Finite Group Databasebeta

← #6854623

Subgroups — #6854623

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36565C211yesmaximal, characteristic
2#36563C211yesmaximal, characteristic
3#33294C211yesmaximal, characteristic
4#32158C214yesmaximal
5#32154C214yesmaximal
6#30310C212yesmaximal
7#29226C211yesmaximal, characteristic
8#29224C212yesmaximal
9#29212C211yesmaximal, characteristic
10#29183C212yesmaximal
11#26446C212yesmaximal
12#26442C212yesmaximal
13#26279C212yesmaximal
14#26277C212yesmaximal
15#22250C212yesmaximal
16C4×#501C211yesmaximal, characteristic
17#21345C211yesmaximal, characteristic
18C2×#501C2211yescharacteristic
19#2868C2211yescharacteristic
20#2862C2212yes—
21#2859C2212yes—
22#2859C2211yescharacteristic
23#2586—22no—
24#2586—22no—
25#2582—21no—
26#2567C2212yes—
27#2552—21no—
28#2551C2211yescharacteristic
29#2551C2211yescharacteristic
30#2550—22no—
31#2550C2212yes—
32#2550C2212yes—
33#2507C2214yes—
34#2451—22no—
35#2451—21no—
36#2451—21no—
37#2369—21no—
38#2369—21no—
39#2369C2212yes—
40#2355—21no—
41#2355C2211yescharacteristic
42#2355C2211yescharacteristic
43Q8×C4⋊C4—21no—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2212yes—
46C4×#125C2211yescharacteristic
47C4×#125C2212yes—
48C42×Q8C2211yescharacteristic
49#2101C2214yes—
50#2101C2214yes—
51#2099C2212yes—
52#2099C2212yes—
53#2088C2214yes—
54#2087C2214yes—
55#2085C2214yes—
56#2085C2214yes—
57#2059C2214yes—
58#2058C2214yes—
59#2057C2214yes—
60#1993C2212yes—
61#1993C2211yescharacteristic
62#1992C2214yes—
63#1992C2212yes—
64#1992C2211yescharacteristic
65#1990C2212yes—
66#1990C2212yes—
67#1988C2214yes—
68#1987C2214yes—
69#1987C2212yes—
70#1987C2212yes—
71#1984C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1944C2212yes—
75#1943C2212yes—
76#1939C2212yes—
77#1921C2214yes—
78#1919C2214yes—
79#1919C2214yes—
80#1919C2214yes—
81#1918C2214yes—
82#1918C2214yes—
83#1917C2214yes—
84#1847C2212yes—
85#1847C2211yescharacteristic
86#1836C2212yes—
87#1836C2212yes—
88#1836C2212yes—
89C4×#107C2212yes—
90C4×#106C2212yes—
91#1767C2211yescharacteristic
92#1767C2212yes—
93#1745C2212yes—
94C4×#97C2211yescharacteristic
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×#125—22no—
104C2×#125C2311yescharacteristic
105C2×#125C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—22no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111#501Q818yescomplemented
112C2×#107C2312yes—
113C2×#107C2314yes—
114C2×#107C2312yes—
115C2×#106C2312yes—
116C2×#106C2314yes—
117C2×#106C2312yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2314yes—
120C2×#97C2311yescharacteristic
121C2×#97C2312yes—
122C2×#97C2312yes—
123C2×#97C2311yescharacteristic
124C2×C4×C8C2312yes—
125#398—24no—
126#398—22no—
127#398—22no—
128#398—24no—
129#395—24no—
130#395—22no—
131#395—22no—
132#395—22no—
133#395—22no—
134#391—22no—
135#391—21no—
136#391—21no—
137#391—22no—
138#391—21no—
139#391—22no—
140#391—22no—
141#391—22no—
142#391—22no—
143#391—22no—
144#391—22no—
145#391—24no—
146#389—22no—
147#389—21no—
148#389—21no—
149#389—21no—
150#389—22no—
151#389—22no—
152#389—21no—
153#389—21no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389—22no—
159#389—24no—
160#389C2312yes—
161#389C2312yes—
162#389C2311yescharacteristic
163#389C2311yescharacteristic
164#389C2314yes—
165#389C2312yes—
166#389C2314yes—
167#389C2314yes—
168#387—22no—
169#387—22no—
170#387—24no—
171#387—21no—
172#387—21no—
173#387—22no—
174#387—22no—
175#384—22no—
176#384—21no—
177#384—21no—
178#384C2311yescharacteristic
179#384C2312yes—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
183#382—22no—
184#382C2312yes—
185#382C2311yescharacteristic
186#382C2312yes—
187#382C2312yes—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—21no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4D411yescharacteristic
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2311yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2314yes—
