Finite Group Databasebeta

← #6857034

Subgroups — #6857034

Order 512.

1357 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2#34950C211yesmaximal, characteristic
3#33198C211yesmaximal, characteristic
4#32478C214yesmaximal
5#31808C214yesmaximal
6#29830C212yesmaximal
7#29347C212yesmaximal
8#29290C212yesmaximal
9#29218C211yesmaximal, characteristic
10#28284C211yesmaximal, characteristic
11#28187C212yesmaximal
12#26026C212yesmaximal
13#25471C212yesmaximal
14#24682C212yesmaximal
15Q8×C4⋊C8C211yesmaximal, characteristic
16#22113C212yesmaximal
17C4×#462C211yesmaximal, characteristic
18C2×#477C2212yes—
19C2×#477C2212yes—
20C2×#462C2211yescharacteristic
21C2×C8×Q8C2212yes—
22C2×C4×Q16C2212yes—
23#2868C2211yescharacteristic
24#2862C2212yes—
25#2586—22no—
26#2582C2211yescharacteristic
27#2582C2212yes—
28#2571—21no—
29#2570—22no—
30#2568—21no—
31#2567C2211yescharacteristic
32#2567C2211yescharacteristic
33#2551C2211yescharacteristic
34#2511C2214yes—
35#2507—21no—
36#2507C2212yes—
37#2451—21no—
38#2451C2212yes—
39#2446—22no—
40#2399—22no—
41#2365—21no—
42#2355—21no—
43#2355C2212yes—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2212yes—
48C4×C4⋊Q8C2211yescharacteristic
49#2295—21no—
50#2295C2212yes—
51C42×Q8C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53#2100C2214yes—
54#2087C2214yes—
55#2087C2214yes—
56#2072C2214yes—
57#2068C2214yes—
58#2068C2214yes—
59#2068C2214yes—
60#2056C2214yes—
61#2056C2214yes—
62#2024C2214yes—
63#1990C2212yes—
64#1987C2212yes—
65#1987C2212yes—
66#1987C2214yes—
67#1987C2212yes—
68#1981C2212yes—
69#1981C2211yescharacteristic
70#1947C2212yes—
71#1947C2214yes—
72#1947C2211yescharacteristic
73#1947C2212yes—
74#1947C2212yes—
75#1942C2214yes—
76#1939C2214yes—
77#1939C2212yes—
78#1933C2214yes—
79#1933C2214yes—
80#1933C2214yes—
81#1928C2214yes—
82#1928C2212yes—
83#1921C2214yes—
84#1878C2212yes—
85#1842C2212yes—
86#1842C2211yescharacteristic
87#1834C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1803C2211yescharacteristic
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—21no—
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113#477—28no—
114#477—44no—
115#462—24no—
116C8×Q8—44no—
117C4×Q16—44no—
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×#107C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2312yes—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2312yes—
128C2×#103C2314yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2314yes—
137#400—24no—
138#400—24no—
139#398—24no—
140#398—22no—
141#398—22no—
142#398—22no—
143#398—22no—
144#395—22no—
145#395—22no—
146#395—22no—
147#395—22no—
148#391—21no—
149#391—22no—
150#391—21no—
151#391—22no—
152#391—21no—
153#391—22no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391C2312yes—
158#391C2312yes—
159#391C2312yes—
160#391C2314yes—
161#391C2314yes—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389C2312yes—
174#389C2312yes—
175#389C2312yes—
176#389C2314yes—
177#389C2312yes—
178#389C2312yes—
179#387—22no—
180#387—22no—
181#387—24no—
182#387—22no—
183#387—22no—
184#387—24no—
185#387—24no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382—22no—
195#382—22no—
196#382—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D412yes—
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218#376—22no—
219C43—21no—
220C43C2312yes—
221C43C2311yescharacteristic
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2412yes—
238C22×Q8C2411yescharacteristic
239C2×Q16—24no—
240C2×Q16—42no—
241C2×Q16—42no—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2414yes—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—24no—
251C4⋊Q8—22no—
252C4⋊Q8—42no—
253C4⋊Q8—44no—
254C4⋊Q8—44no—
255C4⋊Q8—82no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×Q8—24no—
262C4×Q8—22no—
263C4×Q8—24no—
