Finite Group Databasebeta

← #6852767

Subgroups — #6852767

Order 512.

1305 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2#34951C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic
4#32144C214yesmaximal
5#30961C214yesmaximal
6#29988C212yesmaximal
7#29286C211yesmaximal, characteristic
8#29274C212yesmaximal
9#29206C211yesmaximal, characteristic
10#28811C212yesmaximal
11#28691C212yesmaximal
12#27816C212yesmaximal
13#26281C212yesmaximal
14#25061C212yesmaximal
15#22821C211yesmaximal, characteristic
16#22112C212yesmaximal
17C4×#475C211yesmaximal, characteristic
18C2×#499C2214yes—
19C2×#477C2212yes—
20C2×#475C2211yescharacteristic
21#2868C2211yescharacteristic
22#2859C2212yes—
23#2585—22no—
24#2582—21no—
25#2582C2211yescharacteristic
26#2568C2211yescharacteristic
27#2567—21no—
28#2552C2211yescharacteristic
29#2552C2212yes—
30#2550C2214yes—
31#2546C2211yescharacteristic
32#2511—21no—
33#2509C2212yes—
34#2507—22no—
35#2453C2212yes—
36#2451—22no—
37#2446—21no—
38#2402—22no—
39#2369—21no—
40#2355C2212yes—
41Q8×C4⋊C4—21no—
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#125C2212yes—
47#2295—21no—
48#2295C2212yes—
49C42×Q8C2211yescharacteristic
50C42×Q8C2211yescharacteristic
51#2101C2214yes—
52#2101C2214yes—
53#2097C2214yes—
54#2086C2214yes—
55#2085C2214yes—
56#2085C2214yes—
57#2059C2214yes—
58#2059C2214yes—
59#2058C2214yes—
60#2026C2214yes—
61#1991C2214yes—
62#1990C2212yes—
63#1988C2211yescharacteristic
64#1988C2212yes—
65#1988C2212yes—
66#1987C2212yes—
67#1984C2212yes—
68#1948C2211yescharacteristic
69#1948C2212yes—
70#1948C2212yes—
71#1947C2212yes—
72#1947C2212yes—
73#1947C2212yes—
74#1947C2212yes—
75#1941C2214yes—
76#1933C2214yes—
77#1927C2214yes—
78#1921C2214yes—
79#1920C2214yes—
80#1919C2214yes—
81#1919C2212yes—
82#1916C2214yes—
83#1865C2212yes—
84#1844C2212yes—
85#1842C2211yescharacteristic
86#1831C2212yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90C4×#106C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×#125C2312yes—
102C2×#125C2314yes—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8—22no—
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112#499—48no—
113#477—44no—
114#475Q818yescomplemented
115C2×#107C2314yes—
116C2×#107C2314yes—
117C2×#106C2312yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×#106C2312yes—
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2311yescharacteristic
126C2×#103C2314yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×C4×C8C2312yes—
130C2×C4×C8C2314yes—
131#400—22no—
132#400—22no—
133#400—22no—
134#400—22no—
135#398—24no—
136#398—22no—
137#398—22no—
138#398—22no—
139#398—22no—
140#398—24no—
141#395—24no—
142#391—21no—
143#391—22no—
144#391—21no—
145#391—21no—
146#391—21no—
147#391—22no—
148#391—21no—
149#391—21no—
150#391—24no—
151#391C2312yes—
152#391C2312yes—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—22no—
165#389—24no—
166#389C2312yes—
167#389C2312yes—
168#389C2312yes—
169#389C2314yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#387—24no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#384—21no—
181#384—22no—
182#384—21no—
183#384—21no—
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2311yescharacteristic
187#384C2312yes—
188#382C2312yes—
189#382C2314yes—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2312yes—
212#376—22no—
213C43—21no—
214C43C2312yes—
215C43C2311yescharacteristic
216#340C2314yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#336C2312yes—
228#336C2314yes—
229#336C2314yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8C2×D411yescharacteristic
235C22×Q8C2411yescharacteristic
236C22×C8C2412yes—
237C22×C8C2414yes—
238C22×C8C2412yes—
239C4⋊Q8—22no—
240C4⋊Q8—22no—
241C4⋊Q8—22no—
242C4⋊Q8—42no—
243C4⋊Q8—42no—
244C4⋊Q8—44no—
245C4⋊Q8—44no—
