Finite Group Databasebeta

← #497385

Subgroups — #497385

Order 512.

1114 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3285C211yesmaximal, characteristic
2C2×#2169C211yesmaximal, characteristic, complemented
3C2×#2139C211yesmaximal, characteristic, complemented
4C2×#2141C211yesmaximal, characteristic, complemented
5#22656C211yesmaximal, characteristic, complemented
6#13254C211yesmaximal, characteristic, complemented
7#13204C211yesmaximal, characteristic, complemented
8#13062C211yesmaximal, characteristic, complemented
9#13061C211yesmaximal, characteristic, complemented
10#12993C211yesmaximal, characteristic, complemented
11#12992C211yesmaximal, characteristic, complemented
12#12814C211yesmaximal, characteristic, complemented
13#12733C211yesmaximal, characteristic, complemented
14#12732C211yesmaximal, characteristic, complemented
15C4×#358C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C23×Q16C2211yescharacteristic
18C2×#548—21no—
19C22×M32C2211yescharacteristic
20C22×M32C2211yescharacteristic
21#3285—24no—
22#3285C418yescomplemented
23C2×#495C2211yescharacteristic
24C2×#467C2211yescharacteristic
25C2×#466C2211yescharacteristic
26C2×#464—21no—
27C2×#452—21no—
28C2×#448—21no—
29C2×C4×Q16C2211yescharacteristic
30C2×C8×D4C2211yescharacteristic
31C2×#427C2211yescharacteristic
32C22×#107C2211yescharacteristic
33#2352—21no—
34#2169—24no—
35C2×C4⋊C16C2211yescharacteristic
36#2141—24no—
37#2139—24no—
38C2×#358C2211yescharacteristic
39C2×#358C2211yescharacteristic
40C2×#358C2211yescharacteristic
41C2×#358C2211yescharacteristic
42C2×#358C2211yescharacteristic
43C2×#358C2211yescharacteristic
44C2×#358C2211yescharacteristic
45C2×#358C2211yescharacteristic
46C2×#357C2211yescharacteristic
47C2×#357C2211yescharacteristic
48C2×#357C2211yescharacteristic
49C2×#357C2211yescharacteristic
50C2×C22⋊C16C2211yescharacteristic
51C2×C22⋊C16C2211yescharacteristic
52C2×C4×C16C2211yescharacteristic
53#1949—21no—
54#1854—21no—
55#1853—21no—
56#1846C2211yescharacteristic
57#1808C2211yescharacteristic
58#1808C2211yescharacteristic
59C4×#107C2211yescharacteristic
60#1380C2211yescharacteristic
61#1380C2211yescharacteristic
62#1380C2211yescharacteristic
63#1380C2211yescharacteristic
64C22×#41—21no—
65C23×Q8—21no—
66C2×#135—22no—
67C2×#135—22no—
68C2×#135—22no—
69C2×#135—41no—
70C2×#135—41no—
71C2×#135C2×C414yes—
72C22×Q16—22no—
73C22×Q16—22no—
74C22×Q16—21no—
75C22×Q16—22no—
76C22×Q16—41no—
77C22×Q16—41no—
78C22×Q16C2×C414yescomplemented
79C22×Q16C2311yescharacteristic
80C22×Q16C2311yescharacteristic
81C22×SD16—21no—
82C22×SD16—21no—
83C22×D8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85C23×C8C2311yescharacteristic
86#548—28no—
87#548—44no—
88C2×#124—21no—
89C2×#124—41no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—41no—
92C2×#121—21no—
93C2×#121—41no—
94C2×C4×Q8—21no—
95C2×C4×D4—41no—
96C2×C4×D4D411yescharacteristic
97C2×#117—21no—
98C22×C4⋊C4—21no—
99C2×M32—22no—
100C2×M32—22no—
101C2×M32—22no—
102C2×M32—22no—
103C2×M32—41no—
104C2×M32—41no—
105C22×C16C2311yescharacteristic
106C22×C16C2311yescharacteristic
107C22×C16C2311yescharacteristic
108C22×C16C2311yescharacteristic
109#495—22no—
110#495C2×C414yescomplemented
111#467—22no—
112#467C2×C414yes—
113#466—22no—
114#466C2×C414yescomplemented
115#464—28no—
116#452—28no—
117#448—28no—
118C4×Q16—22no—
119C4×Q16C2×C414yescomplemented
120C8×D4C2×C414yes—
121C8×D4D414yes—
122#427—22no—
123#427—41no—
124C2×#107—22no—
125C2×#107—21no—
126C2×#107—41no—
127C2×#107C2311yescharacteristic
128C2×#107C2311yescharacteristic
129C2×#107C2311yescharacteristic
130C2×#107C2311yescharacteristic
131C2×#107C2311yescharacteristic
132C2×#106—21no—