207C4×C4⋊C4C2311yescharacteristic
208C4×C4⋊C4C2312yes—
209C43C2312yes—
210C43C2311yescharacteristic
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#336C2314yes—
226#336C2312yes—
227#336C2314yes—
228#336C2314yes—
229C22×Q8—21no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8C2×D411yescharacteristic
234C22×C8C2414yes—
235C22×C8C2412yes—
236C22×C8C2412yes—
237C4⋊Q8—22no—
238C4⋊Q8—28no—
239C4⋊Q8—44no—
240C4⋊Q8—44no—
241C4⋊Q8—44no—
242C4⋊Q8—44no—
243C4⋊Q8—82no—
244C4⋊Q8—82no—
245C4⋊Q8C2×Q814yes—
246#125—22no—
247#125—44no—
248#125—84no—
249#125C2×Q814yes—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—42no—
253C4×Q8—44no—
254C4×Q8—84no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2×D411yescharacteristic
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2411yescharacteristic
291C2×C4⋊C4C2414yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C42—22no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2412yes—
311C2×C42C2411yescharacteristic
312C2×C42C2412yes—
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315#107—42no—
316#107—44no—
317#107C2×Q818yes—
318#106—44no—
319#106—42no—
320#106C2×Q818yes—
321C4⋊C8—24no—
322C4⋊C8—44no—
323#97—22no—
324#97—24no—
325#97—42no—
326#97C2×Q814yes—
327C4×C8—24no—
328#95—24no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—21no—
334#95—21no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—24no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—22no—
347#95—24no—
348#95—22no—
349#95—22no—
350#95C2×D412yes—
351#95C2×D412yes—
352#95C2×D412yes—
353C2×Q8—22no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—82no—
363C2×Q8—82no—
364C2×Q8—82no—
365C2×Q8—82no—
366C2×Q8—82no—
367C2×Q8—82no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C2511yescharacteristic
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—24no—
392C2×C8—22no—
393C2×C8—24no—
394C2×C8—22no—
395C2×C8—24no—
396C2×C8C22×Q814yes—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—28no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—84no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4#13714yes—
439C4⋊C4C22×Q812yes—
440C4⋊C4C22×Q814yes—
441C4⋊C4C22×Q812yes—
442C42—22no—
443C42—24no—
444C42—22no—
445C42—24no—
446C42—22no—
447C42—24no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—41no—
452C42—41no—
453C42—42no—
454C42—44no—
455C42—42no—
456C42—82no—
457C42—82no—
458C42—84no—
459C42C22×Q812yes—
460C422−1+414yes—
461C422−1+412yes—
462C23C23×D411yescharacteristic
463Q8—48no—
464Q8—44no—
465Q8—164no—
466Q8—164no—
467Q8—164no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4#53612yes—
519C2×C4#53612yes—
520C2×C4D4×Q812yes—
521C2×C4D4×Q812yes—
522C2×C4C2×#13714yes—
523C2×C4C2×#13712yes—
524C2×C4#57512yes—
525C2×C4C23×Q811yescharacteristic
526C2×C4C2×2−1+412yes—
527C2×C4C2×2−1+411yescharacteristic
528C2×C4#58512yes—
529C8—24no—
530C8—42no—
531C8—44no—
532C22C2×#53611yescharacteristic
533C22C2×D4×Q811yescharacteristic
534C22#350011yescharacteristic
535C22C22×#13711yescharacteristic
536C22C2×#57511yescharacteristic
537C22#358511yescharacteristic
538C22#358511yescharacteristic
539C4—24no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—41no—
544C4—41no—
545C4—42no—
546C4—42no—
547C4—42no—
548C4—44no—
549C4—42no—
550C4—42no—
551C4—82no—
552C4—81no—
553C4—82no—
554C4—82no—
555C4—81no—
556C4—84no—
557C4#323714yes—
558C4#350212yes—
559C4#355512yes—
560C2#3961911yescharacteristic
561C2C2×#323711yescharacteristic
562C2#6088311yescharacteristic
563C2#6136711yescharacteristic
564C2Q8×#13711yescharacteristic
565C2#6258211yescharacteristic
566C2#6259211yescharacteristic
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
111. #501Q8semidirect

Maximal subgroups

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

subgroupindex
#365652
#365632
#332942
#321582
#321542
#303102
#292262
#292242
#292122
#291832
#264462
#264422
#262792
#262772
#222502
C4×#5012
#213452