Finite Group Databasebeta

← #626977

Subgroups — #626977

Order 512.

2328 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic, complemented
2C2×#2628C211yesmaximal, characteristic
3#15690C214yesmaximal, complemented
4#15636C214yesmaximal
5#15499C214yesmaximal
6#15332C214yesmaximal, complemented
7#15232C212yesmaximal, complemented
8#15125C212yesmaximal
9#14714C211yesmaximal, characteristic, complemented
10#14340C211yesmaximal, characteristic
11#14282C211yesmaximal, characteristic, complemented
12#14053C212yesmaximal, complemented
13#13875C212yesmaximal
14#13779C211yesmaximal, characteristic, complemented
15#13761C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×#117C2211yescharacteristic
19#2862—28no—
20#2628—28no—
21C2×#400C2212yes—
22C2×#399C2212yes—
23C2×#395C2212yes—
24C2×#393C2212yes—
25C2×#393C2214yes—
26C2×#391C2214yes—
27C2×#391C2214yes—
28C2×#391C2214yes—
29C2×#391C2214yes—
30C2×#389C2212yes—
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2212yes—
34C2×#382C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×#376C2211yescharacteristic
41C2×C43C2211yescharacteristic
42#1446C2214yes—
43#1446C2214yes—
44#1446C2214yes—
45#1445C2214yes—
46#1445C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2212yes—
53#1441C2212yes—
54#1441C2214yes—
55#1440C2214yes—
56#1439C2212yes—
57#1439C2212yes—
58#1439C2214yes—
59#1437C2212yes—
60#1437C2214yes—
61#1437C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2212yes—
65#1436C2214yes—
66#1436C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2214yes—
74#1434C2214yes—
75C4×#95C2211yescharacteristic
76C4×#95C2211yescharacteristic
77C23×Q8C2314yes—
78C23×Q8C2312yes—
79C24×C4C2311yescharacteristic
80C2×C4⋊Q8—28no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—44no—
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—24no—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115#400—28no—
116#399—28no—
117#395—28no—
118#393—28no—
119#393—48no—
120#391—216no—
121#391—216no—
122#391—48no—
123#391—48no—
124#389—28no—
125#384—24no—
126#382—28no—
127#382—24no—
128#382D418yes—
129#380—28no—
130#380—44no—
131C4×C4⋊C4—44no—
132C4×C22⋊C4—24no—
133C4×C22⋊C4Q818yescomplemented
134#376—24no—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C43—24no—
166C25C2411yescharacteristic
167C22×Q8—28no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—28no—
171C22×Q8—24no—
172C22×Q8—28no—
173C22×Q8—28no—
174C22×Q8—44no—
175C22×Q8—44no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8—44no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4C2×Q814yes—
186C23×C4C2411yescharacteristic
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C4⋊Q8—432no—
201C4⋊Q8—416no—
202#117—416no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D418yes—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4C2×Q818yes—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42C2×D414yes—
274C2×C42C2×Q814yes—
275C2×C42C2×Q814yes—
276C2×C42#4114yes—
277C2×C42#4114yes—
278C2×C42#4114yes—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—28no—
290#95—28no—
291#95—24no—
292#95—44no—
293#95—42no—
294#95—42no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95C2×D418yes—
307#95C2×D418yes—
308#95C2×Q818yes—
309C24—21no—
310C24—21no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—21no—
315C24—22no—
316C24—21no—
317C24—22no—
318C24—22no—
319C24C22×Q812yes—
320C24C2511yescharacteristic
321C2×Q8—416no—
322C2×Q8—416no—
323C2×Q8—416no—
324C2×Q8—416no—
325C2×Q8—48no—
326C2×Q8—416no—
327C2×Q8—88no—
328C2×Q8—84no—
329C2×Q8—88no—
330C2×Q8—84no—
331C2×Q8—88no—
332C2×Q8—88no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—21no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4C22×D418yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×Q812yes—
415C22×C4C22×Q814yes—
416C22×C4C22×Q814yes—
417C22×C4C22×Q812yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4114yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4H(5,𝔽2)14yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C42−1+414yes—
429C22×C42−1+412yes—
430C4⋊C4—216no—
431C4⋊C4—416no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—88no—
441C22⋊C4—48no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—48no—
445C22⋊C4—88no—
446C22⋊C4—88no—
447C42—28no—
448C42—28no—
449C42—28no—
450C42—48no—
451C42—48no—
452C42—44no—
453C42—44no—
454C42—88no—
455C42#124116yes—
456C23—22no—
457C23—24no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—41no—
462C23—42no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23C23×D411yescharacteristic
480C23C23×Q811yescharacteristic
481C23C22×#4111yescharacteristic
482C23C22×#4111yescharacteristic
483C23C22×#4111yescharacteristic
484C23C2×H(5,𝔽2)11yescharacteristic
485C23C2×H(5,𝔽2)12yes—
486C23C2×2−1+411yescharacteristic
487C23C2×2−1+412yes—
488C23#58512yes—
489C23#58512yes—
490Q8—832no—
491Q8—816no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—28no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—416no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4C2×#12414yes—
574C2×C4C2×#12414yes—
575C2×C4C2×#12414yes—
576C2×C4#53314yes—
577C2×C4#54218yes—
578C2×C4#54214yes—
579C2×C4#54614yes—
580C2×C4D4×Q814yes—
581C2×C4#55414yes—
582C2×C4#55614yes—
583C22—42no—
584C22—44no—
585C22—44no—
586C22—42no—
587C22—42no—
588C22—81no—
589C22—81no—
590C22—82no—
591C22—82no—
592C22—81no—
593C22—81no—
594C22C22×#12411yescharacteristic
595C22C2×#53311yescharacteristic
596C22C2×#54211yescharacteristic
597C22C2×#54212yes—
598C22#345812yes—
599C22C2×#54611yescharacteristic
600C22C2×D4×Q811yescharacteristic
601C22C2×#55411yescharacteristic
602C22C2×#55611yescharacteristic
603C22#348812yes—
604C22#348912yes—
605C22#350312yes—
606C22#350612yes—
607C22#350712yes—
608C22#350811yescharacteristic
609C22#350911yescharacteristic
610C22#351011yescharacteristic
611C22#351011yescharacteristic
612C22#351012yes—
613C22#354712yes—
614C22#355512yes—
615C22#355812yes—
616C22#356112yes—
617C4—24no—
618C4—28no—
619C4—24no—
620C4—44no—
621C4—48no—
622C4—44no—
623C4—42no—
624C4—81no—
625C4—84no—
626C4—82no—
627C4—84no—
628C4—81no—
629C4—84no—
630C4#249218yes—
631C2—82no—
632C2C2×#249211yescharacteristic
633C2#3528311yescharacteristic
634C2#3534411yescharacteristic
635C2#3555611yescharacteristic
636C2#3562811yescharacteristic
637C2#3732211yescharacteristic
638C2#3739611yescharacteristic
639C2#3769012yes—
640C2#3769612yes—
641C2#3949312yes—
642C2#3951512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2862C2semidirect
3. #15690C2semidirect
6. #15332C2semidirect
7. #15232C2semidirect
9. #14714C2semidirect
11. #14282C2semidirect
12. #14053C2semidirect
14. #13779C2semidirect
133. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28622
C2×#26282
#156902
#156362
#154992
#153322
#152322
#151252
#147142
#143402
#142822
#140532
#138752
#137792
#137612