Finite Group Databasebeta

← #626969

Subgroups — #626969

Order 512.

2408 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic, complemented
2C2×#2628C211yesmaximal, characteristic
3C2×#2628C211yesmaximal, characteristic
4C2×#2331C211yesmaximal, characteristic, complemented
5#15858C214yesmaximal, complemented
6#15855C214yesmaximal
7#15503C214yesmaximal, complemented
8#15499C214yesmaximal
9#15225C212yesmaximal, complemented
10#15220C212yesmaximal
11#14707C211yesmaximal, characteristic, complemented
12#14282C211yesmaximal, characteristic, complemented
13#14006C212yesmaximal, complemented
14#13875C212yesmaximal
15#13761C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#2862—28no—
20#2628—28no—
21#2628—28no—
22C2×#400C2212yes—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#399C2212yes—
26C2×#395C2212yes—
27C2×#395C2212yes—
28C2×#393C2212yes—
29C2×#393C2212yes—
30#2331—28no—
31C2×#391C2214yes—
32C2×#391C2214yes—
33C2×#391C2214yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#387C2214yes—
39C2×#382C2212yes—
40C2×#382C2212yes—
41C2×#382C2211yescharacteristic
42C2×#382C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×#376C2211yescharacteristic
49C2×C43C2211yescharacteristic
50#1446C2214yes—
51#1445C2214yes—
52#1445C2214yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1437C2214yes—
65#1437C2212yes—
66#1437C2212yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2212yes—
70#1436C2214yes—
71#1436C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2214yes—
76#1434C2214yes—
77C4×#95C2211yescharacteristic
78C4×#95C2211yescharacteristic
79C23×Q8C2312yes—
80C23×Q8C2312yes—
81C24×C4C2311yescharacteristic
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8D418yes—
88C2×#117—24no—
89C2×#117—24no—
90C2×#117—24no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2314yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2311yescharacteristic
118#400—28no—
119#400—28no—
120#399—28no—
121#399—28no—
122#395—28no—
123#395—28no—
124#393—28no—
125#393—28no—
126#391—216no—
127#391—216no—
128#391—216no—
129#389—28no—
130#389—28no—
131#389—28no—
132#389—28no—
133#387—216no—
134#382—28no—
135#382—28no—
136#382—24no—
137#382D418yes—
138#380—28no—
139#380—28no—
140C4×C4⋊C4—28no—
141C4×C22⋊C4—24no—
142C4×C22⋊C4Q818yescomplemented
143#376—24no—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C43—24no—
174C25C2411yescharacteristic
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—44no—
184C22×Q8C2×D418yes—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4C2×Q814yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C4⋊Q8—232no—
207C4⋊Q8—416no—
208#117—416no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
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—28no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×D418yes—
243C2×C4⋊C4C2×D418yes—
244C2×C4⋊C4C2×D418yes—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4C2×Q818yes—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—28no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—42no—
282C2×C42—44no—
283C2×C42—44no—
284C2×C42C2×D414yes—
285C2×C42C2×D414yes—
286C2×C42C2×Q814yes—
287C2×C42C2×Q814yes—
288C2×C42#4114yes—
289C2×C42#4114yes—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—28no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—28no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—42no—
312#95—42no—
313#95—42no—
314#95—44no—
315#95—44no—
316#95C2×D418yes—
317#95C2×D418yes—
318#95C2×Q818yes—
319C24—21no—
320C24—21no—
321C24—21no—
322C24—21no—
323C24—21no—
324C24—21no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24C22×Q812yes—
330C24C2511yescharacteristic
331C2×Q8—216no—
332C2×Q8—416no—
333C2×Q8—416no—
334C2×Q8—48no—
335C2×Q8—88no—
336C2×Q8—88no—
337C2×Q8—84no—
338C2×Q8—84no—
339C2×Q8—84no—
340C2×Q8—84no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—44no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4C22×D418yes—
420C22×C4C22×D418yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×Q812yes—
428C22×C4C22×Q814yes—
429C22×C4C22×Q814yes—
430C22×C4C22×Q812yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C42−1+412yes—
437C22×C42−1+412yes—
438C4⋊C4—216no—
439C4⋊C4—216no—
440C4⋊C4—216no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—88no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C22⋊C4—48no—
451C22⋊C4—48no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—88no—
455C22⋊C4—88no—
456C42—28no—
457C42—28no—
458C42—28no—
459C42—48no—
460C42—48no—
461C42—44no—
462C42—44no—
463C42—88no—
464C42#121116yes—
465C23—22no—
466C23—24no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23C23×D411yescharacteristic
489C23C23×D411yescharacteristic
490C23C23×Q811yescharacteristic
491C23C22×#4111yescharacteristic
492C23C22×#4111yescharacteristic
493C23C2×2−1+411yescharacteristic
494C23C2×2−1+411yescharacteristic
495C23#58512yes—
496C23#58512yes—
497C23#58512yes—
498C23#58512yes—
499Q8—816no—
500Q8—816no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—28no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—416no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4C2×#12114yes—
584C2×C4C2×#12114yes—
585C2×C4C2×#12114yes—
586C2×C4#53314yes—
587C2×C4#54114yes—
588C2×C4#54714yes—
589C2×C4#54814yes—
590C2×C4D4×Q814yes—
591C2×C4D4×Q814yes—
592C22—42no—
593C22—42no—
594C22—44no—
595C22—42no—
596C22—44no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—82no—
601C22—81no—
602C22—82no—
603C22C22×#12111yescharacteristic
604C22C2×#53311yescharacteristic
605C22C2×#54111yescharacteristic
606C22#345512yes—
607C22#345812yes—
608C22C2×#54711yescharacteristic
609C22C2×#54811yescharacteristic
610C22C2×D4×Q811yescharacteristic
611C22C2×D4×Q811yescharacteristic
612C22#348512yes—
613C22#348712yes—
614C22#350012yes—
615C22#350012yes—
616C22#350412yes—
617C22#351011yescharacteristic
618C22#351011yescharacteristic
619C22#351011yescharacteristic
620C22#351011yescharacteristic
621C22#351012yes—
622C22#356512yes—
623C22#356512yes—
624C22#356512yes—
625C22#356512yes—
626C4—24no—
627C4—24no—
628C4—48no—
629C4—44no—
630C4—44no—
631C4—42no—
632C4—44no—
633C4—48no—
634C4—81no—
635C4—84no—
636C4—81no—
637C4—84no—
638C4—82no—
639C4#242918yes—
640C2—82no—
641C2C2×#242911yescharacteristic
642C2#3533711yescharacteristic
643C2#3534411yescharacteristic
644C2#3561711yescharacteristic
645C2#3567711yescharacteristic
646C2#3743311yescharacteristic
647C2#3743411yescharacteristic
648C2#3769112yes—
649C2#3771112yes—
650C2#3936612yes—
651C2#3937412yes—
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
4. C2×#2331C2semidirect
5. #15858C2semidirect
7. #15503C2semidirect
9. #15225C2semidirect
11. #14707C2semidirect
12. #14282C2semidirect
13. #14006C2semidirect
142. 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
C2×#23312
#158582
#158552
#155032
#154992
#152252
#152202
#147072
#142822
#140062
#138752
#137612