Finite Group Databasebeta

← #6627454

Subgroups — #6627454

Order 512.

1453 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2C4×Q82C211yesmaximal, characteristic
3#33202C212yesmaximal
4#33196C211yesmaximal, characteristic
5#29290C211yesmaximal, characteristic
6#29286C212yesmaximal
7#29224C212yesmaximal
8#29219C211yesmaximal, characteristic
9#26520C214yesmaximal
10#26507C212yesmaximal
11#26120C214yesmaximal
12#26109C212yesmaximal
13#22743C212yesmaximal
14#22722C211yesmaximal, characteristic
15#22525C212yesmaximal
16#21615C212yesmaximal
17C4×#477C211yesmaximal, characteristic
18C2×Q82C2211yescharacteristic
19C2×#477C2211yescharacteristic
20C2×#475C2212yes—
21C2×#446C2212yes—
22#2868C2211yescharacteristic
23#2862C2212yes—
24#2582—21no—
25#2571C2212yes—
26#2570C2214yes—
27#2568—21no—
28#2511—22no—
29#2446—22no—
30#2369—21no—
31#2369—22no—
32#2369C2211yescharacteristic
33#2369C2212yes—
34#2365—21no—
35#2355—21no—
36#2355C2212yes—
37Q8×C4⋊C4—21no—
38Q8×C4⋊C4C2211yescharacteristic
39Q8×C4⋊C4C2212yes—
40Q8×C4⋊C4C2211yescharacteristic
41C4×C4⋊Q8—21no—
42C4×C4⋊Q8—22no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C42×Q8—21no—
48C42×Q8C2211yescharacteristic
49C42×Q8C2212yes—
50C42×Q8C2211yescharacteristic
51#2101C2214yes—
52#2101C2214yes—
53#2099C2214yes—
54#2087C2214yes—
55#2086C2214yes—
56#2086C2214yes—
57#1993C2214yes—
58#1990C2212yes—
59#1988C2212yes—
60#1988C2212yes—
61#1988C2212yes—
62#1988C2214yes—
63#1987C2211yescharacteristic
64#1987C2212yes—
65#1987C2212yes—
66#1949C2214yes—
67#1949C2214yes—
68#1948C2212yes—
69#1948C2212yes—
70#1947C2214yes—
71#1947C2211yescharacteristic
72#1947C2212yes—
73#1947C2212yes—
74#1940C2214yes—
75#1939C2212yes—
76#1921C2212yes—
77#1920C2214yes—
78#1863C2212yes—
79#1863C2214yes—
80#1842C2212yes—
81#1842C2211yescharacteristic
82#1836C2214yes—
83#1834C2212yes—
84#1805C2214yes—
85#1805C2214yes—
86#1803C2212yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1776C2214yes—
90C4×#107C2212yes—
91#1767C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93#1763C2214yes—
94C4×#103C2212yes—
95Q82—24no—
96Q82Q818yescomplemented
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8—21no—
110C2×C4×Q8—22no—
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119#477—24no—
120#475—28no—
121#446—28no—
122C2×#107C2312yes—
123C2×#107C2312yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2311yescharacteristic
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2312yes—
132C2×C4⋊C8C2314yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#97C2314yes—
137C2×C4×C8C2312yes—
138#400—24no—
139#400—24no—
140#398—24no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—21no—
145#391—21no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#389—21no—
154#389—22no—
155#389—21no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—21no—
160#389—22no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389C2312yes—
166#389C2314yes—
167#389C2312yes—
168#389C2312yes—
169#389C2312yes—
170#387—24no—
171#387—24no—
172#387—24no—
173#384—21no—
174#384—22no—
175#384—21no—
176#384—21no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2312yes—
180#384C2312yes—
181#384C2312yes—
182#382—22no—
183#382C2314yes—
184#382C2314yes—
185C4×C4⋊C4—21no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D412yes—
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C43—22no—
215C43—21no—
216C43C2312yes—
217C43C2311yescharacteristic
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8—22no—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—21no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2411yescharacteristic
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2414yes—
242C4⋊Q8—216no—
243C4⋊Q8—28no—
244C4⋊Q8—22no—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8—22no—
248C4⋊Q8—24no—
249C4⋊Q8—22no—
250C4⋊Q8—44no—
251C4⋊Q8—82no—
252C4⋊Q8C2×Q814yes—
253C4⋊Q8C2×Q818yes—
254C4×Q8—28no—
255C4×Q8—24no—
256C4×Q8—22no—
