Finite Group Databasebeta

← #6434810

Subgroups — #6434810

Order 512.

1408 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#35265C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#30104C214yesmaximal, complemented
4#30103C214yesmaximal, complemented
5#30096C212yesmaximal, complemented
6#30096C212yesmaximal, complemented
7#30092C212yesmaximal, complemented
8#30091C212yesmaximal, complemented
9#23748C212yesmaximal
10#23745C211yesmaximal, characteristic
11C2×#2099C211yesmaximal, characteristic
12#23662C212yesmaximal
13#23554C211yesmaximal, characteristic
14#23551C211yesmaximal, characteristic
15#21039C212yesmaximal
16#21036C212yesmaximal
17#20774C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#428C2212yes—
20C2×#427C2212yes—
21#2864C2212yes—
22#2859C2212yes—
23#2647C2212yes—
24#2647C2212yes—
25#2509—21no—
26#2508—21no—
27#2507—21no—
28#2506—21no—
29#2505—22no—
30#2473—22no—
31#2360—22no—
32#2360—22no—
33#2355—21no—
34#2355—21no—
35Q8×C4⋊C4—21no—
36Q8×C4⋊C4—21no—
37#2330C2214yes—
38#2317C2211yescharacteristic
39#2317C2212yes—
40#2317C2211yescharacteristic
41#2317C2211yescharacteristic
42#2316C2211yescharacteristic
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45#2296C2211yescharacteristic
46C2×#389C2211yescharacteristic
47C2×#384C2211yescharacteristic
48#2274C2212yes—
49C4×#117C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2100C2214yes—
53#2100C2212yes—
54#2100C2212yes—
55#2099C2212yes—
56#2099C2212yes—
57#2099C2214yes—
58#2099C2212yes—
59#2099C2212yes—
60#2099C2214yes—
61#2085C2214yes—
62#2085C2214yes—
63#2084C2214yes—
64#2083C2214yes—
65#2059C2214yes—
66#2059C2212yes—
67#2059C2212yes—
68#2058C2212yes—
69#2058C2211yescharacteristic
70#2058C2211yescharacteristic
71#2057C2212yes—
72#2057C2211yescharacteristic
73#2057C2211yescharacteristic
74#1921C2214yes—
75#1921C2214yes—
76#1920C2214yes—
77#1920C2214yes—
78#1919C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1918C2214yes—
82#1840C2214yes—
83#1839C2212yes—
84#1839C2212yes—
85#1736C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1735C2212yes—
89C2×#340C2212yes—
90C2×#340C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×M16C2312yes—
96C23×C8C2312yes—
97#533—28no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#125—22no—
101C2×#125C2312yes—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8—21no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120#428—28no—
121#427—28no—
122C2×#107C2312yes—
123C2×#107C2314yes—
124C2×#107C2312yes—
125C2×#106C2312yes—
126C2×#106C2314yes—
127C2×#106C2312yes—
128#400—22no—
129#400—22no—
130#400—22no—
131#400—22no—
132#398—22no—
133#398—22no—
134#398—22no—
135#398—22no—
136#391—22no—
137#391—22no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389—21no—
143#389—21no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—21no—
149#389—21no—
150#389—24no—
151#389—24no—
152#389C2314yes—
153#389C2314yes—
154#389C2311yescharacteristic
155#389C2312yes—
156#389C2311yescharacteristic
157#389C2312yes—
158#389C2311yescharacteristic
159#389C2311yescharacteristic
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—24no—
169#387—24no—
170#385—21no—
171#385—21no—
172#385—24no—
173#385—22no—
174#385—22no—
175#385—22no—
176#384—22no—
177#384—21no—
178#384—21no—
179#384—21no—
180#384C2312yes—
181#384C2312yes—
182#384C2312yes—
183#384C2312yes—
184#382—22no—
185#382—21no—
186#382—22no—
187#382C2312yes—
188#382C2312yes—
189#380—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4D411yescharacteristic
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4D411yescharacteristic
199C4×C4⋊C4D411yescharacteristic
200C4×C4⋊C4C2314yes—
201C4×C4⋊C4C2314yes—
202C4×C22⋊C4—24no—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4D412yes—
205#376C2312yes—
206C2×#95—21no—
207C43C2312yes—
208#340C2312yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2314yes—
213#340C2312yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#336C2311yescharacteristic
223#336C2312yes—
224#336C2311yescharacteristic
225#336C2312yes—
226#336C2312yes—
227#336C2314yes—
228#336C2314yes—
