Finite Group Databasebeta

← #364787

Subgroups — #364787

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×Q8C211yesmaximal, characteristic, complemented
2C2×#2580C211yesmaximal, characteristic
3C2×#1804C211yesmaximal, characteristic
4C2×C4×#102C211yesmaximal, characteristic
5#13925C211yesmaximal, characteristic, complemented
6#11228C212yesmaximal, complemented
7#11076C212yesmaximal
8#10925C212yesmaximal
9#10880C212yesmaximal, complemented
10#10850C211yesmaximal, characteristic, complemented
11#10842C211yesmaximal, characteristic, complemented
12C22×C4×Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×C8×Q8—24no—
15C2×C8×Q8—24no—
16C2×C8×Q8—24no—
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2211yescharacteristic
22C22×C4×C8C2212yes—
23C22×C4×C8C2211yescharacteristic
24#2580—24no—
25#2580C418yescomplemented
26C2×#399—21no—
27C2×#393—21no—
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#385—22no—
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1804—24no—
35C4×#102—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1446—21no—
44#1445—21no—
45#1436—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8D411yescharacteristic
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8C2×C414yes—
59C2×C4×Q8D414yes—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4C2×C414yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D412yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—22no—
77C22×C42—21no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C8×Q8—232no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8D414yes—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—42no—
95C2×#102Q814yescomplemented
96C2×C4×C8—24no—
97C2×C4×C8—24no—
98C2×C4×C8—24no—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8D414yes—
102#399—28no—
103#393—28no—
104#390—24no—
105#390C2×C418yes—
106#389—24no—
107#389C2×C418yes—
108#385—216no—
109#382—22no—
110#382—22no—
111#382—22no—
112#382C2×C414yes—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C2×#95—21no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125#340—24no—
126#340—22no—
127#340—22no—
128#340—24no—
129#340—24no—
130#340—42no—
131#340—42no—
132C25—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—24no—
136C22×Q8—24no—
137C22×Q8C22⋊C414yes—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C22×C412yes—
150C22×D4C2×Q812yes—
151C22×D4#4112yes—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2411yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8C2×D414yes—
184C22×C8C2×D412yes—
185C22×C8C2×D412yes—
186C22×C8C2×Q812yes—
187C22×C8#4112yes—
188C4×Q8—28no—
189C4×Q8—28no—
190C4×Q8—28no—
191C4×Q8—28no—
192C4×D4—28no—
193C4×D4—28no—
194C4×D4—44no—
195C4×D4—44no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4—82no—
229C2×C4⋊C4C22⋊C418yes—
230C2×C4⋊C4C22⋊C414yes—
231C2×C4⋊C4C22×C412yes—
232C2×C4⋊C4C22×C414yes—
233C2×C4⋊C4C2×Q812yes—
234C2×C4⋊C4#4112yes—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—42no—
249C2×C42—82no—
250C2×C42C22×C414yes—
251C2×C42C22×C412yes—
252C2×C42C2×D414yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42#4112yes—
257C2×C42#4112yes—
258C4⋊C8—216no—
259C4⋊C8—28no—
260#102—28no—
261#102—28no—
262#102—48no—
263C4×C8—216no—
264C4×C8—28no—
265#95—24no—
266#95—24no—
267#95—28no—
268#95—42no—
269#95—42no—
270#95—42no—
271#95—44no—
272#95—44no—
273#95—82no—
274#95—82no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24C22×D411yescharacteristic
287C2×Q8—28no—
288C2×Q8—28no—
289C2×Q8—216no—
290C2×Q8—28no—
291C2×Q8—216no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4C4×Q814yes—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—48no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—88no—
365C22×C4—84no—
366C22×C4—84no—
367C22×C4—88no—
368C22×C4C2×C22⋊C414yes—
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C412yes—
371C22×C4C2×C22⋊C414yes—
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4#13514yes—
377C22×C4#13514yes—
