Finite Group Databasebeta

← #6627449

Subgroups — #6627449

Order 512.

1443 conjugacy classes of subgroups.

The subgroup lattice has 609 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic
2C4×#557C211yesmaximal, characteristic
3#33202C212yesmaximal
4#33196C211yesmaximal, characteristic
5#29288C212yesmaximal
6#29287C211yesmaximal, characteristic
7#29222C212yesmaximal
8#29219C211yesmaximal, characteristic
9#26514C214yesmaximal
10#26513C212yesmaximal
11#26114C214yesmaximal
12#26113C212yesmaximal
13#22741C212yesmaximal
14Q8×C4⋊C8C211yesmaximal, characteristic
15#22521C212yesmaximal
16C4×#477C211yesmaximal, characteristic
17C4×#475C212yesmaximal
18C2×#557C2211yescharacteristic
19C2×#477C2211yescharacteristic
20C2×#475C2212yes—
21C2×C8×Q8C2212yes—
22#2868C2211yescharacteristic
23#2862C2212yes—
24#2582—21no—
25#2571C2212yes—
26#2570C2214yes—
27#2568—21no—
28#2511—22no—
29#2446—22no—
30#2370—22no—
31#2370C2212yes—
32#2365—21no—
33#2365—21no—
34#2365C2211yescharacteristic
35#2355—21no—
36#2355—21no—
37#2355C2211yescharacteristic
38#2355C2212yes—
39#2351C2212yes—
40Q8×C4⋊C4C2211yescharacteristic
41C4×C4⋊Q8—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×#125—22no—
46C4×#125C2212yes—
47C42×Q8—21no—
48C42×Q8C2212yes—
49C42×Q8C2211yescharacteristic
50C42×Q8C2211yescharacteristic
51#2101C2214yes—
52#2101C2214yes—
53#2099C2214yes—
54#2088C2214yes—
55#2088C2214yes—
56#2087C2214yes—
57#1991C2214yes—
58#1990C2212yes—
59#1989C2214yes—
60#1988C2212yes—
61#1988C2212yes—
62#1988C2212yes—
63#1988C2212yes—
64#1987C2211yescharacteristic
65#1987C2212yes—
66#1949C2214yes—
67#1949C2214yes—
68#1949C2214yes—
69#1948C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1948C2212yes—
73#1947C2211yescharacteristic
74#1939C2212yes—
75#1939C2214yes—
76#1921C2212yes—
77#1919C2214yes—
78#1865C2214yes—
79#1863C2212yes—
80#1842C2211yescharacteristic
81#1842C2212yes—
82#1835C2214yes—
83#1834C2212yes—
84#1805C2214yes—
85#1805C2214yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89C4×#107C2212yes—
90C4×#106C2214yes—
91C4×C4⋊C8C2211yescharacteristic
92C4×C4⋊C8C2212yes—
93C4×#103C2212yes—
94C4×#103C2214yes—
95#557—24no—
96#557Q818yescomplemented
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—22no—
104C2×#125C2312yes—
105C2×C4×Q8—21no—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118#477—24no—
119#475—28no—
120C8×Q8—28no—
121C2×#107C2312yes—
122C2×#107C2312yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×C4×C8C2312yes—
136C2×C4×C8C2314yes—
137#400—24no—
138#400—24no—
139#398—24no—
140#391—21no—
141#391—22no—
142#391—21no—
143#391—22no—
144#391—21no—
145#391—21no—
146#391—22no—
147#391—22no—
148#391—22no—
149#391—22no—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2314yes—
162#389C2312yes—
163#387—22no—
164#387—22no—
165#387—24no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—24no—
171#387—24no—
172#384—21no—
173#384—21no—
174#384—21no—
175#384C2311yescharacteristic
176#384C2311yescharacteristic
177#384C2312yes—
178#384C2312yes—
179#382—22no—
180#382—22no—
181#382C2312yes—
182#382C2314yes—
183#382C2314yes—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4—21no—
187C4×C4⋊C4—21no—
188C4×C4⋊C4—21no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
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—22no—
199C4×C4⋊C4—22no—
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⋊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#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2411yescharacteristic
238C22×C8C2412yes—
239C22×C8C2414yes—
240C22×C8C2412yes—
241C4⋊Q8—28no—
242C4⋊Q8—22no—
243C4⋊Q8—24no—
244C4⋊Q8—22no—
245C4⋊Q8—24no—
246C4⋊Q8—22no—
247C4⋊Q8—44no—
248C4⋊Q8—82no—
249C4⋊Q8C2×Q814yes—
250#125—216no—
251#125—24no—
252#125C2×Q818yes—
253C4×Q8—28no—
254C4×Q8—24no—
255C4×Q8—24no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—24no—
260C4×Q8—42no—