229#336C2314yes—
230#336C2314yes—
231C22×Q8—21no—
232C22×Q8—21no—
233C23×C4—22no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×M16—24no—
240C2×M16—24no—
241C2×M16—24no—
242C22×C8—24no—
243C22×C8—24no—
244C22×C8—24no—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2414yes—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250#125—24no—
251#125—24no—
252#125—84no—
253C22⋊Q8—28no—
254C22⋊Q8—28no—
255C4×Q8—28no—
256C4×Q8—28no—
257C4×Q8—84no—
258C4×Q8—84no—
259#117—24no—
260#117—24no—
261#117—22no—
262#117—22no—
263#117—24no—
264#117—22no—
265#117—22no—
266#117#4118yes—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—82no—
299C2×C4⋊C4—82no—
300C2×C4⋊C4—82no—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2414yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2412yes—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—21no—
322C2×C42—21no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D412yes—
330C2×C42C2×D412yes—
331C2×C42C2412yes—
332C2×C42C2412yes—
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335#107—24no—
336#107—24no—
337#107—28no—
338#106—24no—
339#106—24no—
340#106—28no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—21no—
346#95—24no—
347#95—24no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95C2×D412yes—
360#95C2×D412yes—
361#95C2×D412yes—
362#95C2×D412yes—
363C24C22×D411yescharacteristic
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—82no—
391C22×C4—81no—
392C22×C4—82no—
393C22×C4—81no—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4114yes—
404C22×C42−1+414yes—
405C22×C42−1+412yes—
406C22×C42−1+412yes—
407C22×C4C2511yescharacteristic
408M16—48no—
409C2×C8—22no—
410C2×C8—24no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—48no—
419C4⋊C4—28no—
420C4⋊C4—28no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—28no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—82no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4—84no—
457C4⋊C4—84no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4C2×#4114yes—
463C4⋊C4C2×#4114yes—
464C22⋊C4—28no—
465C22⋊C4—24no—
466C22⋊C4—24no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C42—22no—
470C42—22no—
471C42—28no—
472C42—24no—
473C42—24no—
474C42—22no—
475C42—22no—
476C42—22no—
477C42—22no—
478C42—24no—
479C42—42no—
480C42—42no—
481C42—42no—
482C42—42no—
483C42—84no—
484C42—84no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23#52512yes—
490C23#53612yes—
491C23#53612yes—
492C23C23×D411yescharacteristic
493Q8—48no—
494Q8—48no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—84no—
547C2×C4—162no—
548C2×C4—164no—
549C2×C4#52512yes—
550C2×C4#53612yes—
551C2×C4#53612yes—
552C2×C4#54118yes—
553C2×C4#54118yes—
554C2×C4#57612yes—
555C2×C4#57712yes—
556C2×C4C22×#4112yes—
557C2×C4C2×2−1+412yes—
558C2×C4C2×2−1+411yescharacteristic
559C2×C4C2×2−1+411yescharacteristic
560C8—42no—
561C8—42no—
562C8—44no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22#263414yes—
570C22C2×#52511yescharacteristic
571C22C2×#53611yescharacteristic
572C22C2×#53611yescharacteristic
573C22C2×#57611yescharacteristic
574C22C2×#57711yescharacteristic
575C22#358611yescharacteristic
576C22#358611yescharacteristic
577C4—22no—
578C4—22no—
579C4—41no—
580C4—44no—
581C4—41no—
582C4—44no—
583C4—42no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—81no—
588C4—82no—
589C4—81no—
590C4—84no—
591C4—81no—
592C4—82no—
593C4—81no—
594C4#263414yes—
595C4C2×#54112yes—
596C4C2×#54112yes—
597C2—24no—
598C2C2×#263411yescharacteristic
599C2#6058411yescharacteristic
600C2#6058511yescharacteristic
601C2#6087711yescharacteristic
602C2#6088411yescharacteristic
603C2#6088811yescharacteristic
604C2#6088911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #30104C2semidirect
4. #30103C2semidirect
5. #30096C2semidirect
6. #30096C2semidirect
7. #30092C2semidirect
8. #30091C2semidirect

Maximal subgroups

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

subgroupindex
#352652
#342142
#301042
#301032
#300962
#300922
#300912
#237482
#237452
C2×#20992
#236622
#235542
#235512
#210392
#210362
#207742