Finite Group Databasebeta

← #949111

Subgroups — #949111

Order 512.

2482 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C212yesmaximal, complemented
2#17185C214yesmaximal, complemented
3#16521C214yesmaximal, complemented
4#16405C214yesmaximal, complemented
5#15439C214yesmaximal
6#15221C212yesmaximal, complemented
7#15135C212yesmaximal, complemented
8#15131C212yesmaximal, complemented
9#14283C211yesmaximal, characteristic, complemented
10#14276C211yesmaximal, characteristic, complemented
11#13881C212yesmaximal, complemented
12#13872C212yesmaximal
13#13847C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C23×C4⋊C4C2212yes—
17#2442—216no—
18C2×#400C2214yes—
19C2×#399C2214yes—
20C2×#398C2214yes—
21C2×#396C2214yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#386C2214yes—
26C2×#385C2214yes—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2212yes—
30C2×#382C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2214yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×#376C2211yescharacteristic
39#1445C2214yes—
40#1445C2212yes—
41#1445C2214yes—
42#1444C2214yes—
43#1444C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1438C2212yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C23×D4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2314yes—
71C2×#125—42no—
72C2×#125—42no—
73C2×#125D418yescomplemented
74C2×C4×D4—42no—
75C2×C4×D4—42no—
76C2×C4×D4Q818yes—
77C22×C4⋊C4—28no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4—44no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106#400—216no—
107#399—48no—
108#398—216no—
109#396—48no—
110#389—28no—
111#389—28no—
112#389—44no—
113#386—48no—
114#385—216no—
115#384—24no—
116#384—42no—
117#382—28no—
118#382—42no—
119#380—216no—
120#380—28no—
121#380—44no—
122#380—44no—
123C4×C4⋊C4—42no—
124C4×C22⋊C4—28no—
125C4×C22⋊C4—28no—
126#376—42no—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C25C2412yes—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—41no—
155C22×D4—41no—
156C22×D4C2×Q814yes—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—28no—
164C23×C4—28no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4C2×Q818yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2414yes—
174C23×C4C2412yes—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2411yescharacteristic
179C23×C4C2412yes—
180C23×C4C2411yescharacteristic
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183#125—416no—
184C4×D4—416no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
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—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—432no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C2×D418yes—
224C2×C4⋊C4C2×D4116yes—
225C2×C4⋊C4C2×Q814yes—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4C2×Q818yes—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42C2×D414yes—
260C2×C42C2×Q814yes—
261C2×C42#4114yes—
262#95—24no—
263#95—24no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—28no—
269#95—24no—
270#95—24no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—42no—
276#95—42no—
277#95—42no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24C22×Q814yes—
299C24H(5,𝔽2)14yes—
300C242−1+414yes—
301C24C2511yescharacteristic
302C2×D4—216no—
303C2×D4—44no—
304C2×D4—48no—
305C2×D4—44no—
306C2×D4—44no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
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—24no—
339C22×C4—216no—
340C22×C4—216no—
341C22×C4—216no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—416no—
378C22×C4—416no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—88no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4C22×D414yes—
389C22×C4C22×D418yes—
390C22×C4C22×D412yes—
391C22×C4C22×Q812yes—
392C22×C4C22×Q812yes—
393C22×C4C22×Q814yes—
394C22×C4C22×Q812yes—
395C22×C4C2×#4114yes—
396C22×C4C2×#4112yes—
397C22×C4H(5,𝔽2)12yes—
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C42−1+412yes—
401C4⋊C4—216no—
402C4⋊C4—416no—
403C4⋊C4—44no—
404C4⋊C4—416no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—88no—
411C4⋊C4—88no—
412C4⋊C4—88no—
413C4⋊C4—88no—
414C4⋊C4—88no—
415C22⋊C4—48no—
416C22⋊C4—48no—
417C22⋊C4—416no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C42—28no—
424C42—28no—
425C42—28no—
426C42—48no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23#54418yes—
462C23C23×D411yescharacteristic
463C23C23×Q811yescharacteristic
464C23C22×#4111yescharacteristic
465C23C2×H(5,𝔽2)11yescharacteristic
466C23C2×H(5,𝔽2)11yescharacteristic
467C23C2×H(5,𝔽2)11yescharacteristic
468C23C2×H(5,𝔽2)12yes—
469C23C2×2−1+411yescharacteristic
470C23C2×2−1+412yes—
471C23#58512yes—
472C23#58512yes—
473D4—416no—
474C2×C4—24no—
475C2×C4—28no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—432no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—832no—
549C2×C4#53414yes—
550C2×C4#53914yes—
551C2×C4#54314yes—
552C2×C4D4×Q814yes—
553C2×C4#55014yes—
554C2×C4#55414yes—
555C2×C4#56214yes—
556C22—28no—
557C22—28no—
558C22—28no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22C2×#53411yescharacteristic
572C22C2×#53911yescharacteristic
573C22C2×#54311yescharacteristic
574C22C2×#54412yes—
575C22C2×D4×Q811yescharacteristic
576C22C2×#55011yescharacteristic
577C22C2×#55411yescharacteristic
578C22#348812yes—
579C22#348912yes—
580C22#349412yes—
581C22#349512yes—
582C22#350412yes—
583C22C2×#56211yescharacteristic
584C22#353612yes—
585C22#353711yescharacteristic
586C22#353811yescharacteristic
587C22#353911yescharacteristic
588C22#353911yescharacteristic
589C22#355112yes—
590C22#355812yes—
591C22#355912yes—
592C22#356112yes—
593C22#356312yes—
594C4—42no—
595C4—42no—
596C4—44no—
597C4—44no—
598C4—48no—
599C4—44no—
600C4—82no—
601C4—84no—
602C4—82no—
603C4—84no—
604C4—84no—
605C4#273218yes—
606C2—48no—
607C2#3742511yescharacteristic
608C2#3747711yescharacteristic
609C2#3768712yes—
610C2#3787512yes—
611C2#4295012yes—
612C2#4295612yes—
613C2#4542311yescharacteristic
614C2#4543211yescharacteristic
615C2C2×#273211yescharacteristic
616C2#5298511yescharacteristic
617C2#5303011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2442C2semidirect
2. #17185C2semidirect
3. #16521C2semidirect
4. #16405C2semidirect
6. #15221C2semidirect
7. #15135C2semidirect
8. #15131C2semidirect
9. #14283C2semidirect
10. #14276C2semidirect
11. #13881C2semidirect
73. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
#171852
#165212
#164052
#154392
#152212
#151352
#151312
#142832
#142762
#138812
#138722
#138472