Finite Group Databasebeta

← #822378

Subgroups — #822378

Order 512.

3142 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C212yesmaximal
2C2×#2551C212yesmaximal
3C2×#2551C212yesmaximal
4#16522C212yesmaximal
5#15762C212yesmaximal
6#15363C212yesmaximal
7#15318C214yesmaximal
8#15143C212yesmaximal
9#15128C211yesmaximal, characteristic
10#14816C214yesmaximal
11#14046C212yesmaximal
12#14005C212yesmaximal
13#13910C211yesmaximal, characteristic
14C4×#395C211yesmaximal, characteristic
15#13821C212yesmaximal
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2212yes—
18C22×C4⋊Q8C2214yes—
19C22×C4⋊Q8C2214yes—
20C22×C4×Q8C2211yescharacteristic
21#2551—216no—
22#2551—216no—
23#2546—216no—
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#395C2212yes—
27C2×#395C2214yes—
28C2×#395C2214yes—
29C2×#395C2214yes—
30C2×#395C2214yes—
31C2×#395C2211yescharacteristic
32C2×#391C2214yes—
33C2×#389C2214yes—
34C2×#389C2214yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2214yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#384C2212yes—
43C2×#384C2212yes—
44C2×#384C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2214yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51#1446C2212yes—
52#1446C2211yescharacteristic
53#1444C2211yescharacteristic
54#1444C2212yes—
55#1444C2214yes—
56#1444C2212yes—
57#1444C2212yes—
58#1444C2214yes—
59#1444C2212yes—
60#1444C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2212yes—
64#1441C2211yescharacteristic
65#1441C2212yes—
66#1441C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76C4×#95C2212yes—
77C4×#95C2211yescharacteristic
78C23×Q8C2314yes—
79C23×Q8C2311yescharacteristic
80C23×Q8C2312yes—
81C2×C4⋊Q8—216no—
82C2×C4⋊Q8—28no—
83C2×C4⋊Q8—216no—
84C2×C4⋊Q8—216no—
85C2×C4⋊Q8—216no—
86C2×C4⋊Q8—28no—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—44no—
89C2×C4⋊Q8—42no—
90C2×C4⋊Q8—48no—
91C2×C4⋊Q8—48no—
92C2×C4×Q8—28no—
93C2×C4×Q8—42no—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2314yes—
119C22×C42C2314yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124#398—216no—
125#398—216no—
126#395—28no—
127#395—216no—
128#395—216no—
129#395—216no—
130#395—48no—
131#395D418yes—
132#391—216no—
133#389—28no—
134#389—28no—
135#389—216no—
136#389—28no—
137#389—28no—
138#389—216no—
139#389—216no—
140#389D4116yes—
141#389Q8116yescomplemented
142#384—28no—
143#384—28no—
144#384—42no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—216no—
149C4×C4⋊C4—42no—
150C4×C4⋊C4Q818yescomplemented
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2311yescharacteristic
175C2×#95C2314yes—
176C2×#95C2312yes—
177C22×Q8—24no—
178C22×Q8—28no—
179C22×Q8—24no—
180C22×Q8—28no—
181C22×Q8—28no—
182C22×Q8—28no—
183C22×Q8—24no—
184C22×Q8—28no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8—42no—
190C22×Q8—44no—
191C22×Q8—44no—
192C22×Q8—42no—
193C22×Q8C2×D414yes—
194C23×C4C2414yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2411yescharacteristic
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C4⋊Q8—432no—
210C4⋊Q8—464no—
211C4⋊Q8—416no—
212C4⋊Q8—464no—
213C4×Q8—416no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
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—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D4116yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×Q818yes—
274C2×C4⋊C4C2×Q8116yes—
275C2×C4⋊C4C2×Q818yes—
276C2×C4⋊C4C2×Q818yes—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—28no—
288C2×C42—28no—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42C2×D418yes—
296C2×C42C2×Q814yes—
297C2×C42C2×Q818yes—
