Finite Group Databasebeta

← #951774

Subgroups — #951774

Order 512.

2262 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17234C212yesmaximal, complemented
2#17107C212yesmaximal, complemented
3#16837C214yesmaximal, complemented
4#16781C214yesmaximal, complemented
5#15782C212yesmaximal, complemented
6#15569C212yesmaximal, complemented
7#15147C212yesmaximal, complemented
8#15144C212yesmaximal, complemented
9#15127C212yesmaximal
10#14282C211yesmaximal, characteristic, complemented
11#14272C211yesmaximal, characteristic
12#14022C212yesmaximal, complemented
13#14002C212yesmaximal, complemented
14#13885C212yesmaximal, complemented
15#13848C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#127C2211yescharacteristic
18C2×#397C2214yes—
19C2×#394C2214yes—
20C2×#394C2214yes—
21C2×#393C2214yes—
22C2×#393C2214yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#388C2214yes—
30C2×#382C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2212yes—
36C2×#376C2211yescharacteristic
37#1446C2212yes—
38#1446C2212yes—
39#1445C2214yes—
40#1445C2212yes—
41#1445C2212yes—
42#1444C2211yescharacteristic
43#1444C2211yescharacteristic
44#1443C2214yes—
45#1442C2211yescharacteristic
46#1442C2211yescharacteristic
47#1441C2212yes—
48#1441C2212yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2212yes—
61#1438C2212yes—
62#1438C2212yes—
63#1438C2212yes—
64#1437C2214yes—
65#1437C2214yes—
66#1437C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74C4×#95C2212yes—
75C23×Q8C2312yes—
76C23×D4C2312yes—
77C23×D4C2312yes—
78C23×D4C2312yes—
79C2×C4⋊Q8—24no—
80C2×C4⋊Q8—42no—
81C2×C4⋊Q8—42no—
82C2×#127—24no—
83C2×#127—42no—
84C2×#127—42no—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2314yes—
112#397—48no—
113#394—48no—
114#394—48no—
115#393—48no—
116#393—48no—
117#390—28no—
118#390—28no—
119#390—44no—
120#390—44no—
121#390—44no—
122#390—44no—
123#388—48no—
124#382—42no—
125#382—42no—
126C4×C4⋊C4—28no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—42no—
129C4×C4⋊C4—44no—
130#376—42no—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2311yescharacteristic
157C2×#95C2311yescharacteristic
158C2×#95C2314yes—
159C25C2414yes—
160C22×Q8—24no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—44no—
164C22×Q8—44no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—44no—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C4⋊Q8—416no—
197#127—416no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
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—24no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42#4114yes—
275C2×C42#4114yes—
276C2×C42#4114yes—
277#95—22no—
278#95—24no—
279#95—24no—
280#95—22no—
281#95—28no—
282#95—24no—
283#95—24no—
284#95—28no—
285#95—44no—
286#95—42no—
287#95—42no—
288#95—41no—
289#95—41no—
290#95—41no—
291#95—41no—
292#95—42no—
293#95—44no—
294#95—42no—
295#95—42no—
296#95—42no—
297#95—42no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24C2511yescharacteristic
321C2×Q8—48no—
322C2×Q8—88no—
323C2×Q8—84no—
324C2×Q8—88no—
325C2×Q8—84no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—48no—
329C2×D4—48no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—88no—
336C2×D4—88no—
337C2×D4—84no—
338C2×D4—84no—
339C2×D4—88no—
340C2×D4—84no—
341C2×D4—84no—
342C2×D4—88no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
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—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4C2×#4112yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4H(5,𝔽2)14yes—
424C22×C4H(5,𝔽2)14yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C4H(5,𝔽2)12yes—
428C22×C42−1+412yes—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C22⋊C4—416no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C22⋊C4—88no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C42—28no—
451C42—28no—
452C42—28no—
453C42—48no—
454C42—48no—
455C42—88no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23C22×#4111yescharacteristic
492C23C22×#4111yescharacteristic
493C23C22×#4111yescharacteristic
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×H(5,𝔽2)11yescharacteristic
496C23C2×H(5,𝔽2)11yescharacteristic
497C23C2×H(5,𝔽2)12yes—
498C23C2×H(5,𝔽2)12yes—
499C23C2×2−1+411yescharacteristic
500C23#58512yes—
501C23#58512yes—
502Q8—816no—
503D4—816no—
504D4—816no—
505D4—816no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4#53814yes—
586C2×C4#53814yes—
587C2×C4#53914yes—
588C2×C4#55514yes—
589C2×C4#55614yes—
590C2×C4#55914yes—
591C2×C4#56214yes—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22C2×#53811yescharacteristic
608C22C2×#53811yescharacteristic
609C22C2×#53911yescharacteristic
610C22#346012yes—
611C22C2×#55511yescharacteristic
612C22C2×#55611yescharacteristic
613C22C2×#55911yescharacteristic
614C22#349412yes—
615C22#349412yes—
616C22#349412yes—
617C22#350812yes—
618C22#351012yes—
619C22C2×#56211yescharacteristic
620C22#353711yescharacteristic
621C22#354011yescharacteristic
622C22#354312yes—
623C22#354612yes—
624C22#354911yescharacteristic
625C22#355111yescharacteristic
626C22#355412yes—
627C22#355412yes—
628C22#355812yes—
629C22#356212yes—
630C4—44no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—42no—
635C4—42no—
636C4—44no—
637C4—42no—
638C4—84no—
639C4—82no—
640C4—84no—
641C4—82no—
642C4—84no—
643C4#280818yes—
644C2—164no—
645C2#3728911yescharacteristic
646C2#3752311yescharacteristic
647C2#3795412yes—
648C2#3981012yes—
649C2#4296111yescharacteristic
650C2#4297511yescharacteristic
651C2#4297711yescharacteristic
652C2#4298211yescharacteristic
653C2#4543012yes—
654C2C2×#280811yescharacteristic
655C2#5299512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17234C2semidirect
2. #17107C2semidirect
3. #16837C2semidirect
4. #16781C2semidirect
5. #15782C2semidirect
6. #15569C2semidirect
7. #15147C2semidirect
8. #15144C2semidirect
10. #14282C2semidirect
12. #14022C2semidirect
13. #14002C2semidirect
14. #13885C2semidirect
15. #13848C2semidirect

Maximal subgroups

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

subgroupindex
#172342
#171072
#168372
#167812
#157822
#155692
#151472
#151442
#151272
#142822
#142722
#140222
#140022
#138852
#138482