Finite Group Databasebeta

← #6585872

Subgroups — #6585872

Order 512.

1538 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2Q8×C4⋊Q8C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic
4#33198C211yesmaximal, characteristic
5#33198C211yesmaximal, characteristic
6#28789C212yesmaximal
7#27815C212yesmaximal
8#26446C212yesmaximal
9#26282C212yesmaximal
10#26112C212yesmaximal
11#26026C212yesmaximal
12#25471C212yesmaximal
13#24817C212yesmaximal
14#24333C212yesmaximal
15#22965C212yesmaximal
16#22771C212yesmaximal
17Q8×#103C212yesmaximal
18C8×C4⋊Q8C211yesmaximal, characteristic
19C42×Q16C211yesmaximal, characteristic
20C2×Q82C2212yes—
21C2×#477C2212yes—
22C2×#477C2212yes—
23C2×C8×Q8C2212yes—
24C2×C4×Q16C2211yescharacteristic
25C2×C4×Q16C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27#2868—21no—
28#2868C2211yescharacteristic
29#2868C2211yescharacteristic
30#2859—21no—
31#2582C2212yes—
32#2567—22no—
33#2567C2212yes—
34#2567C2212yes—
35#2552C2212yes—
36#2551—22no—
37#2551—22no—
38#2551C2212yes—
39#2551C2212yes—
40#2550—22no—
41#2507—22no—
42#2451—22no—
43#2369C2212yes—
44#2355C2212yes—
45Q8×C4⋊C4C2212yes—
46Q8×C4⋊C4C2212yes—
47Q8×C4⋊C4C2212yes—
48C4×C4⋊Q8—21no—
49C4×C4⋊Q8—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C42×Q8C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53#2100C2214yes—
54#2087C2214yes—
55#2083C2214yes—
56#2068C2214yes—
57#2057C2214yes—
58#2024C2214yes—
59#1990C2212yes—
60#1990C2212yes—
61#1990C2212yes—
62#1990C2212yes—
63#1990C2214yes—
64#1987C2212yes—
65#1987C2214yes—
66#1987C2212yes—
67#1984C2212yes—
68#1947C2212yes—
69#1947C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1942C2214yes—
73#1939C2212yes—
74#1939C2212yes—
75#1933C2214yes—
76#1921C2214yes—
77#1921C2214yes—
78#1878C2214yes—
79#1863C2214yes—
80#1842C2211yescharacteristic
81#1834C2212yes—
82#1834C2212yes—
83#1803C2212yes—
84#1803C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88C4×#107C2211yescharacteristic
89C4×#107C2211yescharacteristic
90C4×#107C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C8×C4⋊C4C2212yes—
93C4×#103C2211yescharacteristic
94C4×#103C2211yescharacteristic
95C4×#103C2211yescharacteristic
96C4×#103C2211yescharacteristic
97C4×#103C2211yescharacteristic
98C4×#103C2211yescharacteristic
99C4×#103C2214yes—
100C42×C8C2211yescharacteristic
101C22×Q16C2311yescharacteristic
102Q82—28no—
103Q82—44no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8—24no—
106C2×C4⋊Q8—22no—
107C2×C4⋊Q8—22no—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8D411yescharacteristic
113C2×C4⋊Q8C2312yes—
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2314yes—
116C2×#125—22no—
117C2×C4×Q8—22no—
118C2×C4×Q8—22no—
119C2×C4×Q8C2311yescharacteristic
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2314yes—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×Q8C2311yescharacteristic
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2312yes—
127#477—28no—
128#477—44no—
129C8×Q8—44no—
130C4×Q16—24no—
131C4×Q16—24no—
132C4×Q16Q818yescomplemented
133C2×#107C2314yes—
134C2×#107C2311yescharacteristic
135C2×#107C2311yescharacteristic
136C2×#107C2314yes—
137C2×#107C2311yescharacteristic
138C2×C4⋊C8C2312yes—
139C2×C4⋊C8C2312yes—
140C2×C4⋊C8C2312yes—
141C2×C4⋊C8C2314yes—
142C2×#103C2314yes—
143C2×#103C2311yescharacteristic
144C2×#103C2311yescharacteristic