246C4⋊Q8—82no—
247C4⋊Q8C2×Q814yes—
248#125—24no—
249#125—42no—
250#125—44no—
251#125—44no—
252C4×Q8—24no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—42no—
260C4×Q8—41no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—84no—
264C4×Q8C2×Q814yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D412yes—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C42—21no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—24no—
312C2×C42—21no—
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D412yes—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D412yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2412yes—
323C2×C42C2412yes—
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326#107—28no—
327#107—44no—
328#106—24no—
329#106—44no—
330#106—44no—
331#106C2×Q818yes—
332C4⋊C8—24no—
333C4⋊C8—22no—
334C4⋊C8—24no—
335C4⋊C8—42no—
336C4⋊C8C2×Q814yes—
337#103—42no—
338#103—44no—
339#103C2×Q818yes—
340C4×C8—24no—
341C4×C8—44no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—22no—
350#95—24no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366C2×Q8—22no—
367C2×Q8—24no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—44no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—82no—
382C2×Q8—82no—
383C2×Q8C22×Q812yes—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2511yescharacteristic
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8C22×Q814yes—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—21no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—42no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—81no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4C22×Q814yes—
458C4⋊C4C22×Q812yes—
459C4⋊C4H(5,𝔽2)12yes—
460C42—24no—
461C42—24no—
462C42—22no—
463C42—24no—
464C42—22no—
465C42—22no—
466C42—21no—
467C42—22no—
468C42—22no—
469C42—21no—
470C42—22no—
471C42—41no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—42no—
476C42—44no—
477C42—44no—
478C42—42no—
479C42—42no—
480C42—81no—
481C42—84no—
482C42#13514yes—
483C42C22×Q812yes—
484C422−1+412yes—
485C422−1+412yes—
486C422−1+412yes—
487C23C23×D411yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—48no—
491Q8—44no—
492Q8—41no—
493Q8—41no—
494Q8—84no—
495Q8—164no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4#53612yes—
545C2×C4#53612yes—
546C2×C4D4×Q812yes—
547C2×C4D4×Q812yes—
548C2×C4C2×#13512yes—
549C2×C4C2×#13512yes—
550C2×C4C2×#13512yes—
551C2×C4#57512yes—
552C2×C4#57712yes—
553C2×C4C23×Q811yescharacteristic
554C2×C4C2×H(5,𝔽2)11yescharacteristic
555C2×C4C2×2−1+411yescharacteristic
556C2×C4C2×2−1+411yescharacteristic
557C2×C4C2×2−1+411yescharacteristic
558C2×C4#58511yescharacteristic
559C8—24no—
560C8—44no—
561C8—42no—
562C22C2×#53611yescharacteristic
563C22C2×D4×Q811yescharacteristic
564C22#348811yescharacteristic
565C22C22×#13511yescharacteristic
566C22C2×#57511yescharacteristic
567C22C2×#57711yescharacteristic
568C22#358611yescharacteristic
569C4—22no—
570C4—22no—
571C4—22no—
572C4—42no—
573C4—42no—
574C4—41no—
575C4—42no—
576C4—44no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—82no—
582C4—82no—
583C4—84no—
584C4—82no—
585C4—81no—
586C4#323314yes—
587C4#338114yes—
588C4#350211yescharacteristic
589C4#350311yescharacteristic
590C4#355011yescharacteristic
591C4#355511yescharacteristic
592C2#3959711yescharacteristic
593C2C2×#323311yescharacteristic
594C2#6088911yescharacteristic
595C2C2×#338111yescharacteristic
596C2#6136911yescharacteristic
597C2#6195411yescharacteristic
598C2#6198611yescharacteristic
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
114. #475Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#349512
#331982
#321442
#309612
#299882
#292862
#292742
#292062
#288112
#286912
#278162
#262812
#250612
#228212
#221122
C4×#4752