298C2×C42C2×Q818yes—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—41no—
316#95—41no—
317#95—41no—
318#95—44no—
319#95—44no—
320#95C2×D418yes—
321#95C2×D414yes—
322#95C2×D4116yes—
323#95C2×Q818yes—
324#95C2×Q8116yes—
325C24C2511yescharacteristic
326C2×Q8—216no—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—48no—
330C2×Q8—44no—
331C2×Q8—416no—
332C2×Q8—416no—
333C2×Q8—48no—
334C2×Q8—416no—
335C2×Q8—416no—
336C2×Q8—416no—
337C2×Q8—88no—
338C2×Q8—88no—
339C2×Q8—88no—
340C2×Q8—88no—
341C2×Q8—84no—
342C2×Q8—84no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4C22×D414yes—
412C22×C4C22×D412yes—
413C22×C4C22×D418yes—
414C22×C4C22×D412yes—
415C22×C4C22×D418yes—
416C22×C4C22×D414yes—
417C22×C4C22×D414yes—
418C22×C4C22×D412yes—
419C22×C4C22×D414yes—
420C22×C4C22×D414yes—
421C22×C4C22×Q812yes—
422C22×C4C22×Q814yes—
423C22×C4C22×Q818yes—
424C22×C4C22×Q818yes—
425C22×C4C22×Q818yes—
426C22×C4C22×Q812yes—
427C22×C4C22×Q814yes—
428C22×C4C22×Q812yes—
429C22×C4C22×Q814yes—
430C22×C4H(5,𝔽2)14yes—
431C22×C4H(5,𝔽2)12yes—
432C22×C42−1+412yes—
433C22×C42−1+412yes—
434C22×C42−1+414yes—
435C22×C42−1+412yes—
436C4⋊C4—216no—
437C4⋊C4—216no—
438C4⋊C4—232no—
439C4⋊C4—216no—
440C4⋊C4—232no—
441C4⋊C4—216no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—416no—
447C4⋊C4—416no—
448C4⋊C4—416no—
449C4⋊C4—416no—
450C4⋊C4—48no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—48no—
454C4⋊C4—84no—
455C4⋊C4—88no—
456C4⋊C4—84no—
457C4⋊C4C4⋊Q8132yescomplemented
458C42—28no—
459C42—216no—
460C42—216no—
461C42—216no—
462C42—416no—
463C42—416no—
464C42—44no—
465C42—44no—
466C23C23×D412yes—
467C23C23×D411yescharacteristic
468C23C23×Q811yescharacteristic
469C23C23×Q812yes—
470C23C2×H(5,𝔽2)11yescharacteristic
471C23C2×H(5,𝔽2)12yes—
472C23C2×2−1+411yescharacteristic
473C23C2×2−1+412yes—
474C23C2×2−1+411yescharacteristic
475C23C2×2−1+411yescharacteristic
476C23#58511yescharacteristic
477Q8—416no—
478Q8—832no—
479Q8—816no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—28no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4C2×C4⋊Q8116yes—
565C2×C4C2×C4⋊Q818yes—
566C2×C4#54414yes—
567C2×C4D4218yes—
568C2×C4D4214yes—
569C2×C4D4×Q814yes—
570C2×C4D4×Q818yes—
571C2×C4D4×Q818yes—
572C2×C4Q8214yes—
573C22C22×C4⋊Q812yes—
574C22C2×#54411yescharacteristic
575C22C2×D4211yescharacteristic
576C22C2×D4212yes—
577C22C2×D4×Q811yescharacteristic
578C22C2×D4×Q812yes—
579C22C2×D4×Q812yes—
580C22C2×Q8211yescharacteristic
581C22#348612yes—
582C22#348812yes—
583C22#349611yescharacteristic
584C22#349912yes—
585C22#349911yescharacteristic
586C22#350211yescharacteristic
587C22#350212yes—
588C22#350312yes—
589C22#350312yes—
590C22#350312yes—
591C22#353711yescharacteristic
592C22#353811yescharacteristic
593C22#355011yescharacteristic
594C22#355212yes—
595C22#356311yescharacteristic
596C4—216no—
597C4—28no—
598C4—24no—
599C4—28no—
600C4—44no—
601C4—42no—
602C4—42no—
603C4—48no—
604C4—42no—
605C4—42no—
606C4—44no—
607C4—48no—
608C4—81no—
609C4—84no—
610C4#255118yescomplemented
611C2C2×#255111yescharacteristic
612C2#3593712yes—
613C2D4×C4⋊Q812yes—
614C2#3936811yescharacteristic
615C2#4293311yescharacteristic
616C2#4527212yes—
617C2#4532111yescharacteristic
618C2#4543511yescharacteristic
619C2#5899711yescharacteristic
620C2#5912712yes—
621C2#5983211yescharacteristic
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
141. #389Q8semidirect
150. C4×C4⋊C4Q8semidirect
457. C4⋊C4C4⋊Q8semidirect
610. C4#2551semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#25512
#165222
#157622
#153632
#153182
#151432
#151282
#148162
#140462
#140052
#139102
C4×#3952
#138212