257C4×Q8—24no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—24no—
261C4×Q8—42no—
262C4×Q8—41no—
263C4×Q8—42no—
264C4×Q8—44no—
265C4×Q8—44no—
266C4×Q8—84no—
267C4×Q8—82no—
268C4×Q8—84no—
269C4×Q8C2×Q814yes—
270C4×Q8C2×Q818yes—
271C4×Q8C2×Q814yes—
272C4×Q8#4114yes—
273C4×Q8#4114yes—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C42—22no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42—24no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338#107—24no—
339#107—42no—
340#107—44no—
341#106—28no—
342C4⋊C8—24no—
343C4⋊C8—28no—
344C4⋊C8—24no—
345C4⋊C8—22no—
346C4⋊C8—22no—
347C4⋊C8—42no—
348C4⋊C8—44no—
349#103—28no—
350#103—24no—
351#103—24no—
352#97—28no—
353C4×C8—24no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—21no—
359#95—22no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—24no—
364#95—24no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C2×Q8—24no—
375C2×Q8—28no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—82no—
388C2×Q8—81no—
389C2×Q8—82no—
390C2×Q8—81no—
391C2×Q8C22⋊Q814yes—
392C2×Q8C22×Q812yes—
393C2×Q8C2×#4112yes—
394C2×Q82−1+412yes—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C2511yescharacteristic
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C4⋊C4—28no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—28no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—22no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—82no—
458C4⋊C4—81no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—84no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4C22⋊Q818yes—
466C4⋊C4C22⋊Q814yes—
467C4⋊C4C22×Q812yes—
468C4⋊C4C22×Q814yes—
469C4⋊C4C2×#4112yes—
470C42—28no—
471C42—24no—
472C42—24no—
473C42—24no—
474C42—22no—
475C42—24no—
476C42—22no—
477C42—21no—
478C42—22no—
479C42—21no—
480C42—22no—
481C42—41no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—44no—
486C42—44no—
487C42—42no—
488C42—44no—
489C42—42no—
490C42—44no—
491C42—81no—
492C42—82no—
493C42—84no—
494C42#13714yes—
495C42C22×Q814yes—
496C42C22×Q812yes—
497C42C2×#4112yes—
498C42C2×#4114yes—
499C42C2×#4112yes—
500C422−1+412yes—
501C23C23×D411yescharacteristic
502Q8—28no—
503Q8—22no—
504Q8—22no—
505Q8—22no—
506Q8—88no—
507Q8—84no—
508Q8—84no—
509Q8—88no—
510Q8#55714yes—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4C2×C22⋊Q814yes—
558C2×C4C2×C22⋊Q812yes—
559C2×C4C2×C22⋊Q814yes—
560C2×C4C2×C22⋊Q812yes—
561C2×C4#54812yes—
562C2×C4#54812yes—
563C2×C4C2×#13712yes—
564C2×C4C2×#13712yes—
565C2×C4C2×#13712yes—
566C2×C4#57512yes—
567C2×C4#57612yes—
568C2×C4C23×Q811yescharacteristic
569C2×C4C22×#4111yescharacteristic
570C2×C4C22×#4111yescharacteristic
571C2×C4C2×2−1+411yescharacteristic
572C2×C4C2×2−1+411yescharacteristic
573C2×C4#58511yescharacteristic
574C8—42no—
575C8—44no—
576C8—42no—
577C22C22×C22⋊Q811yescharacteristic
578C22C2×#54811yescharacteristic
579C22#348911yescharacteristic
580C22C22×#13711yescharacteristic
581C22C2×#57511yescharacteristic
582C22C2×#57611yescharacteristic
583C22#358611yescharacteristic
584C4—24no—
585C4—24no—
586C4—22no—
587C4—22no—
588C4—22no—
589C4—24no—
590C4—44no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—84no—
597C4—82no—
598C4—82no—
599C4—81no—
600C4—82no—
601C4#307814yes—
602C4#337514yes—
603C4C2×#55711yescharacteristic
604C4#348011yescharacteristic
605C4#350211yescharacteristic
606C4#352311yescharacteristic
607C2#3657711yescharacteristic
608C2C2×#307811yescharacteristic
609C2#6049411yescharacteristic
610C2C2×#337511yescharacteristic
611C2#6132311yescharacteristic
612C2#6211311yescharacteristic
613C2#6213611yescharacteristic
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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
96. Q82Q8semidirect

Maximal subgroups

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

subgroupindex
#365982
C4×Q822
#332022
#331962
#292902
#292862
#292242
#292192
#265202
#265072
#261202
#261092
#227432
#227222
#225252
#216152
C4×#4772