133C2×#106—21no—
134C2×C4⋊C8D411yescharacteristic
135C2×#103—21no—
136C2×#103—21no—
137C2×#102—21no—
138C2×C22⋊C8D411yescharacteristic
139C2×C22⋊C8D411yescharacteristic
140C2×C4×C8C2311yescharacteristic
141#388—41no—
142#387—41no—
143#386—41no—
144#382—21no—
145#380—21no—
146#380—21no—
147C4×C4⋊C4—21no—
148C4×C22⋊C4—41no—
149C4⋊C16—22no—
150#358—22no—
151#358—22no—
152#358—22no—
153#358—22no—
154#358—22no—
155#358—41no—
156#358—41no—
157#358D414yescomplemented
158#357—22no—
159#357—22no—
160#357—41no—
161#357—41no—
162C22⋊C16—22no—
163C22⋊C16—22no—
164C4×C16—22no—
165#340—21no—
166#340—21no—
167#340—21no—
168C2×#41—24no—
169C2×#41—42no—
170C2×#41—42no—
171C2×#41—42no—
172C2×#41—81no—
173C2×#41—81no—
174C22×Q8—24no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—42no—
178C22×Q8—42no—
179C22×Q8—41no—
180C22×Q8—42no—
181C22×Q8—81no—
182C22×Q8—81no—
183C22×D4—21no—
184C22×D4—41no—
185C22×D4—41no—
186C22×D4D811yescharacteristic
187C23×C4—41no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192#135—28no—
193#135—44no—
194#135—44no—
195#135—44no—
196C2×Q16—28no—
197C2×Q16—24no—
198C2×Q16—21no—
199C2×Q16—21no—
200C2×Q16—21no—
201C2×Q16—21no—
202C2×Q16—21no—
203C2×Q16—21no—
204C2×Q16—21no—
205C2×Q16—21no—
206C2×Q16—44no—
207C2×Q16—44no—
208C2×Q16—44no—
209C2×Q16—42no—
210C2×Q16—42no—
211C2×Q16C22×C412yes—
212C2×Q16C22×C412yes—
213C2×Q16C2×D412yes—
214C2×Q16#4112yes—
215C2×SD16—24no—
216C2×SD16—24no—
217C2×SD16—42no—
218C2×SD16—42no—
219C2×SD16—81no—
220C2×SD16—81no—
221C2×D8—21no—
222C2×D8—21no—
223C2×D8—21no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8C22×C412yes—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8C22×C412yes—
236C22×C8C22×C412yes—
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239C22×C8C2×D411yescharacteristic
240C22×C8C2×D412yes—
241C22×C8C2×D412yes—
242C22×C8C2411yescharacteristic
243#124—24no—
244#124—48no—
245#124—42no—
246C22⋊Q8—24no—
247C22⋊Q8—42no—
248C22⋊Q8—48no—
249#121—24no—
250#121—48no—
251#121—42no—
252C4×Q8—24no—
253C4×Q8—42no—
254C4×D4—48no—
255C4×D4C22⋊C414yes—
256C4×D4SD1614yes—
257#117—42no—
258#117—81no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—81no—
269C2×C4⋊C4D811yescharacteristic
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4D811yescharacteristic
278C2×C22⋊C4D811yescharacteristic
279C2×C42—41no—
280C2×C42—41no—
281C2×C42—41no—
282C2×C42C2×D411yescharacteristic
283M32—44no—
284M32—44no—
285C2×C16—21no—
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—21no—
289C2×C16—21no—
290C2×C16—21no—
291C2×C16—21no—
292C2×C16—21no—
293C2×C16—21no—
294C2×C16—21no—
295C2×C16—21no—
296C2×C16C2×D412yes—
297#107—21no—
298#107—21no—
299#107—21no—
300#107—21no—
301#107—21no—
302#107—21no—
303#107—21no—
304#107—21no—
305#107—42no—
306#107C22×C412yes—
307#107C2×D412yes—
308#106—42no—
309#106—81no—
310C4⋊C8C22⋊C414yes—
311#103—24no—
312#103—24no—
313#102—24no—
314C22⋊C8C22⋊C414yes—
315C22⋊C8C22⋊C414yes—
316C4×C8C22×C412yes—
317C4×C8C2×D412yes—
318#95—41no—
319#95—41no—
320#95—41no—
321#95—41no—
322#95—41no—
323C24—41no—
324C24C2×D811yescharacteristic
325C24C2×D811yescharacteristic
326#41—48no—
327#41—84no—
328#41—84no—
329#41—84no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—48no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—44no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—82no—
345C2×Q8—82no—
346C2×Q8—84no—
347C2×Q8—84no—
348C2×Q8—84no—
349C2×D4—22no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—82no—
360C2×D4—82no—
361C2×D4—82no—
362C2×D4—82no—
363C2×D4#10214yes—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4—82no—
386C22×C4—82no—
387C22×C4—161no—
