Finite Group Databasebeta

← #880149

Subgroups — #880149

Order 512.

2078 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2805C212yesmaximal
2#17079C212yesmaximal
3#17074C212yesmaximal
4#15893C214yesmaximal
5#15382C214yesmaximal
6#15156C212yesmaximal
7#15146C212yesmaximal
8#15128C212yesmaximal
9#14714C211yesmaximal, characteristic
10#14712C211yesmaximal, characteristic
11#14441C212yesmaximal
12#14049C212yesmaximal
13#14046C212yesmaximal
14#13884C212yesmaximal
15C4×#389C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17#2805—216no—
18C2×#400C2214yes—
19C2×#400C2214yes—
20C2×#395C2214yes—
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#389C2214yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2211yescharacteristic
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#387C2214yes—
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C43C2211yescharacteristic
42#1446C2214yes—
43#1445C2214yes—
44#1445C2214yes—
45#1444C2214yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1436C2212yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C4×#95C2212yes—
76C23×Q8C2312yes—
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—42no—
79C2×C4⋊Q8—42no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C42C2312yes—
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#400—216no—
111#400—216no—
112#395—216no—
113#391—28no—
114#391—44no—
115#389—216no—
116#389—28no—
117#389—28no—
118#389—24no—
119#389—28no—
120#389—44no—
121#389—44no—
122#389D418yes—
123#387—216no—
124#387—216no—
125#387—48no—
126#384—42no—
127#384—42no—
128#384Q818yescomplemented
129C4×C4⋊C4—28no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—42no—
132C4×C4⋊C4—44no—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C43—42no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—42no—
168C22×Q8—42no—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2411yescharacteristic
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C4⋊Q8—416no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4C2×D418yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×Q818yes—
243C2×C4⋊C4C2×Q818yes—
244C2×C4⋊C4C2×Q818yes—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—28no—
262C2×C42—41no—
263C2×C42—41no—
264C2×C42—41no—
265C2×C42—41no—
266C2×C42—41no—
267C2×C42—41no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42C2×D414yes—
271C2×C42C2×Q814yes—
272C2×C42#4114yes—
273C2×C42#4114yes—
274C2×C42#4114yes—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—24no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—24no—
288#95—28no—
289#95—24no—
290#95—28no—
291#95—24no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—42no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95C2×D418yes—
303C24C2511yescharacteristic
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—48no—
307C2×Q8—84no—
308C2×Q8—84no—
309C2×Q8—84no—
310C2×Q8—84no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4C22×D414yes—
391C22×C4C22×D414yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×Q818yes—
396C22×C4C22×Q812yes—
397C22×C4C22×Q812yes—
398C22×C4C22×Q812yes—
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4H(5,𝔽2)12yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C42−1+412yes—
412C22×C42−1+412yes—
413C22×C42−1+412yes—
414C22×C42−1+412yes—
415C22×C42−1+412yes—
416C4⋊C4—28no—
417C4⋊C4—216no—
418C4⋊C4—216no—
419C4⋊C4—216no—
420C4⋊C4—216no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4—416no—
426C4⋊C4—48no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—88no—
435C4⋊C4—88no—
436C42—28no—
437C42—28no—
438C42—28no—
439C42—44no—
440C42—44no—
441C42—44no—
442C42—44no—
443C42—48no—
444C42—48no—
445C42—88no—
446C23C23×D411yescharacteristic
447C23C23×Q811yescharacteristic
448C23C22×#4111yescharacteristic
449C23C22×#4111yescharacteristic
450C23C22×#4111yescharacteristic
451C23C2×H(5,𝔽2)11yescharacteristic
452C23C2×H(5,𝔽2)11yescharacteristic
453C23C2×H(5,𝔽2)11yescharacteristic
454C23C2×2−1+411yescharacteristic
455C23C2×2−1+411yescharacteristic
456C23C2×2−1+411yescharacteristic
457C23C2×2−1+411yescharacteristic
458C23C2×2−1+411yescharacteristic
459C23#58511yescharacteristic
460C23#58511yescharacteristic
461Q8—816no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4#53214yes—
556C2×C4#53814yes—
557C2×C4#53914yes—
558C2×C4#54414yes—
559C2×C4#54714yes—
560C2×C4D4×Q814yes—
561C2×C4#55414yes—
562C2×C4#55414yes—
563C2×C4#55614yes—
564C2×C4#55614yes—
565C2×C4#55714yes—
566C2×C4#56314yes—
567C22C2×#53211yescharacteristic
568C22C2×#53811yescharacteristic
569C22C2×#53911yescharacteristic
570C22#345511yescharacteristic
571C22C2×#54411yescharacteristic
572C22C2×#54711yescharacteristic
573C22C2×D4×Q811yescharacteristic
574C22C2×#55411yescharacteristic
575C22C2×#55411yescharacteristic
576C22C2×#55611yescharacteristic
577C22C2×#55611yescharacteristic
578C22C2×#55711yescharacteristic
579C22#349611yescharacteristic
580C22#349711yescharacteristic
581C22#349811yescharacteristic
582C22#349911yescharacteristic
583C22#350211yescharacteristic
584C22#350211yescharacteristic
585C22#350511yescharacteristic
586C22#350611yescharacteristic
587C22#350711yescharacteristic
588C22#350711yescharacteristic
589C22#350911yescharacteristic
590C22#350911yescharacteristic
591C22#351111yescharacteristic
592C22#351111yescharacteristic
593C22C2×#56311yescharacteristic
594C22#353611yescharacteristic
595C22#353811yescharacteristic
596C22#355011yescharacteristic
597C22#355611yescharacteristic
598C22#355911yescharacteristic
599C22#356111yescharacteristic
600C22#356111yescharacteristic
601C22#356311yescharacteristic
602C4—24no—
603C4—24no—
604C4—24no—
605C4—24no—
606C4—24no—
607C4—44no—
608C4—42no—
609C4—48no—
610C4—44no—
611C4—82no—
612C4—82no—
613C4—84no—
614C4—84no—
615C4—84no—
616C4#280518yescomplemented
617C2#3500611yescharacteristic
618C2#3618811yescharacteristic
619C2#3655911yescharacteristic
620C2#3732311yescharacteristic
621C2#3936811yescharacteristic
622C2#3937511yescharacteristic
623C2#3949611yescharacteristic
624C2#3950911yescharacteristic
625C2#3962911yescharacteristic
626C2#3976011yescharacteristic
627C2#4599811yescharacteristic
628C2#4602211yescharacteristic
629C2C2×#280511yescharacteristic
630C2#5982811yescharacteristic
631C2#5987011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
128. #384Q8semidirect
616. C4#2805semidirect

Maximal subgroups

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

subgroupindex
C2×#28052
#170792
#170742
#158932
#153822
#151562
#151462
#151282
#147142
#147122
#144412
#140492
#140462
#138842
C4×#3892