145C2×#103C2311yescharacteristic
146C2×#103C2311yescharacteristic
147C2×#103C2314yes—
148C2×#103C2311yescharacteristic
149C2×#103C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2314yes—
156C2×C4×C8C2311yescharacteristic
157#398—24no—
158#398—24no—
159#395—24no—
160#395—24no—
161#395—24no—
162#395—24no—
163#391—22no—
164#391—22no—
165#391—24no—
166#391D412yes—
167#391C2314yes—
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389—22no—
176#389—24no—
177#389—24no—
178#389—24no—
179#389D412yes—
180#389D412yes—
181#389C2314yes—
182#389C2314yes—
183#389C2314yes—
184#387—24no—
185#384—22no—
186#384—22no—
187#384—22no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384—21no—
192#384D411yescharacteristic
193#384C2314yes—
194#384C2312yes—
195#384C2312yes—
196#384C2312yes—
197#384C2312yes—
198#382—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2312yes—
217C4×C4⋊C4C2312yes—
218C43—21no—
219C43—21no—
220C43C2311yescharacteristic
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2311yescharacteristic
228#340C2314yes—
229#340C2314yes—
230#340C2311yescharacteristic
231#340C2311yescharacteristic
232#340C2311yescharacteristic
233#340C2311yescharacteristic
234#340C2311yescharacteristic
235#336C2314yes—
236#336C2314yes—
237C22×Q8—24no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8C2×D412yes—
241C22×Q8C2411yescharacteristic
242C22×Q8C2411yescharacteristic
243C2×Q16—22no—
244C2×Q16—22no—
245C2×Q16C2×Q814yes—
246C22×C8C2411yescharacteristic
247C22×C8C2414yes—
248C22×C8C2411yescharacteristic
249C22×C8C2411yescharacteristic
250C22×C8C2411yescharacteristic
251C4⋊Q8—28no—
252C4⋊Q8—24no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—44no—
257C4⋊Q8—48no—
258C4⋊Q8—416no—
259C4⋊Q8—82no—
260C4⋊Q8—82no—
261C4⋊Q8—82no—
262C4⋊Q8—82no—
263C4⋊Q8—84no—
264C4⋊Q8—84no—
265C4⋊Q8Q1618yescomplemented
266#125—84no—
267C4×Q8—24no—
268C4×Q8—22no—
269C4×Q8—24no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—28no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×Q8—44no—
279C4×Q8—48no—
280C4×Q8—84no—
281C4×Q8C2×Q814yes—
282C4×Q8C2×Q814yes—
283C4×Q8#4114yes—
284C4×Q8#4114yes—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D414yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2414yes—
315C2×C4⋊C4C2411yescharacteristic
316C2×C4⋊C4C2411yescharacteristic
317C2×C4⋊C4C2414yes—
318C2×C4⋊C4C2411yescharacteristic
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2411yescharacteristic
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—24no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—24no—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2414yes—
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343C2×C42C2411yescharacteristic
344#107—22no—
345#107—22no—
346#107—28no—
347#107—44no—
348#107C2×Q814yes—
349C4⋊C8—24no—
350C4⋊C8—24no—
351C4⋊C8—24no—
352C4⋊C8—44no—
353#103—22no—
354#103—22no—
355#103—28no—
356#103—22no—
357#103—22no—
358#103—44no—
359#103C2×Q814yes—
360#103C2×Q814yes—
361C4×C8—22no—
362C4×C8—22no—
363C4×C8—22no—
364C4×C8—22no—
365C4×C8—22no—
366C4×C8—44no—
367C4×C8C2×Q814yes—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95C2×D414yes—
381#95C2×D414yes—
382C2×Q8—24no—
383C2×Q8—24no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—48no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—84no—
400C2×Q8—84no—
