Finite Group Databasebeta

← #6591286

Subgroups — #6591286

Order 512.

1380 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic
4#33198C211yesmaximal, characteristic
5#33196C211yesmaximal, characteristic
6#28196C212yesmaximal
7#27738C212yesmaximal
8#27737C212yesmaximal
9#26507C212yesmaximal
10#26506C212yesmaximal
11#24995C212yesmaximal
12#24994C212yesmaximal
13#24683C212yesmaximal
14#24682C212yesmaximal
15#23069C212yesmaximal
16#23068C212yesmaximal
17#22743C212yesmaximal
18#21712C211yesmaximal, characteristic
19C42×Q16C211yesmaximal, characteristic
20C2×#477C2212yes—
21C2×#475C2212yes—
22C2×#446C2212yes—
23C2×C4×Q16C2211yescharacteristic
24C2×C4×Q16C2211yescharacteristic
25C2×C4×Q16C2211yescharacteristic
26#2868C2211yescharacteristic
27#2868C2211yescharacteristic
28#2862—21no—
29#2859—21no—
30#2582C2212yes—
31#2582C2212yes—
32#2571C2212yes—
33#2570—22no—
34#2568C2212yes—
35#2567C2212yes—
36#2552C2212yes—
37#2550—22no—
38#2511—22no—
39#2507—22no—
40#2451—22no—
41#2446—22no—
42#2369C2212yes—
43#2365C2212yes—
44#2355C2212yes—
45#2355C2212yes—
46#2355C2212yes—
47Q8×C4⋊C4C2212yes—
48C4×C4⋊Q8—21no—
49C4×C4⋊Q8—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C42×Q8C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53#2101C2214yes—
54#2099C2214yes—
55#2086C2214yes—
56#2074C2214yes—
57#2072C2214yes—
58#2056C2214yes—
59#1990C2212yes—
60#1990C2212yes—
61#1990C2212yes—
62#1990C2212yes—
63#1988C2212yes—
64#1987C2212yes—
65#1984C2212yes—
66#1949C2214yes—
67#1948C2212yes—
68#1948C2212yes—
69#1947C2212yes—
70#1947C2214yes—
71#1947C2212yes—
72#1939C2212yes—
73#1939C2212yes—
74#1935C2214yes—
75#1935C2214yes—
76#1928C2214yes—
77#1927C2214yes—
78#1920C2214yes—
79#1918C2214yes—
80#1842C2211yescharacteristic
81#1836C2214yes—
82#1834C2212yes—
83#1834C2212yes—
84#1803C2212yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1776C2214yes—
89C4×#107C2211yescharacteristic
90C4×#107C2211yescharacteristic
91C4×#107C2211yescharacteristic
92#1770C2212yes—
93#1770C2212yes—
94C4×#103C2211yescharacteristic
95C4×#103C2211yescharacteristic
96C4×#103C2211yescharacteristic
97C4×#103C2211yescharacteristic
98C4×#103C2211yescharacteristic
99C4×#103C2211yescharacteristic
100C42×C8C2211yescharacteristic
101C22×Q16C2311yescharacteristic
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8D411yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×#125—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119#477—28no—
120#475—28no—
121#446—28no—
122C4×Q16—24no—
123C4×Q16—24no—
124C4×Q16—24no—
125C2×#107C2314yes—
126C2×#107C2311yescharacteristic
127C2×#107C2311yescharacteristic
128C2×#107C2311yescharacteristic
129C2×#106C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×C4⋊C8C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2311yescharacteristic
137C2×#103C2311yescharacteristic
138C2×#103C2311yescharacteristic
139C2×#103C2311yescharacteristic
140C2×#103C2311yescharacteristic
141C2×#103C2311yescharacteristic
142C2×#97C2314yes—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149#400—24no—
150#400—24no—
151#398—24no—
152#398—24no—
153#398—24no—
154#395—24no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—24no—
162#391D412yes—
163#391C2314yes—
164#391C2314yes—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—24no—
174#389D412yes—
175#389D412yes—
176#389C2314yes—
177#389C2314yes—
178#387—24no—
179#387—24no—
180#387—24no—
181#387—24no—
182#384—21no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384D411yescharacteristic
187#384C2314yes—
188#384C2312yes—
189#384C2312yes—
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#382—22no—
194#382—22no—
195#382C2314yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2314yes—
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C43—21no—
217C43—21no—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2311yescharacteristic
228#340C2311yescharacteristic
229#340C2311yescharacteristic
230#340C2311yescharacteristic