378C22×C4C23×C411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×Q811yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4H(5,𝔽2)11yescharacteristic
389C22×C42−1+411yescharacteristic
390C22×C42−1+411yescharacteristic
391C22×C42−1+411yescharacteristic
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—24no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—41no—
412C2×C8C22⋊Q814yes—
413C4⋊C4—216no—
414C4⋊C4—28no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—88no—
436C22⋊C4—48no—
437C22⋊C4—416no—
438C22⋊C4—88no—
439C42—24no—
440C42—24no—
441C42—24no—
442C42—24no—
443C42—84no—
444C42C4×D414yes—
445C42C4×D414yes—
446C42#12014yes—
447C42#12314yes—
448C23—24no—
449C23—44no—
450C23—44no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—81no—
466C23—81no—
467C23—82no—
468C23—81no—
469C23—81no—
470C23—82no—
471C23C22×C22⋊C411yescharacteristic
472C23C2×#12111yescharacteristic
473C23C2×#12111yescharacteristic
474C23#54811yescharacteristic
475C23#54811yescharacteristic
476C23D4×Q811yescharacteristic
477C23#55011yescharacteristic
478C23C2×#13512yes—
479C23C2×#13512yes—
480C23#57612yes—
481C23#57712yes—
482Q8—232no—
483D4—24no—
484D4—24no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—42no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—816no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4#38614yes—
543C2×C4#38614yes—
544C2×C4#41618yes—
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C2×C4×Q811yescharacteristic
548C2×C4C4×#4111yescharacteristic
549C2×C4#51911yescharacteristic
550C2×C4#52011yescharacteristic
551C2×C4C2×#12011yescharacteristic
552C2×C4C2×C22⋊Q811yescharacteristic
553C2×C4C2×#12311yescharacteristic
554C2×C4#52511yescharacteristic
555C2×C4#52911yescharacteristic
556C2×C4#53311yescharacteristic
557C2×C4#53611yescharacteristic
558C2×C4#53711yescharacteristic
559C2×C4#54111yescharacteristic
560C2×C4#54211yescharacteristic
561C2×C4#54611yescharacteristic
562C2×C4#54711yescharacteristic
563C2×C4#54811yescharacteristic
564C2×C4#54811yescharacteristic
565C2×C4#54811yescharacteristic
566C2×C4D4×Q811yescharacteristic
567C2×C4D4×Q811yescharacteristic
568C2×C4#55011yescharacteristic
569C2×C4#55211yescharacteristic
570C2×C4#55211yescharacteristic
571C2×C4#55411yescharacteristic
572C2×C4#55611yescharacteristic
573C8—24no—
574C8—42no—
575C8—44no—
576C22—44no—
577C22—48no—
578C22—84no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—84no—
584C22—84no—
585C22—82no—
586C22—82no—
587C22C2×#38611yescharacteristic
588C22Q8×C22⋊C411yescharacteristic
589C22#234111yescharacteristic
590C22#242611yescharacteristic
591C22#242911yescharacteristic
592C22#253611yescharacteristic
593C22#253811yescharacteristic
594C22C2×#41612yes—
595C22#289712yes—
596C22#305011yescharacteristic
597C22#305011yescharacteristic
598C22#305012yes—
599C22#305711yescharacteristic
600C22#305811yescharacteristic
601C22#306212yes—
602C22#336012yes—
603C22#336112yes—
604C22#336112yes—
605C22#336512yes—
606C22#338012yes—
607C22#338112yes—
608C22#339412yes—
609C22#339612yes—
610C4—24no—
611C4—28no—
612C4—24no—
613C4—44no—
614C4—84no—
615C4—82no—
616C4—82no—
617C4—82no—
618C4—162no—
619C4C4×C22⋊Q811yescharacteristic
620C4#232311yescharacteristic
621C4#236011yescharacteristic
622C4#236211yescharacteristic
623C4#240711yescharacteristic
624C4#240911yescharacteristic
625C4#242111yescharacteristic
626C4#242311yescharacteristic
627C2—84no—
628C2#1396911yescharacteristic
629C2#2096711yescharacteristic
630C2#2097511yescharacteristic
631C2#2251912yes—
632C2#2254312yes—
633C2#2529612yes—
634C2#2547312yes—
635C2#2818811yescharacteristic
636C2#2818911yescharacteristic
637C2#2826611yescharacteristic
638C2#2826711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×C8×Q8C2semidirect
5. #13925C2semidirect
6. #11228C2semidirect
9. #10880C2semidirect
10. #10850C2semidirect
11. #10842C2semidirect
25. #2580C4semidirect
95. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C8×Q82
C2×#25802
C2×#18042
C2×C4×#1022
#139252
#112282
#110762
#109252
#108802
#108502
#108422