Finite Group Databasebeta

← #6852806

Subgroups — #6852806

Order 512.

1319 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36563C211yesmaximal, characteristic
2#34950C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4#32154C214yesmaximal
5#30969C214yesmaximal
6#29988C212yesmaximal
7#29290C211yesmaximal, characteristic
8#29273C212yesmaximal
9#29206C211yesmaximal, characteristic
10#28817C212yesmaximal
11#28693C212yesmaximal
12#27824C212yesmaximal
13#26277C212yesmaximal
14#25061C212yesmaximal
15#22825C211yesmaximal, characteristic
16#22119C212yesmaximal
17C4×#477C211yesmaximal, characteristic
18C2×#501C2214yes—
19C2×#477C2211yescharacteristic
20C2×#475C2212yes—
21#2868C2211yescharacteristic
22#2859C2212yes—
23#2586—22no—
24#2582—21no—
25#2582C2211yescharacteristic
26#2567—21no—
27#2567C2211yescharacteristic
28#2567C2211yescharacteristic
29#2552C2212yes—
30#2551C2211yescharacteristic
31#2550C2214yes—
32#2507—21no—
33#2507—22no—
34#2507C2212yes—
35#2451—22no—
36#2451—21no—
37#2451C2212yes—
38#2399—22no—
39#2369—21no—
40#2355C2212yes—
41Q8×C4⋊C4—21no—
42C4×C4⋊Q8—21no—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47#2295—21no—
48#2295C2212yes—
49C42×Q8C2211yescharacteristic
50C42×Q8C2211yescharacteristic
51#2101C2214yes—
52#2099C2214yes—
53#2097C2214yes—
54#2087C2214yes—
55#2085C2214yes—
56#2085C2214yes—
57#2059C2214yes—
58#2058C2214yes—
59#2058C2214yes—
60#2026C2214yes—
61#1992C2214yes—
62#1990C2212yes—
63#1988C2212yes—
64#1987C2211yescharacteristic
65#1987C2212yes—
66#1987C2212yes—
67#1984C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1947C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1943C2214yes—
76#1935C2214yes—
77#1927C2214yes—
78#1921C2212yes—
79#1919C2214yes—
80#1919C2214yes—
81#1918C2214yes—
82#1917C2214yes—
83#1863C2212yes—
84#1844C2212yes—
85#1842C2211yescharacteristic
86#1831C2212yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90C4×#107C2212yes—
91#1770C2212yes—
92#1767C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#125C2314yes—
103C2×C4×Q8—21no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113#501—48no—
114#477Q818yescomplemented
115#475—44no—
116C2×#107C2312yes—
117C2×#107C2314yes—
118C2×#107C2312yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2311yescharacteristic
127C2×#103C2314yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×#97C2314yes—
131C2×C4×C8C2312yes—
132#398—24no—
133#398—22no—
134#398—22no—
135#398—22no—
136#398—22no—
137#398—24no—
138#395—24no—
139#395—22no—
140#395—22no—
141#395—22no—
142#395—22no—
143#391—22no—
144#391—21no—
145#391—21no—
146#391—22no—
147#391—21no—
148#391—22no—
149#391—21no—
150#391—22no—
151#391—24no—
152#391C2312yes—
153#391C2312yes—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—21no—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389—24no—
169#389C2312yes—
170#389C2312yes—
171#389C2314yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#387—22no—
177#387—22no—
178#387—24no—
179#387—22no—
180#387—22no—
181#384—21no—
182#384—22no—
183#384—21no—
184#384—21no—
185#384C2312yes—
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2311yescharacteristic
189#384C2312yes—
190#382C2314yes—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4D412yes—
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#340C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#336C2312yes—
228#336C2314yes—
229#336C2314yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8C2×D411yescharacteristic
236C22×Q8C2411yescharacteristic
237C22×C8C2412yes—
238C22×C8C2414yes—
239C22×C8C2412yes—
240C4⋊Q8—24no—
241C4⋊Q8—22no—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244C4⋊Q8—42no—
245C4⋊Q8—42no—
246C4⋊Q8—42no—
247C4⋊Q8—44no—
248C4⋊Q8—44no—
249C4⋊Q8—82no—
250C4⋊Q8C2×Q814yes—
