Finite Group Databasebeta

← #196659

Subgroups — #196659

Order 512.

1410 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2925C211yesmaximal, characteristic, complemented
2#35273C211yesmaximal, characteristic
3#21847C211yesmaximal, characteristic, complemented
4C2×#1547C211yesmaximal, characteristic, complemented
5C2×#1540C211yesmaximal, characteristic, complemented
6#9142C212yesmaximal, complemented
7#9140C212yesmaximal, complemented
8#9134C212yesmaximal, complemented
9#9132C212yesmaximal, complemented
10#8043C211yesmaximal, characteristic, complemented
11C2×#1296C211yesmaximal, characteristic, complemented
12C2×#537—21no—
13C2×#532C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15#2925—24no—
16#2925—24no—
17C2×#433—22no—
18C2×#432C2212yes—
19C2×#431C2211yescharacteristic
20C2×#431C2212yes—
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C2×#408—21no—
25C2×#405—21no—
26#2857C2212yes—
27#2856C2212yes—
28#2620—22no—
29#2538—21no—
30#2538C412yes—
31#2537—21no—
32#2537C412yes—
33#2416—22no—
34#2416C414yes—
35#2311—21no—
36C2×#384C2211yescharacteristic
37#1984—21no—
38#1954—21no—
39#1869—22no—
40#1797C2212yes—
41#1797C2212yes—
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—21no—
58C23×C8—21no—
59#537—44no—
60#537—44no—
61#532—22no—
62#532—22no—
63#532—22no—
64#532—22no—
65C2×C4⋊Q8C2311yescharacteristic
66C2×#127C2311yescharacteristic
67C2×#124—21no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×#121—24no—
71C2×#121—22no—
72C2×#121—22no—
73C2×#121—21no—
74C2×#121—22no—
75C2×#121C2×C412yes—
76C2×#121C2×C414yes—
77C4×#41—22no—
78C4×#41—22no—
79C4×#41—22no—
80C4×#41—22no—
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8D411yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4C2×C412yes—
85C2×C4×D4C2×C412yes—
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4D412yes—
88C2×C4×D4D411yescharacteristic
89C2×#117D411yescharacteristic
90C2×#117D412yes—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4C2×C412yes—
94C22×C42C2312yes—
95C22×C42C2311yescharacteristic
96#433—216no—
97#432—24no—
98#432—24no—
99#431—24no—
100#431—24no—
101#431—22no—
102#431—22no—
103#423—22no—
104#423D414yescomplemented
105C2×C4⋊C8—21no—
106C2×C4⋊C8—22no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—22no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8D414yescomplemented
112C2×C4⋊C8C2312yes—
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2312yes—
118#408—28no—
119C2×C22⋊C8—22no—
120C2×C22⋊C8—22no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—22no—
123C2×C22⋊C8—22no—
124#405—28no—
125C2×#97—22no—
126C2×C4×C8—22no—
127C2×C4×C8—21no—
128#396—24no—
129#396—24no—
130#394—24no—
131#394—24no—
132#391—22no—
133#390—22no—
134#390—22no—
135#390C2×C414yes—
136#389—22no—
137#389C2×C414yes—
138#386—24no—
139#384—21no—
140#384—21no—
141#384C2×C412yes—
142#384C2312yes—
143#380—22no—
144#380—22no—
145#380—22no—
146#380C2×C414yes—
147#336—22no—
148#336—22no—
149#332—22no—
150#332—22no—
151#332—24no—
152#331—22no—
153#331—22no—
154#331—24no—
155C2×#41—22no—
156C2×#41—22no—
157C2×#41—22no—
158C2×#41—22no—
159C2×#41—22no—
160C2×#41—22no—
161C2×#41—42no—
162C2×#41—42no—
163C2×#41—42no—
164C2×#41—42no—
165C2×#41—42no—
166C2×#41—42no—
167C22×Q8—21no—
168C22×Q8C22⋊C411yescharacteristic
169C22×D4—22no—
170C22×D4—24no—
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C22⋊C412yes—
174C22×D4C22⋊C411yescharacteristic
175C22×D4C22×C412yes—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4C22×C412yes—
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C2×M16—24no—
185C2×M16—42no—
186C2×M16—42no—
187C22×C8—24no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—42no—
195C22×C8—42no—
196C22×C8—42no—
197C22×C8—81no—
198C22×C8—81no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203#127—21no—
204#127—21no—
205#127—21no—
206#127—21no—
207#124—42no—
208#124—42no—
209C22⋊Q8—44no—
210C22⋊Q8—44no—
211C22⋊Q8—48no—
212#121—24no—
213#121—24no—
214#121—44no—
215#121—44no—
216#121—416no—
217#121—48no—
218#121—42no—
219#121—42no—
220#121—82no—
221#121—82no—
222#121—84no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×D4—24no—
228C4×D4—24no—
229C4×D4—24no—
230C4×D4—24no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—42no—
238C4×D4—42no—
239#117—22no—
240#117—22no—
241#117—24no—
242#117—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—81no—
253C2×C4⋊C4—81no—
254C2×C4⋊C4—82no—
255C2×C4⋊C4C22⋊C411yescharacteristic
256C2×C4⋊C4C22⋊C412yes—
257C2×C4⋊C4C22⋊C411yescharacteristic
258C2×C4⋊C4C22⋊C412yes—
259C2×C4⋊C4C22×C412yes—
260C2×C4⋊C4C22×C414yes—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4C22⋊C412yes—
268C2×C22⋊C4C22⋊C412yes—
269C2×C22⋊C4C22⋊C412yes—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42C22×C412yes—
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42#4112yes—
