Finite Group Databasebeta

← #6627609

Subgroups — #6627609

Order 512.

1447 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2C4×#557C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic
4#33027C212yesmaximal
5#29357C212yesmaximal
6#29275C212yesmaximal
7#29218C211yesmaximal, characteristic
8#28283C211yesmaximal, characteristic
9#27580C214yesmaximal
10#27152C214yesmaximal
11#24995C212yesmaximal
12#24332C212yesmaximal
13#23068C212yesmaximal
14#22825C211yesmaximal, characteristic
15#22763C212yesmaximal
16#21616C212yesmaximal
17C4×#462C211yesmaximal, characteristic
18C2×#557C2211yescharacteristic
19C2×#477C2212yes—
20C2×#475C2212yes—
21C2×#462C2211yescharacteristic
22C2×C4×Q16C2212yes—
23#2868C2211yescharacteristic
24#2862C2212yes—
25#2582C2212yes—
26#2571—21no—
27#2570—22no—
28#2568—21no—
29#2511C2214yes—
30#2446—22no—
31#2370—22no—
32#2370C2212yes—
33#2365—21no—
34#2365—21no—
35#2365C2211yescharacteristic
36#2355—21no—
37#2355—21no—
38#2355C2212yes—
39#2355C2211yescharacteristic
40#2351C2212yes—
41Q8×C4⋊C4C2211yescharacteristic
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#125—22no—
47C4×#125C2212yes—
48C42×Q8—21no—
49C42×Q8C2212yes—
50C42×Q8C2211yescharacteristic
51C42×Q8C2211yescharacteristic
52#2101C2214yes—
53#2100C2214yes—
54#2074C2214yes—
55#2066C2214yes—
56#2059C2214yes—
57#2026C2214yes—
58#1990C2212yes—
59#1987C2212yes—
60#1987C2212yes—
61#1987C2212yes—
62#1984C2212yes—
63#1981C2212yes—
64#1981C2211yescharacteristic
65#1948C2212yes—
66#1947C2212yes—
67#1947C2212yes—
68#1947C2211yescharacteristic
69#1935C2214yes—
70#1935C2214yes—
71#1928C2214yes—
72#1928C2214yes—
73#1928C2212yes—
74#1920C2214yes—
75#1918C2214yes—
76#1918C2214yes—
77#1917C2214yes—
78#1878C2214yes—
79#1878C2214yes—
80#1878C2212yes—
81#1865C2214yes—
82#1842C2211yescharacteristic
83#1834C2212yes—
84#1831C2212yes—
85#1803C2212yes—
86#1803C2211yescharacteristic
87#1803C2212yes—
88#1803C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#103C2212yes—
95C22×Q16C2312yes—
96#557—24no—
97#557—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×#125—22no—
105C2×#125C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119#477—28no—
120#475—28no—
121#462—24no—
122C4×Q16—28no—
123C2×#107C2314yes—
124C2×#107C2312yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#103C2314yes—
137C2×#103C2312yes—
138C2×#97C2314yes—
139C2×C4×C8C2312yes—
140#400—24no—
141#400—24no—
142#398—24no—
143#391—21no—
144#391—22no—
145#391—21no—
146#391—22no—
147#391—21no—
148#391—22no—
149#391—22no—
150#391C2312yes—
151#391C2314yes—
152#391C2314yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—21no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389C2312yes—
163#389C2312yes—
164#389C2314yes—
165#389C2312yes—
166#387—22no—
167#387—22no—
168#387—24no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—24no—
174#387—24no—
175#384—21no—
176#384—21no—
177#384—21no—
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2312yes—
181#384C2312yes—
182#382—22no—
183#382—22no—
184#382—22no—
185#382—22no—
186#382C2312yes—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4D412yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C43—22no—
218C43—21no—
219C43C2312yes—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2312yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2314yes—
229#340C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2312yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2412yes—
238C22×Q8C2411yescharacteristic
239C2×Q16—24no—
240C2×Q16—24no—
