Finite Group Databasebeta

← #414862

Subgroups — #414862

Order 512.

1166 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic, complemented
3C2×#1987C211yesmaximal, characteristic, complemented
4C2×#1950C211yesmaximal, characteristic, complemented
5#20629C211yesmaximal, characteristic
6C4×#417C211yesmaximal, characteristic
7#11290C212yesmaximal, complemented
8#11249C212yesmaximal, complemented
9#11192C212yesmaximal
10#11105C212yesmaximal
11#10847C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#3079—24no—
15#3079—24no—
16C2×#477—21no—
17C2×#475—21no—
18C2×#476—21no—
19C2×#474—21no—
20C22×#108C2212yes—
21C2×#428C2212yes—
22C2×#423C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C2×#417C2212yes—
26C2×#417C2211yescharacteristic
27C2×#417C2211yescharacteristic
28C2×#408C2212yes—
29C2×C4×M16C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1997—24no—
32#1987—22no—
33#1987—22no—
34#1950—24no—
35#1798—22no—
36#1797—22no—
37#1763—21no—
38#1762—21no—
39C4×#103—21no—
40C4×#102—21no—
41C2×#343C2212yes—
42C2×#343C2212yes—
43C2×#343C2212yes—
44C2×#343C2212yes—
45C2×#341C2212yes—
46#1736C2211yescharacteristic
47#1736C2211yescharacteristic
48C2×#340C2212yes—
49C2×#340C2212yes—
50#1729C2212yes—
51C22×#41C2311yescharacteristic
52C23×Q8—21no—
53C22×M16C2312yes—
54C22×M16C2312yes—
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C23×C8C2312yes—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8D414yes—
66C4×#41—22no—
67C4×#41—22no—
68C4×#41—22no—
69C4×#41—22no—
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4—21no—
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×#117—21no—
77C2×#117C2311yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85#477—28no—
86#476—28no—
87#475—28no—
88#474—28no—
89C2×#108—24no—
90C2×#108—24no—
91C2×#108—42no—
92#428—24no—
93#428—24no—
94C2×#107—22no—
95C2×#106—22no—
96#423—22no—
97#423—22no—
98#423—22no—
99#423D414yes—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—22no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8D412yes—
107C2×C4⋊C8D412yes—
108#417—22no—
109#417—24no—
110#417—22no—
111#417—24no—
112#417—22no—
113#417Q814yescomplemented
114C2×#103—22no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×#102—22no—
120#408—24no—
121#408—42no—
122C2×C22⋊C8—24no—
123C4×M16—22no—
124C4×M16—22no—
125C2×#97—21no—
126C2×C4×C8—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C2×#95—21no—
132#343—24no—
133#343—24no—
134#343—24no—
135#343—42no—
136#341—24no—
137#340—24no—
138#340—24no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#336—24no—
144C2×#41—21no—
145C2×#41—21no—
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41C2×Q812yes—
155C2×#41#4112yes—
156C22×Q8—24no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—42no—
160C22×Q8—82no—
161C22×Q8—82no—
162C22×Q8C2×D411yescharacteristic
163C22×D4—21no—
164C22×D4C2×D411yescharacteristic
165C23×C4—21no—
166C23×C4—22no—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4#4111yescharacteristic
172C23×C4#4111yescharacteristic
173C23×C4C2411yescharacteristic
174C2×M16—22no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—22no—
188C2×M16—21no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16C2×D414yes—
196C2×M16C2×Q812yes—
197C2×M16#4112yes—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8C2×D414yes—
209C4⋊Q8—24no—
210C4⋊Q8—24no—
211C4⋊Q8—24no—
212C4⋊Q8—24no—
213C4⋊Q8—42no—
214C4⋊Q8—42no—
215C4⋊Q8—42no—
216C4⋊Q8—42no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—22no—
225C4×D4—42no—
226C4×D4—42no—
227C4×D4—42no—
228C4×D4—42no—
229#117—21no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—42no—
234#117—42no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—41no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×Q812yes—
260C2×C4⋊C4#4112yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—42no—
268C2×C42—42no—
269C2×C42—41no—
270C2×C42—41no—
271C2×C42—82no—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42#4112yes—
279C2×C42#4112yes—
280#108—48no—
281#107—28no—
282#106—28no—
283C4⋊C8—24no—
284C4⋊C8—24no—
285C4⋊C8—24no—
286C4⋊C8—24no—
287C4⋊C8—42no—
288C4⋊C8—42no—
289#103—24no—
290#103—28no—
291#103—24no—
292#102—24no—
293#102—28no—
294#102—24no—
295C22⋊C8—48no—
296#97—42no—
297C4×C8—42no—
298#95—41no—
299#95—41no—
300C24—21no—
301C24C2×#4111yescharacteristic
302#41—22no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—22no—