251#125—44no—
252#125—44no—
253C4×Q8—24no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—42no—
261C4×Q8—41no—
262C4×Q8—44no—
263C4×Q8—84no—
264C4×Q8—82no—
265C4×Q8—84no—
266C4×Q8C2×Q814yes—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—24no—
315C2×C42—21no—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D412yes—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2411yescharacteristic
325C2×C42C2412yes—
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329#107—24no—
330#107—44no—
331#107C2×Q818yes—
332#106—28no—
333#106—44no—
334#106—44no—
335C4⋊C8—24no—
336C4⋊C8—22no—
337C4⋊C8—24no—
338C4⋊C8—42no—
339C4⋊C8C2×Q814yes—
340#103—42no—
341#103—44no—
342#103C2×Q818yes—
343#97—44no—
344C4×C8—24no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—21no—
350#95—21no—
351#95—21no—
352#95—24no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—24no—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367C2×Q8—22no—
368C2×Q8—24no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—44no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—44no—
380C2×Q8—42no—
381C2×Q8—44no—
382C2×Q8—81no—
383C2×Q8—82no—
384C2×Q8—81no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×Q8C22×Q812yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C2511yescharacteristic
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8C22×Q814yes—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—21no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—82no—
451C4⋊C4—82no—
452C4⋊C4—81no—
453C4⋊C4—84no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4C22×Q814yes—
463C4⋊C4C22×Q812yes—
464C4⋊C4H(5,𝔽2)12yes—
465C42—24no—
466C42—24no—
467C42—22no—
468C42—22no—
469C42—24no—
470C42—22no—
471C42—21no—
472C42—22no—
473C42—22no—
474C42—21no—
475C42—22no—
476C42—42no—
477C42—42no—
478C42—41no—
479C42—42no—
480C42—42no—
481C42—44no—
482C42—42no—
483C42—42no—
484C42—44no—
485C42—81no—
486C42—84no—
487C42#13714yes—
488C42C22×Q812yes—
489C422−1+412yes—
490C422−1+412yes—
491C422−1+412yes—
492C23C23×D411yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—48no—
496Q8—44no—
497Q8—41no—
498Q8—41no—
499Q8—84no—
500Q8—88no—
501Q8—164no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4#53612yes—
551C2×C4#53612yes—
552C2×C4D4×Q812yes—
553C2×C4D4×Q812yes—
554C2×C4C2×#13712yes—
555C2×C4C2×#13712yes—
556C2×C4C2×#13712yes—
557C2×C4#57512yes—
558C2×C4#57712yes—
559C2×C4C23×Q811yescharacteristic
560C2×C4C2×H(5,𝔽2)11yescharacteristic
561C2×C4C2×2−1+411yescharacteristic
562C2×C4C2×2−1+411yescharacteristic
563C2×C4C2×2−1+411yescharacteristic
564C2×C4#58511yescharacteristic
565C8—24no—
566C8—44no—
567C8—42no—
568C22C2×#53611yescharacteristic
569C22C2×D4×Q811yescharacteristic
570C22#348811yescharacteristic
571C22C22×#13711yescharacteristic
572C22C2×#57511yescharacteristic
573C22C2×#57711yescharacteristic
574C22#358511yescharacteristic
575C4—22no—
576C4—22no—
577C4—22no—
578C4—42no—
579C4—41no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—42no—
584C4—42no—
585C4—44no—
586C4—42no—
587C4—84no—
588C4—82no—
589C4—82no—
590C4—81no—
591C4—82no—
592C4#323714yes—
593C4#338714yes—
594C4#350211yescharacteristic
595C4#350311yescharacteristic
596C4#355011yescharacteristic
597C4#355511yescharacteristic
598C2#3959711yescharacteristic
599C2C2×#323711yescharacteristic
600C2#6089211yescharacteristic
601C2C2×#338711yescharacteristic
602C2#6136711yescharacteristic
603C2#6196511yescharacteristic
604C2#6199611yescharacteristic
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. #477Q8semidirect

Maximal subgroups

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

subgroupindex
#365632
#349502
#331962
#321542
#309692
#299882
#292902
#292732
#292062
#288172
#286932
#278242
#262772
#250612
#228252
#221192
C4×#4772