Finite Group Databasebeta

← #6856529

Subgroups — #6856529

Order 512.

1401 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#36593C211yesmaximal, characteristic
2#34950C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4#32312C214yesmaximal
5#31456C214yesmaximal
6#29866C212yesmaximal
7#29287C211yesmaximal, characteristic
8#29276C212yesmaximal
9#29219C212yesmaximal
10#29219C211yesmaximal, characteristic
11#28791C212yesmaximal
12#27729C212yesmaximal
13#26280C212yesmaximal
14#24994C212yesmaximal
15#22821C211yesmaximal, characteristic
16#22112C212yesmaximal
17C4×#475C211yesmaximal, characteristic
18C2×#557—21no—
19C2×#475C2212yes—
20C2×#475C2211yescharacteristic
21#2868C2212yes—
22#2868C2211yescharacteristic
23#2586—22no—
24#2582C2211yescharacteristic
25#2567C2211yescharacteristic
26#2567C2211yescharacteristic
27#2552—22no—
28#2552—21no—
29#2552C2214yes—
30#2552C2212yes—
31#2551C2211yescharacteristic
32#2546—21no—
33#2546—22no—
34#2507—21no—
35#2507C2212yes—
36#2451—21no—
37#2451C2212yes—
38#2399—22no—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4C2212yes—
41C4×C4⋊Q8—21no—
42C4×C4⋊Q8C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46#2295—21no—
47#2295C2212yes—
48C42×Q8C2211yescharacteristic
49C42×Q8C2211yescharacteristic
50#2100C2214yes—
51#2099C2214yes—
52#2099C2214yes—
53#2099C2214yes—
54#2088C2214yes—
55#2084C2214yes—
56#2084C2214yes—
57#2072C2214yes—
58#2072C2214yes—
59#2058C2214yes—
60#1990C2212yes—
61#1990C2214yes—
62#1988C2212yes—
63#1988C2211yescharacteristic
64#1988C2212yes—
65#1988C2212yes—
66#1988C2212yes—
67#1987C2212yes—
68#1984C2212yes—
69#1948C2212yes—
70#1948C2211yescharacteristic
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1941C2214yes—
75#1933C2214yes—
76#1927C2214yes—
77#1920C2214yes—
78#1920C2214yes—
79#1920C2214yes—
80#1919C2212yes—
81#1916C2214yes—
82#1865C2212yes—
83#1842C2211yescharacteristic
84#1834C2214yes—
85#1834C2212yes—
86#1831C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90C4×#106C2212yes—
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95#557—48no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—22no—
99C2×C4⋊Q8—22no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×#125—24no—
106C2×#125—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8—22no—
110C2×C4×Q8—22no—
111C2×C4×Q8—21no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118#475—28no—
119#475—24no—
120C2×#106C2312yes—
121C2×#106C2314yes—
122C2×#106C2312yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2312yes—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×C4×C8C2312yes—
133C2×C4×C8C2314yes—
134#398—22no—
135#398—22no—
136#398—22no—
137#398—22no—
138#398—24no—
139#398—24no—
140#398—24no—
141#395—22no—
142#395—22no—
143#395—22no—
144#395—22no—
145#391—21no—
146#391—21no—
147#391—21no—
148#391—22no—
149#391—22no—
150#391—21no—
151#391—22no—
152#391C2312yes—
153#391C2312yes—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—24no—
158#389—22no—
159#389—22no—
160#389—24no—
161#389—21no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—22no—
166#389—22no—
167#389C2312yes—
168#389C2314yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#384—21no—
179#384—21no—
180#384—22no—
181#384—21no—
182#384—22no—
183#384C2311yescharacteristic
184#384C2312yes—
185#384C2311yescharacteristic
186#384C2312yes—
187#384C2312yes—
188#384C2314yes—
189#384C2314yes—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4D412yes—
200C4×C4⋊C4D412yes—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4C2311yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2312yes—
210#376—22no—
211C43—21no—
212C43C2312yes—
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#336C2314yes—
228#336C2314yes—
229#336C2312yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—21no—
235C22×Q8C2×D411yescharacteristic
236C22×Q8C2411yescharacteristic
237C22×C8C2412yes—
238C22×C8C2412yes—
239C22×C8C2414yes—
240C4⋊Q8—24no—
241C4⋊Q8—24no—
242C4⋊Q8—22no—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—24no—
246C4⋊Q8—22no—
247C4⋊Q8—42no—
248C4⋊Q8—44no—
249C4⋊Q8—48no—
250C4⋊Q8—44no—
251C4⋊Q8—84no—