283C2×C42#4112yes—
284C2×C42C2411yescharacteristic
285C4⋊C8—24no—
286C4⋊C8—24no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—22no—
290C4⋊C8—22no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—22no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—42no—
297C4⋊C8C2×D412yes—
298C4⋊C8C2×D412yes—
299C4⋊C8C2×D414yes—
300C22⋊C8—28no—
301C22⋊C8—28no—
302C22⋊C8—42no—
303C22⋊C8—82no—
304C22⋊C8—82no—
305#97—28no—
306C4×C8—28no—
307C4×C8—42no—
308#95—24no—
309#95—24no—
310C24—22no—
311C24C2×C22⋊C412yes—
312C24C2×C22⋊C411yescharacteristic
313#41—44no—
314#41—44no—
315#41—44no—
316#41—44no—
317#41—84no—
318#41—84no—
319#41—84no—
320#41—84no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—48no—
344C2×D4—41no—
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—84no—
357C2×D4—84no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
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—24no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—82no—
387C22×C4—82no—
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C4×D412yes—
395C22×C4C4×D412yes—
396C22×C4#12012yes—
397C22×C4#12112yes—
398C22×C4#12112yes—
399C22×C4#12312yes—
400C22×C4#13512yes—
401C22×C4#13512yes—
402C22×C4AGL(1,ℤ/8)12yes—
403C22×C4AGL(1,ℤ/8)12yes—
404C22×C4C23×C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407M16—48no—
408C2×C8—22no—
409C2×C8—24no—
410C2×C8—22no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—42no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—42no—
417C2×C8—48no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—42no—
421C2×C8—41no—
422C2×C8—42no—
423C2×C8—41no—
424C2×C8—42no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—84no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—82no—
444C4⋊C4—82no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—84no—
453C22⋊C4—84no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42C4×D412yes—
461C42C4×D412yes—
462C42#12012yes—
463C42#12112yes—
464C42#12112yes—
465C42#12312yes—
466C42C22×D411yescharacteristic
467C42C22×D411yescharacteristic
468C42C2×#4111yescharacteristic
469C42C2×#4111yescharacteristic
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23#38612yes—
489C23#41612yes—
490C23#41812yes—
491C23C22×C22⋊C411yescharacteristic
492Q8—44no—
493Q8—81no—
494Q8—81no—
495Q8—81no—
496Q8—81no—
497D4—48no—
498D4—44no—
499D4—82no—
500D4—82no—
501D4—88no—
502D4—81no—
503D4—81no—
504D4—81no—
505D4—81no—
506D4—81no—
507D4—81no—
508D4—81no—
509D4—81no—
510D4—168no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
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—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—88no—
549C2×C4—164no—
550C2×C4#38612yes—
551C2×C4#38612yes—
552C2×C4#38612yes—
553C2×C4#41612yes—
554C2×C4#41812yes—
555C2×C4#44914yes—
556C2×C4#44914yes—
557C2×C4#48214yes—
558C2×C4#48214yes—
559C2×C4C2×C4×D411yescharacteristic
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C2×#12011yescharacteristic
562C2×C4C2×#12111yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12311yescharacteristic
565C2×C4C2×#13511yescharacteristic
566C2×C4C2×#13511yescharacteristic
567C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
568C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
569C2×C4#57611yescharacteristic
570C2×C4#57711yescharacteristic
571C2×C4#57711yescharacteristic
572C2×C4#57811yescharacteristic
573C8—42no—
574C8—42no—
575C8—44no—
576C8—81no—
577C8—81no—
578C8—82no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22—82no—
598C22—82no—
599C22#187514yes—
600C22C2×#38611yescharacteristic
601C22#288811yescharacteristic
602C22#288911yescharacteristic
603C22C2×#41611yescharacteristic
604C22C2×#41811yescharacteristic
605C22#289711yescharacteristic
606C22#289811yescharacteristic
607C4—21no—
608C4—21no—
609C4—22no—
610C4—22no—
611C4—42no—
612C4—42no—
613C4—42no—
614C4—84no—
615C4—82no—
616C4#187514yes—
617C4C2×#44911yescharacteristic
618C4C2×#44911yescharacteristic
619C4#306011yescharacteristic
620C4#306111yescharacteristic
621C4#306211yescharacteristic
622C4#306311yescharacteristic
623C4C2×#48211yescharacteristic
624C4C2×#48211yescharacteristic
625C2—24no—
626C2—82no—
627C2—162no—
628C2#1772611yescharacteristic
629C2#1772711yescharacteristic
630C2#1775811yescharacteristic
631C2#1775911yescharacteristic
632C2C2×#187511yescharacteristic
633C2#2097611yescharacteristic
634C2#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×#2925C2semidirect
3. #21847C2semidirect
4. C2×#1547C2semidirect
5. C2×#1540C2semidirect
6. #9142C2semidirect
7. #9140C2semidirect
8. #9134C2semidirect
9. #9132C2semidirect
10. #8043C2semidirect
11. C2×#1296C2semidirect
104. #423D4semidirect
111. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29252
#352732
#218472
C2×#15472
C2×#15402
#91422
#91402
#91342
#91322
#80432
C2×#12962