
637C4#306314yes—
638C4C2×#53712yes—
639C4C2×#54212yes—
640C2—24no—
641C2—82no—
642C2—162no—
643C2C2×#262011yescharacteristic
644C2C2×#306211yescharacteristic
645C2C2×#306311yescharacteristic
646C2#6082211yescharacteristic
647C2#6082411yescharacteristic
648C2#6088011yescharacteristic
649C2#6088111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35273C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3141C2semidirect
2. C2×#2922C2semidirect
3. #35279C2semidirect
5. #29857C2semidirect
6. #29845C2semidirect
7. #29844C2semidirect
8. #29843C2semidirect
9. #29842C2semidirect
10. #23616C2semidirect
11. #23485C2semidirect
12. C2×#1990C2semidirect
13. #21195C2semidirect
14. #20800C2semidirect
148. C4×M16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#31412
C2×#29222
#352792
#352732
#298572
#298452
#298442
#298432
#298422
#236162
#234852
C2×#19902
#211952
#208002