261C4×Q8—42no—
262C4×Q8—41no—
263C4×Q8—44no—
264C4×Q8—44no—
265C4×Q8—44no—
266C4×Q8—84no—
267C4×Q8C2×Q818yes—
268C4×Q8C2×Q814yes—
269C4×Q8C2×Q814yes—
270C4×Q8#4114yes—
271C4×Q8#4114yes—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D412yes—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2411yescharacteristic
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2414yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C42—22no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—21no—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335#107—24no—
336#107—42no—
337#106—28no—
338#106—44no—
339C4⋊C8—28no—
340C4⋊C8—24no—
341C4⋊C8—24no—
342C4⋊C8—22no—
343C4⋊C8—22no—
344C4⋊C8—42no—
345C4⋊C8—44no—
346#103—28no—
347#103—24no—
348#103—24no—
349C4×C8—28no—
350C4×C8—24no—
351#95—22no—
352#95—22no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C2×Q8—24no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—44no—
387C2×Q8C22⋊Q814yes—
388C2×Q8C22×Q812yes—
389C2×Q8C2×#4112yes—
390C2×Q82−1+412yes—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4C22×D412yes—
401C22×C4C22×D412yes—
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C2511yescharacteristic
411C2×C8—22no—
412C2×C8—24no—
413C2×C8—22no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—22no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—28no—
423C4⋊C4—24no—
424C4⋊C4—28no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—81no—
456C4⋊C4—82no—
457C4⋊C4—84no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4C22⋊Q814yes—
461C4⋊C4C22⋊Q818yes—
462C4⋊C4C22×Q812yes—
463C4⋊C4C22×Q814yes—
464C4⋊C4C2×#4112yes—
465C42—28no—
466C42—24no—
467C42—24no—
468C42—24no—
469C42—22no—
470C42—24no—
471C42—22no—
472C42—21no—
473C42—22no—
474C42—21no—
475C42—22no—
476C42—41no—
477C42—42no—
478C42—42no—
479C42—42no—
480C42—44no—
481C42—44no—
482C42—44no—
483C42—42no—
484C42—44no—
485C42—42no—
486C42—81no—
487C42—82no—
488C42—84no—
489C42#13514yes—
490C42C22×Q814yes—
491C42C22×Q812yes—
492C42C2×#4114yes—
493C42C2×#4112yes—
494C42C2×#4112yes—
495C422−1+412yes—
496C23C23×D411yescharacteristic
497Q8—28no—
498Q8—22no—
499Q8—22no—
500Q8—22no—
501Q8—48no—
502Q8—84no—
503Q8—88no—
504Q8#55714yes—
505C2×C4—21no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4C2×C22⋊Q814yes—
552C2×C4C2×C22⋊Q812yes—
553C2×C4C2×C22⋊Q814yes—
554C2×C4C2×C22⋊Q812yes—
555C2×C4#54812yes—
556C2×C4#54812yes—
557C2×C4C2×#13512yes—
558C2×C4C2×#13512yes—
559C2×C4C2×#13512yes—
560C2×C4#57512yes—
561C2×C4#57712yes—
562C2×C4C23×Q811yescharacteristic
563C2×C4C22×#4111yescharacteristic
564C2×C4C22×#4111yescharacteristic
565C2×C4C2×2−1+411yescharacteristic
566C2×C4C2×2−1+411yescharacteristic
567C2×C4#58511yescharacteristic
568C8—44no—
569C8—42no—
570C8—42no—
571C22C22×C22⋊Q811yescharacteristic
572C22C2×#54811yescharacteristic
573C22#348911yescharacteristic
574C22C22×#13511yescharacteristic
575C22C2×#57511yescharacteristic
576C22C2×#57711yescharacteristic
577C22#358611yescharacteristic
578C4—24no—
579C4—24no—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—24no—
584C4—44no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—41no—
590C4—84no—
591C4—82no—
592C4—82no—
593C4—81no—
594C4—82no—
595C4#307614yes—
596C4#336514yes—
597C4C2×#55711yescharacteristic
598C4#348011yescharacteristic
599C4#350211yescharacteristic
600C4#352311yescharacteristic
601C2#3657711yescharacteristic
602C2C2×#307611yescharacteristic
603C2#6049711yescharacteristic
604C2C2×#336511yescharacteristic
605C2#6132311yescharacteristic
606C2#6210111yescharacteristic
607C2#6214511yescharacteristic
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. #557Q8semidirect

Maximal subgroups

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

subgroupindex
#365982
C4×#5572
#332022
#331962
#292882
#292872
#292222
#292192
#265142
#265132
#261142
#261132
#227412
Q8×C4⋊C82
#225212
C4×#4772
C4×#4752