252C4⋊Q8—82no—
253#125—416no—
254#125—48no—
255C4×Q8—24no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—41no—
263C4×Q8—48no—
264C4×Q8—48no—
265C4×Q8—44no—
266C4×Q8—84no—
267C4×Q8—82no—
268C4×Q8SD1618yes—
269C4×Q8#4114yes—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D412yes—
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C42—21no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—22no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D412yes—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D412yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D412yes—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2412yes—
330C2×C42C2411yescharacteristic
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#106—24no—
335#106—28no—
336#106—24no—
337#106—44no—
338C4⋊C8—22no—
339C4⋊C8—24no—
340C4⋊C8—24no—
341C4⋊C8—22no—
342C4⋊C8—42no—
343#103—28no—
344#103—24no—
345#103—42no—
346C4×C8—24no—
347C4×C8—44no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—21no—
353#95—21no—
354#95—21no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—22no—
363#95—24no—
364#95—22no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
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—44no—
378C2×Q8—42no—
379C2×Q8—44no—
380C2×Q8—42no—
381C2×Q8—44no—
382C2×Q8—42no—
383C2×Q8—44no—
384C2×Q8—82no—
385C2×Q8—81no—
386C2×Q8—81no—
387C2×Q8C2×SD1614yes—
388C2×Q8C2×#4112yes—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2511yescharacteristic
409C2×C8—24no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—42no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—81no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—84no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—84no—
463C4⋊C4C2×SD1618yes—
464C4⋊C4C2×SD1614yes—
465C4⋊C4#13714yes—
466C4⋊C4C2×#4112yes—
467C4⋊C42−1+412yes—
468C42—24no—
469C42—24no—
470C42—22no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—21no—
476C42—22no—
477C42—22no—
478C42—21no—
479C42—41no—
480C42—42no—
481C42—42no—
482C42—42no—
483C42—48no—
484C42—44no—
485C42—42no—
486C42—42no—
487C42—44no—
488C42—81no—
489C42C2×SD1618yes—
490C42C2×SD1614yes—
491C42C2×#4112yes—
492C422−1+412yes—
493C422−1+412yes—
494C422−1+412yes—
495C23C23×D411yescharacteristic
496Q8—28no—
497Q8—22no—
498Q8—22no—
499Q8—41no—
500Q8—41no—
501Q8—88no—
502Q8—84no—
503Q8—88no—
504Q8—84no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
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—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4#53612yes—
551C2×C4#53612yes—
552C2×C4#54812yes—
553C2×C4#54812yes—
554C2×C4C22×SD1614yes—
555C2×C4C22×SD1612yes—
556C2×C4C22×SD1614yes—
557C2×C4C22×SD1612yes—
558C2×C4C22×SD1612yes—
559C2×C4C2×#13714yes—
560C2×C4C2×#13712yes—
561C2×C4#57612yes—
562C2×C4C22×#4111yescharacteristic
563C2×C4C2×2−1+411yescharacteristic
564C2×C4C2×2−1+411yescharacteristic
565C2×C4C2×2−1+411yescharacteristic
566C2×C4C2×2−1+411yescharacteristic
567C2×C4#58511yescharacteristic
568C8—44no—
569C8—42no—
570C8—42no—
571C22C2×#53611yescharacteristic
572C22C2×#54811yescharacteristic
573C22#348911yescharacteristic
574C22C23×SD1611yescharacteristic
575C22C22×#13711yescharacteristic
576C22C2×#57611yescharacteristic
577C22#358611yescharacteristic
578C4—22no—
579C4—22no—
580C4—24no—
581C4—24no—
582C4—24no—
583C4—22no—
584C4—42no—
585C4—42no—
586C4—41no—
587C4—42no—
588C4—42no—
589C4—84no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—81no—
594C4—82no—
595C4#322914yes—
596C4#336214yes—
597C4#350711yescharacteristic
598C4#350911yescharacteristic
599C4#355511yescharacteristic
600C4#355611yescharacteristic
601C2#3972411yescharacteristic
602C2C2×#322911yescharacteristic
603C2#6088811yescharacteristic
604C2C2×#336211yescharacteristic
605C2#6132211yescharacteristic
606C2#6209911yescharacteristic
607C2#6212711yescharacteristic
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

Maximal subgroups

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

subgroupindex
#365932
#349502
#331962
#323122
#314562
#298662
#292872
#292762
#292192
#287912
#277292
#262802
#249942
#228212
#221122
C4×#4752