401C2×Q8C2×Q1618yes—
402C2×Q8C22×Q812yes—
403C2×Q8C22×Q812yes—
404C2×Q8C2×#4112yes—
405C2×Q8C2×#4112yes—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4C22×D414yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D414yes—
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C2511yescharacteristic
425Q16—28no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—24no—
433C2×C8—21no—
434C2×C8—24no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—24no—
439C2×C8C22×Q812yes—
440C2×C8C22×Q812yes—
441C4⋊C4—24no—
442C4⋊C4—28no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—24no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—24no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—42no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4—82no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4—82no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4C2×Q1618yes—
481C4⋊C4C2×Q1618yes—
482C4⋊C4C22×Q812yes—
483C4⋊C4C22×Q812yes—
484C4⋊C4C2×#4112yes—
485C4⋊C4C2×#4112yes—
486C42—24no—
487C42—24no—
488C42—24no—
489C42—24no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—24no—
494C42—24no—
495C42—48no—
496C42—42no—
497C42—42no—
498C42—42no—
499C42—42no—
500C42—42no—
501C42—42no—
502C42—42no—
503C42—42no—
504C42—84no—
505C42C2×Q1614yes—
506C42C22×Q812yes—
507C42C2×#4112yes—
508C42C2×#4112yes—
509C42H(5,𝔽2)12yes—
510C422−1+412yes—
511C422−1+412yes—
512C23C23×D411yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—22no—
516Q8—216no—
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521Q8—164no—
522Q8—164no—
523Q8—168no—
524Q8#53314yes—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4#54612yes—
567C2×C4#54612yes—
568C2×C4#54812yes—
569C2×C4#54812yes—
570C2×C4D4×Q812yes—
571C2×C4D4×Q812yes—
572C2×C4C22×Q1618yes—
573C2×C4C22×Q1612yes—
574C2×C4C22×Q1612yes—
575C2×C4C22×Q1612yes—
576C2×C4#57612yes—
577C2×C4#57712yes—
578C2×C4#57712yes—
579C2×C4C23×Q811yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C22×#4111yescharacteristic
582C2×C4C2×H(5,𝔽2)11yescharacteristic
583C2×C4C2×2−1+411yescharacteristic
584C2×C4C2×2−1+411yescharacteristic
585C8—22no—
586C8—22no—
587C8—41no—
588C8—41no—
589C8—44no—
590C22C2×#54611yescharacteristic
591C22C2×#54811yescharacteristic
592C22C2×D4×Q811yescharacteristic
593C22C23×Q1611yescharacteristic
594C22C2×#57611yescharacteristic
595C22C2×#57711yescharacteristic
596C22C2×#57711yescharacteristic
597C4—22no—
598C4—24no—
599C4—22no—
600C4—28no—
601C4—22no—
602C4—44no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—81no—
609C4—84no—
610C4—81no—
611C4—84no—
612C4#329914yes—
613C4#336314yes—
614C4Q8×Q1614yes—
615C4C2×#53311yescharacteristic
616C4#350311yescharacteristic
617C4#350911yescharacteristic
618C4#351111yescharacteristic
619C2#3612911yescharacteristic
620C2C2×#329911yescharacteristic
621C2#6121511yescharacteristic
622C2C2×#336311yescharacteristic
623C2#6132911yescharacteristic
624C2C2×Q8×Q1611yescharacteristic
625C2#6137411yescharacteristic
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

Complements

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

NH ≅ G/Ntype
132. C4×Q16Q8semidirect
265. C4⋊Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#365632
Q8×C4⋊Q82
#332892
#331982
#287892
#278152
#264462
#262822
#261122
#260262
#254712
#248172
#243332
#229652
#227712
Q8×#1032
C8×C4⋊Q82
C42×Q162