241C2×Q16—24no—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2414yes—
245C4⋊Q8—28no—
246C4⋊Q8—22no—
247C4⋊Q8—24no—
248C4⋊Q8—22no—
249C4⋊Q8—24no—
250C4⋊Q8—22no—
251C4⋊Q8—44no—
252C4⋊Q8—82no—
253C4⋊Q8#4114yes—
254#125—216no—
255#125—24no—
256#125—24no—
257C4×Q8—28no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—42no—
270C4×Q8—42no—
271C4×Q8—41no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—84no—
275C4×Q8#4114yes—
276C4×Q8#4114yes—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—21no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—21no—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2412yes—
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2411yescharacteristic
340#107—24no—
341#107—28no—
342#106—28no—
343C4⋊C8—22no—
344C4⋊C8—24no—
345C4⋊C8—24no—
346C4⋊C8—22no—
347C4⋊C8—42no—
348#103—24no—
349#103—28no—
350#103—24no—
351#103—24no—
352#103—28no—
353#103—42no—
354#103—44no—
355#97—44no—
356C4×C8—24no—
357#95—22no—
358#95—22no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379C2×Q8—22no—
380C2×Q8—24no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8C2×#4112yes—
398C2×Q8C2×#4112yes—
399C2×Q82−1+412yes—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C2511yescharacteristic
420Q16—48no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—81no—
467C4⋊C4—82no—
468C4⋊C4—84no—
469C4⋊C4—82no—
470C4⋊C4—82no—
471C4⋊C4C2×#4114yes—
472C4⋊C4C2×#4112yes—
473C4⋊C4C2×#4112yes—
474C42—28no—
475C42—24no—
476C42—24no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—24no—
481C42—22no—
482C42—22no—
483C42—21no—
484C42—22no—
485C42—22no—
486C42—21no—
487C42—42no—
488C42—42no—
489C42—41no—
490C42—42no—
491C42—44no—
492C42—44no—
493C42—44no—
494C42—44no—
495C42—42no—
496C42—42no—
497C42—81no—
498C42—82no—
499C42—84no—
500C42#13714yes—
501C42C2×#4112yes—
502C42C2×#4112yes—
503C42C2×#4112yes—
504C422−1+412yes—
505C23C23×D411yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—22no—
509Q8—44no—
510Q8—48no—
511Q8—42no—
512Q8—42no—
513Q8—42no—
514Q8—42no—
515Q8—88no—
516Q8#55614yes—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4#52512yes—
566C2×C4#52512yes—
567C2×C4#54812yes—
568C2×C4#54812yes—
569C2×C4C2×#13712yes—
570C2×C4C2×#13712yes—
571C2×C4C2×#13712yes—
572C2×C4#57512yes—
573C2×C4#57812yes—
574C2×C4C22×#4111yescharacteristic
575C2×C4C22×#4111yescharacteristic
576C2×C4C22×#4111yescharacteristic
577C2×C4C2×2−1+411yescharacteristic
578C2×C4C2×2−1+411yescharacteristic
579C2×C4#58511yescharacteristic
580C8—42no—
581C8—44no—
582C8—42no—
583C22C2×#52511yescharacteristic
584C22C2×#54811yescharacteristic
585C22#348911yescharacteristic
586C22C22×#13711yescharacteristic
587C22C2×#57511yescharacteristic
588C22C2×#57811yescharacteristic
589C22#358511yescharacteristic
590C4—22no—
591C4—22no—
592C4—22no—
593C4—24no—
594C4—42no—
595C4—44no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—41no—
602C4—84no—
603C4—82no—
604C4—82no—
605C4—81no—
606C4—82no—
607C4#311614yes—
608C4#337414yes—
609C4C2×#55611yescharacteristic
610C4#348011yescharacteristic
611C4#350711yescharacteristic
612C4#352311yescharacteristic
613C2#3657911yescharacteristic
614C2C2×#311611yescharacteristic
615C2#6060011yescharacteristic
616C2C2×#337411yescharacteristic
617C2#6132411yescharacteristic
618C2#6211211yescharacteristic
619C2#6214611yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#365982
C4×#5572
#331982
#330272
#293572
#292752
#292182
#282832
#275802
#271522
#249952
#243322
#230682
#228252
#227632
#216162
C4×#4622