307#41—22no—
308#41—22no—
309#41—22no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315#41—41no—
316#41—41no—
317#41—41no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—82no—
326C2×Q8—82no—
327C2×Q8—82no—
328C2×Q8—82no—
329C2×Q8—88no—
330C2×Q8—88no—
331C2×Q8C22⋊Q812yes—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4C22⋊Q812yes—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12111yescharacteristic
381C22×C4#12111yescharacteristic
382C22×C4#12312yes—
383C22×C4#12312yes—
384C22×C4#12412yes—
385C22×C4#12412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×Q811yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4H(5,𝔽2)11yescharacteristic
397C22×C42−1+411yescharacteristic
398C22×C42−1+411yescharacteristic
399C22×C42−1+411yescharacteristic
400M16—22no—
401M16—24no—
402M16—24no—
403M16—22no—
404M16—22no—
405M16—22no—
406M16—41no—
407M16—42no—
408M16—41no—
409M16—41no—
410M16—42no—
411M16—41no—
412M16—42no—
413M16—42no—
414M16—42no—
415M16—42no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C42—41no—
460C42—41no—
461C42—42no—
462C42—42no—
463C42—82no—
464C42—82no—
465C42#12012yes—
466C42#12012yes—
467C42#12112yes—
468C42#12112yes—
469C42#12112yes—
470C42#12112yes—
471C42#12312yes—
472C42#12312yes—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—22no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23C2×#12111yescharacteristic
485C23C2×#12311yescharacteristic
486C23C2×#12411yescharacteristic
487C23#55211yescharacteristic
488C23#55211yescharacteristic
489C23#55411yescharacteristic
490C23#55611yescharacteristic
491Q8—22no—
492Q8—22no—
493Q8—41no—
494Q8—41no—
495Q8—88no—
496Q8—88no—
497D4—22no—
498D4—22no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—41no—
504D4—41no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—84no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—84no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—84no—
545C2×C4#39412yes—
546C2×C4#39412yes—
547C2×C4#39412yes—
548C2×C4#39412yes—
549C2×C4#47112yes—
550C2×C4#47312yes—
551C2×C4C2×#12011yescharacteristic
552C2×C4C2×#12111yescharacteristic
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×C22⋊Q811yescharacteristic
555C2×C4C2×#12311yescharacteristic
556C2×C4#52511yescharacteristic
557C2×C4#52911yescharacteristic
558C2×C4#52911yescharacteristic
559C2×C4#53311yescharacteristic
560C2×C4#53311yescharacteristic
561C2×C4#53611yescharacteristic
562C2×C4#53711yescharacteristic
563C2×C4#54111yescharacteristic
564C2×C4#54111yescharacteristic
565C2×C4#54211yescharacteristic
566C2×C4#54211yescharacteristic
567C2×C4#54611yescharacteristic
568C2×C4#54611yescharacteristic
569C2×C4#54711yescharacteristic
570C2×C4#54711yescharacteristic
571C2×C4#54811yescharacteristic
572C2×C4#54811yescharacteristic
573C2×C4#54811yescharacteristic
574C2×C4#54811yescharacteristic
575C2×C4D4×Q811yescharacteristic
576C2×C4D4×Q811yescharacteristic
577C2×C4D4×Q811yescharacteristic
578C2×C4#55011yescharacteristic
579C8—44no—
580C8—42no—
581C8—42no—
582C8—44no—
583C8—82no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22C2×#39411yescharacteristic
604C22#241511yescharacteristic
605C22#241711yescharacteristic
606C22#242111yescharacteristic
607C22#242311yescharacteristic
608C22#248911yescharacteristic
609C22#249211yescharacteristic
610C22C2×#47111yescharacteristic
611C22C2×#47311yescharacteristic
612C22#306811yescharacteristic
613C22#306811yescharacteristic
614C4—22no—
615C4—22no—
616C4—22no—
617C4—42no—
618C4—82no—
619C4—82no—
620C4—81no—
621C4—81no—
622C4—84no—
623C4#205214yes—
624C4#240711yescharacteristic
625C4#240911yescharacteristic
626C4#242611yescharacteristic
627C4#242711yescharacteristic
628C4#242911yescharacteristic
629C4#242911yescharacteristic
630C4#262511yescharacteristic
631C4#285511yescharacteristic
632C2—21no—
633C2—21no—
634C2—21no—
635C2—21no—
636C2—82no—
637C2#1517911yescharacteristic
638C2C2×#205211yescharacteristic
639C2#2353111yescharacteristic
640C2#2472111yescharacteristic
641C2#2472211yescharacteristic
642C2#2485611yescharacteristic
643C2#2485711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1997C2semidirect
3. C2×#1987C2semidirect
4. C2×#1950C2semidirect
7. #11290C2semidirect
8. #11249C2semidirect
11. #10847C2semidirect
113. #417Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
C2×#19972
C2×#19872
C2×#19502
#206292
C4×#4172
#112902
#112492
#111922
#111052
#108472