Finite Group Databasebeta

← #726111

Subgroups — #726111

Order 512.

2014 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2565C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3#17048C212yesmaximal
4#17031C212yesmaximal
5#15558C214yesmaximal
6#15395C214yesmaximal
7#15192C212yesmaximal
8#15192C212yesmaximal
9#15102C212yesmaximal
10#15080C212yesmaximal
11#14283C211yesmaximal, characteristic
12#14160C211yesmaximal, characteristic
13#13911C211yesmaximal, characteristic
14#13885C212yesmaximal
15#13884C212yesmaximal
16#13856C211yesmaximal, characteristic
17#13786C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20#2565—28no—
21C2×#398C2212yes—
22C2×#398C2212yes—
23#2355—28no—
24C2×#391C2212yes—
25C2×#391C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#387C2214yes—
29C2×#387C2212yes—
30C2×#387C2212yes—
31C2×#387C2214yes—
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#382C2212yes—
36C2×#382C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×#376C2211yescharacteristic
45#1445C2214yes—
46#1445C2214yes—
47#1445C2214yes—
48#1444C2212yes—
49#1444C2211yescharacteristic
50#1444C2211yescharacteristic
51#1444C2212yes—
52#1444C2211yescharacteristic
53#1444C2211yescharacteristic
54#1443C2214yes—
55#1443C2214yes—
56#1443C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2212yes—
62#1442C2212yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2212yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2214yes—
83#1434C2212yes—
84#1434C2212yes—
85C23×Q8C2311yescharacteristic
86C2×#125—24no—
87C2×#125—24no—
88C2×#125—24no—
89C2×C4×Q8—24no—
90C2×C4×Q8—24no—
91C2×C4×Q8D418yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2314yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120#398—28no—
121#398—28no—
122#391—24no—
123#391—24no—
124#391—28no—
125#391—28no—
126#387—216no—
127#387—28no—
128#387—28no—
129#387—48no—
130#384—24no—
131#384—24no—
132#382—24no—
133#382—28no—
134#382—44no—
135C4×C4⋊C4—28no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4—28no—
139C4×C4⋊C4—28no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4—24no—
142#376—24no—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8C2×D414yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201#125—416no—
202C4×Q8—232no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
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—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×D418yes—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—28no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—44no—
276C2×C42—44no—
277C2×C42C2×D418yes—
278C2×C42C2×D414yes—
279C2×C42#4118yes—
280C2×C42#4114yes—
281C2×C42#4114yes—
282#95—28no—
283#95—24no—
284#95—22no—
285#95—22no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318C24C2511yescharacteristic
319C2×Q8—28no—
320C2×Q8—28no—
321C2×Q8—28no—
322C2×Q8—48no—
323C2×Q8—48no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
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—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4C22×D414yes—
410C22×C4C22×D414yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C2×#4114yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4114yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C42−1+414yes—
425C22×C42−1+412yes—
426C22×C42−1+414yes—
427C4⋊C4—28no—
428C4⋊C4—216no—
429C4⋊C4—28no—
430C4⋊C4—216no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—84no—
447C42—216no—
448C42—216no—
449C42—48no—
450C42—44no—
451C42—48no—
452C42—44no—
453C42—88no—
454C42#123116yes—
455C23C23×D411yescharacteristic
456C23C22×#4111yescharacteristic
457C23C22×#4111yescharacteristic
458C23C22×#4112yes—
459C23C2×H(5,𝔽2)11yescharacteristic
460C23C2×2−1+412yes—
461C23C2×2−1+412yes—
462C23C2×2−1+411yescharacteristic
463C23#58512yes—
464C23#58511yescharacteristic
465C23#58511yescharacteristic
466Q8—416no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
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—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
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—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4C2×#12314yes—
564C2×C4C2×#12314yes—
565C2×C4C2×#12314yes—
566C2×C4#53214yes—
567C2×C4#53614yes—
568C2×C4#55218yes—
569C2×C4#55618yes—
570C22C22×#12311yescharacteristic
571C22C2×#53211yescharacteristic
572C22C2×#53611yescharacteristic
573C22#346112yes—
574C22C2×#55212yes—
575C22C2×#55612yes—
576C22#348611yescharacteristic
577C22#348711yescharacteristic
578C22#348811yescharacteristic
579C22#348911yescharacteristic
580C22#349212yes—
581C22#349312yes—
582C22#349512yes—
583C22#349512yes—
584C22#350611yescharacteristic
585C22#350712yes—
586C22#350711yescharacteristic
587C22#351111yescharacteristic
588C22#351211yescharacteristic
589C22#355412yes—
590C22#355512yes—
591C22#355612yes—
592C22#355712yes—
593C4—24no—
594C4—24no—
595C4—44no—
596C4—42no—
597C4—44no—
598C4—44no—
599C4—44no—
600C4—42no—
601C4—44no—
602C4—84no—
603C4—81no—
604C4—82no—
605C4—84no—
606C4—82no—
607C4—82no—
608C4—81no—
609C4#249918yes—
610C2C2×#249911yescharacteristic
611C2#3518211yescharacteristic
612C2#3551711yescharacteristic
613C2#3592611yescharacteristic
614C2#3606511yescharacteristic
615C2#3798212yes—
616C2#3798612yes—
617C2#4281111yescharacteristic
618C2#4288611yescharacteristic
619C2#4515212yes—
620C2#4537012yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25652
C2×#23552
#170482
#170312
#155582
#153952
#151922
#151022
#150802
#142832
#141602
#139112
#138852
#138842
#138562
#137862