Finite Group Databasebeta

← #4462756

Subgroups — #4462756

Order 512.

2386 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2641C211yesmaximal, characteristic
2C2×#2402C211yesmaximal, characteristic, complemented
3#17126C214yesmaximal, complemented
4#17117C214yesmaximal, complemented
5#17103C214yesmaximal, complemented
6#17096C214yesmaximal, complemented
7#15222C212yesmaximal, complemented
8#15218C212yesmaximal, complemented
9#15148C212yesmaximal, complemented
10#15147C212yesmaximal, complemented
11#15134C212yesmaximal, complemented
12#15127C212yesmaximal
13#14348C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#125C2211yescharacteristic
16#2641—28no—
17#2402—28no—
18C2×#400C2212yes—
19C2×#400C2212yes—
20C2×#399C2214yes—
21C2×#398C2212yes—
22C2×#398C2212yes—
23C2×#397C2214yes—
24C2×#396C2214yes—
25C2×#396C2214yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#390C2212yes—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#384C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×#376C2211yescharacteristic
45#1444C2214yes—
46#1444C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1438C2214yes—
64#1438C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C23×D4C2312yes—
70C23×D4C2314yes—
71C2×#127—24no—
72C2×#127—42no—
73C2×#127—42no—
74C2×#125—24no—
75C2×#125—42no—
76C2×#125—42no—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊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×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107#400—28no—
108#400—28no—
109#399—48no—
110#398—28no—
111#398—28no—
112#397—48no—
113#396—48no—
114#396—48no—
115#394—216no—
116#394—216no—
117#394—48no—
118#394—48no—
119#390—28no—
120#390—28no—
121#390—44no—
122#390—44no—
123#390—44no—
124#390—44no—
125#389—44no—
126#389—44no—
127#384—42no—
128#382—42no—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—28no—
132C4×C4⋊C4—28no—
133#376D418yes—
134C2×#95C2314yes—
135C2×#95C2311yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2311yescharacteristic
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C25C2414yes—
156C22×D4—24no—
157C22×D4—28no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2414yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C23×C4C2411yescharacteristic
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2411yescharacteristic
182#127—416no—
183#125—416no—
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—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
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—28no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
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×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×C42C2×D414yes—
272C2×C42C2×D414yes—
273C2×C42C2×D414yes—
274#95—22no—
275#95—22no—
276#95—24no—
277#95—24no—
278#95—22no—
279#95—24no—
280#95—22no—
281#95—24no—
282#95—28no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95C2×D418yes—
294#95C2×D418yes—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24C2511yescharacteristic
311C2×D4—416no—
312C2×D4—48no—
313C2×D4—48no—
314C2×D4—48no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—48no—
318C2×D4—48no—
319C2×D4—88no—
320C2×D4—88no—
321C2×D4—88no—
322C2×D4—88no—
323C2×D4—88no—
324C2×D4—88no—
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—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
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—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D414yes—
395C22×C4C22×D414yes—
396C22×C4C22×D412yes—
397C22×C4H(5,𝔽2)14yes—
398C22×C4H(5,𝔽2)14yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C4H(5,𝔽2)12yes—
402C22×C42−1+412yes—
403C22×C42−1+412yes—
404C22×C42−1+412yes—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—416no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C4⋊C4—88no—
418C22⋊C4—88no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C22⋊C4—88no—
422C22⋊C4—88no—
423C22⋊C4—88no—
424C22⋊C4—88no—
425C22⋊C4—88no—
426C22⋊C4—88no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C22⋊C4—88no—
430C42—28no—
431C42—28no—
432C42—28no—
433C42—48no—
434C42—48no—
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—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23C23×D411yescharacteristic
471C23C2×H(5,𝔽2)11yescharacteristic
472C23C2×H(5,𝔽2)11yescharacteristic
473C23C2×H(5,𝔽2)11yescharacteristic
474C23C2×H(5,𝔽2)12yes—
475C23C2×H(5,𝔽2)12yes—
476C23C2×2−1+411yescharacteristic
477C23C2×2−1+411yescharacteristic
478C23C2×2−1+411yescharacteristic
479C23#58512yes—
480C23#58512yes—
481D4—832no—
482D4—816no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
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—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4#53514yes—
557C2×C4#53614yes—
558C2×C4#53714yes—
559C2×C4#56214yes—
560C2×C4#56214yes—
561C2×C4#56314yes—
562C2×C4#56314yes—
563C22—84no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—84no—
577C22—84no—
578C22C2×#53511yescharacteristic
579C22C2×#53611yescharacteristic
580C22C2×#53711yescharacteristic
581C22#348812yes—
582C22#348812yes—
583C22C2×#56211yescharacteristic
584C22C2×#56211yescharacteristic
585C22C2×#56311yescharacteristic
586C22C2×#56311yescharacteristic
587C22#354312yes—
588C22#354312yes—
589C22#354612yes—
590C22#354712yes—
591C22#354811yescharacteristic
592C22#354811yescharacteristic
593C22#355412yes—
594C22#355412yes—
595C22#355711yescharacteristic
596C22#355711yescharacteristic
597C22#355812yes—
598C22#355912yes—
599C22#356212yes—
600C22#356212yes—
601C4—42no—
602C4—42no—
603C4—44no—
604C4—44no—
605C4—44no—
606C4—44no—
607C4—84no—
608C4—84no—
609C4—82no—
610C4—82no—
611C4—84no—
612C4#282018yes—
613C2—164no—
614C2C2×#282011yescharacteristic
615C2#3727411yescharacteristic
616C2#3727411yescharacteristic
617C2#3727612yes—
618C2#3727812yes—
619C2#5299512yes—
620C2#5299712yes—
621C2#5311711yescharacteristic
622C2#5312511yescharacteristic
623C2#5313211yescharacteristic
624C2#5315411yescharacteristic
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
2. C2×#2402C2semidirect
3. #17126C2semidirect
4. #17117C2semidirect
5. #17103C2semidirect
6. #17096C2semidirect
7. #15222C2semidirect
8. #15218C2semidirect
9. #15148C2semidirect
10. #15147C2semidirect
11. #15134C2semidirect
13. #14348C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26412
C2×#24022
#171262
#171172
#171032
#170962
#152222
#152182
#151482
#151472
#151342
#151272
#143482