Finite Group Databasebeta

← #952115

Subgroups — #952115

Order 512.

2558 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×#2512C212yesmaximal, complemented
2#17123C214yesmaximal, complemented
3#16658C214yesmaximal, complemented
4#16347C214yesmaximal, complemented
5#15939C214yesmaximal, complemented
6#15222C212yesmaximal, complemented
7#15148C212yesmaximal, complemented
8#15140C212yesmaximal
9#14276C211yesmaximal, characteristic, complemented
10#14272C211yesmaximal, characteristic
11#14022C212yesmaximal, complemented
12#14006C212yesmaximal, complemented
13#13848C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#126C2214yes—
16#2512—216no—
17C2×#399C2214yes—
18C2×#397C2214yes—
19C2×#396C2214yes—
20C2×#394C2214yes—
21C2×#394C2214yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#390C2212yes—
26C2×#390C2212yes—
27C2×#390C2212yes—
28C2×#389C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×#376C2211yescharacteristic
41#1444C2212yes—
42#1442C2214yes—
43#1442C2212yes—
44#1441C2214yes—
45#1441C2212yes—
46#1441C2212yes—
47#1441C2214yes—
48#1441C2214yes—
49#1440C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2214yes—
61#1438C2214yes—
62#1437C2214yes—
63#1436C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4×#95C2212yes—
68C23×D4C2312yes—
69C23×D4C2314yes—
70C2×#127—24no—
71C2×#127—42no—
72C2×#127—42no—
73C2×#126—216no—
74C2×#126—48no—
75C2×#126—48no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2312yes—
100C22×C42C2312yes—
101C22×C42C2314yes—
102C22×C42C2311yescharacteristic
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105#399—48no—
106#397—216no—
107#396—216no—
108#394—48no—
109#394—48no—
110#390—28no—
111#390—28no—
112#390—28no—
113#390—28no—
114#390—44no—
115#390—44no—
116#389—28no—
117#389—28no—
118#389—44no—
119#388—216no—
120#388—48no—
121#384D418yescomplemented
122#382—42no—
123#382—42no—
124C4×C4⋊C4—28no—
125C4×C4⋊C4—28no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—42no—
128#376—42no—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2311yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2311yescharacteristic
151C25C2414yes—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—28no—
156C22×D4—24no—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—44no—
161C22×D4—44no—
162C22×D4—44no—
163C22×D4—44no—
164C22×D4—44no—
165C22×D4—44no—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2414yes—
169C23×C4C2412yes—
170C23×C4C2414yes—
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2411yescharacteristic
178#127—416no—
179#126—464no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
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—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4C2×D418yes—
212C2×C4⋊C4C2×D4116yes—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
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×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—28no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—28no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—44no—
264C2×C42C2×D414yes—
265C2×C42#4114yes—
266C2×C42#4114yes—
267#95—24no—
268#95—28no—
269#95—24no—
270#95—28no—
271#95—22no—
272#95—22no—
273#95—28no—
274#95—24no—
275#95—24no—
276#95—42no—
277#95—44no—
278#95—42no—
279#95—42no—
280#95—42no—
281#95—41no—
282#95—41no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24C2511yescharacteristic
305C2×D4—416no—
306C2×D4—48no—
307C2×D4—48no—
308C2×D4—48no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4—48no—
313C2×D4—88no—
314C2×D4—88no—
315C2×D4—88no—
316C2×D4—88no—
317C2×D4—88no—
318C2×D4—88no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4C22×D418yes—
386C22×C4C22×D414yes—
387C22×C4C22×D412yes—
388C22×C4C2×#4112yes—
389C22×C4C2×#4114yes—
390C22×C4C2×#4112yes—
391C22×C4C2×#4112yes—
392C22×C4C2×#4112yes—
393C22×C4H(5,𝔽2)14yes—
394C22×C4H(5,𝔽2)14yes—
395C22×C4H(5,𝔽2)12yes—
396C22×C4H(5,𝔽2)12yes—
397C22×C42−1+412yes—
398C22×C42−1+412yes—
399C4⋊C4—216no—
400C4⋊C4—216no—
401C4⋊C4—416no—
402C4⋊C4—48no—
403C4⋊C4—416no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—88no—
409C4⋊C4—88no—
410C22⋊C4—416no—
411C22⋊C4—416no—
412C22⋊C4—416no—
413C22⋊C4—416no—
414C22⋊C4—416no—
415C22⋊C4—88no—
416C22⋊C4—88no—
417C22⋊C4—88no—
418C22⋊C4—88no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C42—28no—
423C42—28no—
424C42—28no—
425C42—416no—
426C42—48no—
427C42—48no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23C23×D411yescharacteristic
464C23C22×#4111yescharacteristic
465C23C22×#4111yescharacteristic
466C23C2×H(5,𝔽2)11yescharacteristic
467C23C2×H(5,𝔽2)11yescharacteristic
468C23C2×H(5,𝔽2)12yes—
469C23C2×H(5,𝔽2)12yes—
470C23C2×2−1+411yescharacteristic
471C23C2×2−1+411yescharacteristic
472C23#58512yes—
473C23#58512yes—
474D4—832no—
475D4—816no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4#53714yes—
550C2×C4#53814yes—
551C2×C4#53914yes—
552C2×C4#54714yes—
553C2×C4#55014yes—
554C2×C4#55614yes—
555C2×C4#56314yes—
556C22—84no—
557C22—84no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22—84no—
571C22C2×#53711yescharacteristic
572C22C2×#53811yescharacteristic
573C22C2×#53911yescharacteristic
574C22#346012yes—
575C22C2×#54711yescharacteristic
576C22C2×#55011yescharacteristic
577C22C2×#55611yescharacteristic
578C22#348412yes—
579C22#348712yes—
580C22#349412yes—
581C22#350812yes—
582C22#351012yes—
583C22C2×#56311yescharacteristic
584C22#354212yes—
585C22#354612yes—
586C22#354612yes—
587C22#354712yes—
588C22#354811yescharacteristic
589C22#354811yescharacteristic
590C22#355711yescharacteristic
591C22#355711yescharacteristic
592C22#355812yes—
593C22#355912yes—
594C4—48no—
595C4—44no—
596C4—42no—
597C4—42no—
598C4—44no—
599C4—44no—
600C4—44no—
601C4—84no—
602C4—82no—
603C4—84no—
604C4—82no—
605C4#272718yes—
606C2—164no—
607C2#3728611yescharacteristic
608C2#3752011yescharacteristic
609C2#3777212yes—
610C2#3980312yes—
611C2#4247612yes—
612C2#4297612yes—
613C2#4493111yescharacteristic
614C2#4526611yescharacteristic
615C2C2×#272711yescharacteristic
616C2#5312511yescharacteristic
617C2#5313211yescharacteristic
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. C2×#2512C2semidirect
2. #17123C2semidirect
3. #16658C2semidirect
4. #16347C2semidirect
5. #15939C2semidirect
6. #15222C2semidirect
7. #15148C2semidirect
9. #14276C2semidirect
11. #14022C2semidirect
12. #14006C2semidirect
13. #13848C2semidirect
121. #384D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25122
#171232
#166582
#163472
#159392
#152222
#151482
#151402
#142762
#142722
#140222
#140062
#138482