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×D811yescharacteristic
391C22×C4C2×D811yescharacteristic
392C22×C4C2×D811yescharacteristic
393C22×C4C2×D811yescharacteristic
394C22×C4C2×SD1612yes—
395C22×C4C2×SD1612yes—
396C22×C4C22×D411yescharacteristic
397Q16—22no—
398Q16—22no—
399Q16—22no—
400Q16—22no—
401Q16—22no—
402Q16—48no—
403Q16—41no—
404Q16—41no—
405Q16C4×D414yes—
406SD16—48no—
407SD16—84no—
408D8—22no—
409D8—22no—
410D8—41no—
411D8—41no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8C2×C22⋊C412yes—
427C2×C8C2×C22⋊C412yes—
428C2×C8C2×C22⋊C412yes—
429C2×C8C23×C411yescharacteristic
430C2×C8C22×D411yescharacteristic
431C2×C8C22×D411yescharacteristic
432C2×C8C2×#4111yescharacteristic
433C2×C8H(5,𝔽2)11yescharacteristic
434C2×C8H(5,𝔽2)11yescharacteristic
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—82no—
448C4⋊C4#10214yes—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—82no—
454C22⋊C4—82no—
455C22⋊C4—161no—
456C22⋊C4#10214yes—
457C22⋊C4#10214yes—
458C42—44no—
459C42—82no—
460C42—161no—
461C42C2×C22⋊C412yes—
462C42C2×SD1612yes—
463C16—22no—
464C16—41no—
465C16—41no—
466C16—41no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—42no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—81no—
486C23C2×#10212yes—
487C23C2×#10212yes—
488C23C22×D811yescharacteristic
489Q8—24no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—88no—
496Q8—81no—
497Q8—81no—
498D4—44no—
499D4—42no—
500D4—42no—
501D4—88no—
502D4—88no—
503D4—81no—
504D4—81no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—88no—
528C2×C4—88no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—164no—
545C2×C4C2×#10212yes—
546C2×C4C2×#10212yes—
547C2×C4C2×#10212yes—
548C2×C4C2×#10212yes—
549C2×C4C22×C22⋊C411yescharacteristic
550C2×C4D4211yescharacteristic
551C2×C4#54611yescharacteristic
552C2×C4C22×SD1611yescharacteristic
553C2×C4#57611yescharacteristic
554C2×C4#57711yescharacteristic
555C8—22no—
556C8—22no—
557C8—22no—
558C8C2×C4×D411yescharacteristic
559C8#51811yescharacteristic
560C8D4211yescharacteristic
561C8#54611yescharacteristic
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—82no—
577C22—82no—
578C22—82no—
579C22—161no—
580C22—161no—
581C22—161no—
582C22—161no—
583C22C22×#10211yescharacteristic
584C22D4×D811yescharacteristic
585C22#329411yescharacteristic
586C22#341411yescharacteristic
587C22#341411yescharacteristic
588C22#341611yescharacteristic
589C22#341711yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—44no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—161no—
599C4—161no—
600C4—161no—
601C4D4×C22⋊C411yescharacteristic
602C4#289711yescharacteristic
603C4D4×SD1611yescharacteristic
604C4#329511yescharacteristic
605C2—42no—
606C2—42no—
607C2—162no—
608C2D4×#10211yescharacteristic
609C2#3338011yescharacteristic
610C2#3338211yescharacteristic
611C2#3384111yescharacteristic
612C2#3384311yescharacteristic
613C2#3389311yescharacteristic
614C2#3389411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#3285C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2169C2semidirect
3. C2×#2139C2semidirect
4. C2×#2141C2semidirect
5. #22656C2semidirect
6. #13254C2semidirect
7. #13204C2semidirect
8. #13062C2semidirect
9. #13061C2semidirect
10. #12993C2semidirect
11. #12992C2semidirect
12. #12814C2semidirect
13. #12733C2semidirect
14. #12732C2semidirect
15. C4×#358C2semidirect
22. #3285C4semidirect
78. C22×Q16C2×C4semidirect
110. #495C2×C4semidirect
114. #466C2×C4semidirect
119. C4×Q16C2×C4semidirect
157. #358D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32852
C2×#21692
C2×#21392
C2×#21412
#226562
#132542
#132042
#130622
#130612
#129932
#129922
#128142
#127332
#127322
C4×#3582