231#340C2311yescharacteristic
232#340C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8C2×D412yes—
238C22×Q8C2411yescharacteristic
239C22×Q8C2411yescharacteristic
240C2×Q16—22no—
241C2×Q16—22no—
242C2×Q16—22no—
243C22×C8C2411yescharacteristic
244C22×C8C2411yescharacteristic
245C22×C8C2414yes—
246C22×C8C2411yescharacteristic
247C22×C8C2411yescharacteristic
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—44no—
254C4⋊Q8—82no—
255C4⋊Q8—82no—
256C4⋊Q8—82no—
257#125—84no—
258C4×Q8—24no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×Q8—24no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—84no—
271C4×Q8—84no—
272C4×Q8#4114yes—
273C4×Q8#4114yes—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D414yes—
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2414yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330#107—22no—
331#107—22no—
332#107—22no—
333#107—28no—
334#106—28no—
335C4⋊C8—28no—
336C4⋊C8—24no—
337C4⋊C8—24no—
338C4⋊C8—24no—
339#103—22no—
340#103—22no—
341#103—28no—
342#103—28no—
343#103—22no—
344#103—22no—
345#103—22no—
346#103—22no—
347#97—28no—
348C4×C8—22no—
349C4×C8—22no—
350C4×C8—22no—
351C4×C8—22no—
352C4×C8—22no—
353C4×C8#4114yes—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95C2×D414yes—
371#95C2×D414yes—
372C2×Q8—24no—
373C2×Q8—24no—
374C2×Q8—24no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—82no—
390C2×Q8—82no—
391C2×Q8C2×#4112yes—
392C2×Q8C2×#4112yes—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4C22×D414yes—
404C22×C4C22×D414yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C2511yescharacteristic
412Q16—28no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8C2×#4112yes—
427C2×C8C2×#4112yes—
428C4⋊C4—24no—
429C4⋊C4—28no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4—84no—
463C4⋊C4—84no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4C2×#4112yes—
469C4⋊C4C2×#4112yes—
470C42—24no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—24no—
476C42—24no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—84no—
488C42—84no—
489C42#13714yes—
490C42C2×#4112yes—
491C42C2×#4112yes—
492C42C2×#4112yes—
493C422−1+412yes—
494C422−1+412yes—
495C422−1+412yes—
496C23C23×D411yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—22no—
500Q8—22no—
501Q8—48no—
502Q8—41no—
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—88no—
507Q8—164no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4#54712yes—
551C2×C4#54712yes—
552C2×C4#54812yes—
553C2×C4#54812yes—
554C2×C4#54812yes—
555C2×C4#54812yes—
556C2×C4C2×#13712yes—
557C2×C4C2×#13712yes—
558C2×C4C2×#13712yes—
559C2×C4#57612yes—
560C2×C4#57712yes—
561C2×C4#57812yes—
562C2×C4C22×#4111yescharacteristic
563C2×C4C22×#4111yescharacteristic
564C2×C4C22×#4111yescharacteristic
565C2×C4C2×2−1+411yescharacteristic
566C2×C4C2×2−1+411yescharacteristic
567C2×C4C2×2−1+411yescharacteristic
568C8—22no—
569C8—22no—
570C8—44no—
571C8—41no—
572C8—41no—
573C22C2×#54711yescharacteristic
574C22C2×#54811yescharacteristic
575C22C2×#54811yescharacteristic
576C22C22×#13711yescharacteristic
577C22C2×#57611yescharacteristic
578C22C2×#57711yescharacteristic
579C22C2×#57811yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—42no—
584C4—42no—
585C4—44no—
586C4—44no—
587C4—42no—
588C4—42no—
589C4—81no—
590C4—81no—
591C4—84no—
592C4—81no—
593C4—84no—
594C4—81no—
595C4#335614yes—
596C4#337414yes—
597C4#337514yes—
598C4#344411yescharacteristic
599C4#350911yescharacteristic
600C4#350911yescharacteristic
601C4#350911yescharacteristic
602C2#3615111yescharacteristic
603C2C2×#335611yescharacteristic
604C2#6128711yescharacteristic
605C2C2×#337411yescharacteristic
606C2C2×#337511yescharacteristic
607C2#6132911yescharacteristic
608C2#6133011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊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
#365632
#332892
#331982
#331962
#281962
#277382
#277372
#265072
#265062
#249952
#249942
#246832
#246822
#230692
#230682
#227432
#217122
C42×Q162