264C4×Q8—42no—
265C4×Q8—41no—
266C4×Q8—84no—
267C4×Q8—82no—
268C4×Q8—84no—
269C4×Q8Q1618yescomplemented
270C4×Q8#4114yes—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—22no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—24no—
317C2×C42—21no—
318C2×C42C2×D412yes—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D412yes—
326C2×C42C2411yescharacteristic
327C2×C42C2412yes—
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331#107—28no—
332#107—42no—
333#107—44no—
334C4⋊C8—22no—
335C4⋊C8—24no—
336C4⋊C8—24no—
337C4⋊C8—22no—
338C4⋊C8—42no—
339C4⋊C8—44no—
340C4⋊C8—44no—
341#103—28no—
342#103—24no—
343#103—24no—
344#103—44no—
345#103—42no—
346#103—42no—
347#103—44no—
348C4×C8—24no—
349C4×C8—44no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—24no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95C2×D412yes—
372#95C2×D412yes—
373#95C2×D412yes—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—21no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—44no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—81no—
394C2×Q8—82no—
395C2×Q8—81no—
396C2×Q8C2×Q1614yes—
397C2×Q8C2×#4112yes—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C2511yescharacteristic
418Q16—48no—
419C2×C8—24no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—24no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—22no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—24no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—81no—
462C4⋊C4—84no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—82no—
469C4⋊C4C2×Q1618yes—
470C4⋊C4C2×Q1614yes—
471C4⋊C4C2×#4112yes—
472C4⋊C42−1+412yes—
473C42—24no—
474C42—22no—
475C42—22no—
476C42—24no—
477C42—22no—
478C42—22no—
479C42—21no—
480C42—22no—
481C42—21no—
482C42—22no—
483C42—42no—
484C42—42no—
485C42—41no—
486C42—42no—
487C42—42no—
488C42—44no—
489C42—42no—
490C42—42no—
491C42—44no—
492C42—81no—
493C42—84no—
494C42C2×Q1618yes—
495C42C2×Q1614yes—
496C42C2×#4112yes—
497C42H(5,𝔽2)12yes—
498C422−1+412yes—
499C422−1+412yes—
500C23C23×D411yescharacteristic
501Q8—28no—
502Q8—22no—
503Q8—22no—
504Q8—41no—
505Q8—41no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—88no—
511Q8—84no—
512Q8—88no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4#53712yes—
560C2×C4#53712yes—
561C2×C4#54812yes—
562C2×C4#54812yes—
563C2×C4C22×Q1614yes—
564C2×C4C22×Q1612yes—
565C2×C4C22×Q1614yes—
566C2×C4C22×Q1612yes—
567C2×C4C22×Q1612yes—
568C2×C4#57512yes—
569C2×C4#57712yes—
570C2×C4C22×#4111yescharacteristic
571C2×C4C2×H(5,𝔽2)11yescharacteristic
572C2×C4C2×2−1+411yescharacteristic
573C2×C4C2×2−1+411yescharacteristic
574C2×C4C2×2−1+411yescharacteristic
575C2×C4#58511yescharacteristic
576C8—44no—
577C8—42no—
578C8—42no—
579C22C2×#53711yescharacteristic
580C22C2×#54811yescharacteristic
581C22#348911yescharacteristic
582C22C23×Q1611yescharacteristic
583C22C2×#57511yescharacteristic
584C22C2×#57711yescharacteristic
585C22#358511yescharacteristic
586C4—22no—
587C4—22no—
588C4—24no—
589C4—22no—
590C4—24no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—42no—
597C4—82no—
598C4—81no—
599C4—82no—
600C4—82no—
601C4—84no—
602C4—82no—
603C4#324914yes—
604C4#336314yes—
605C4#350611yescharacteristic
606C4#350911yescharacteristic
607C4#355011yescharacteristic
608C4#355511yescharacteristic
609C2#3972711yescharacteristic
610C2C2×#324911yescharacteristic
611C2#6092911yescharacteristic
612C2C2×#336311yescharacteristic
613C2#6132411yescharacteristic
614C2#6210011yescharacteristic
615C2#6214411yescharacteristic
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
269. C4×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#365982
#349502
#331982
#324782
#318082
#298302
#293472
#292902
#292182
#282842
#281872
#260262
#254712
#246822
Q8×C4⋊C82
#221132
C4×#4622