Finite Group Databasebeta

← #6434808

Subgroups — #6434808

Order 512.

1436 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35265C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#30102C214yesmaximal, complemented
4#30100C214yesmaximal, complemented
5#30096C212yesmaximal, complemented
6#30094C212yesmaximal, complemented
7#30093C212yesmaximal, complemented
8#30092C212yesmaximal, complemented
9#23749C212yesmaximal
10#23747C211yesmaximal, characteristic
11#23744C211yesmaximal, characteristic
12#23661C212yesmaximal
13#23552C211yesmaximal, characteristic
14C2×#2058C211yesmaximal, characteristic
15#21038C212yesmaximal
16#21035C212yesmaximal
17#20774C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#428C2212yes—
20C22×#106C2212yes—
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#2369—21no—
32#2369—21no—
33#2365—21no—
34#2365—21no—
35#2345—22no—
36Q8×C22⋊C4—22no—
37#2331C2214yes—
38#2317C2211yescharacteristic
39#2317C2211yescharacteristic
40#2317C2211yescharacteristic
41#2316C2211yescharacteristic
42#2316C2212yes—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45#2296C2211yescharacteristic
46C2×#389C2211yescharacteristic
47#2285C2211yescharacteristic
48#2274C2212yes—
49C4×#117C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2101C2212yes—
53#2101C2214yes—
54#2101C2212yes—
55#2100C2214yes—
56#2100C2211yescharacteristic
57#2100C2212yes—
58#2100C2211yescharacteristic
59#2099C2211yescharacteristic
60#2099C2212yes—
61#2099C2214yes—
62#2099C2211yescharacteristic
63#2085C2214yes—
64#2085C2214yes—
65#2084C2214yes—
66#2084C2214yes—
67#2058C2212yes—
68#2058C2212yes—
69#2058C2212yes—
70#2058C2212yes—
71#2058C2214yes—
72#2057C2212yes—
73#2057C2212yes—
74#1921C2214yes—
75#1920C2214yes—
76#1920C2214yes—
77#1920C2214yes—
78#1919C2214yes—
79#1919C2214yes—
80#1919C2214yes—
81#1918C2214yes—
82#1840C2214yes—
83#1839C2212yes—
84#1839C2212yes—
85#1736C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1735C2212yes—
89#1735C2212yes—
90C2×#340C2212yes—
91#1729C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C23×Q8—21no—
96C22×M16C2312yes—
97C23×C8C2312yes—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2312yes—
103C2×#125—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×#117C2311yescharacteristic
107C2×#117C2312yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117#428—28no—
118C2×#107C2314yes—
119C2×#106—28no—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123#400—22no—
124#400—22no—
125#400—22no—
126#400—22no—
127#398—22no—
128#398—22no—
129#398—22no—
130#398—22no—
131#391—22no—
132#391—22no—
133#391—24no—
134#391—24no—
135#389—22no—
136#389—22no—
137#389—21no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389—21no—
142#389—21no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—21no—
148#389—21no—
149#389C2314yes—
150#389C2314yes—
151#389C2311yescharacteristic
152#389C2311yescharacteristic
153#389C2311yescharacteristic
154#389C2312yes—
155#389C2312yes—
156#389C2311yescharacteristic
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#387—22no—
162#387—22no—
163#387—22no—
164#387—22no—
165#387—24no—
166#387—24no—
167#385—21no—
168#385—21no—
169#385—22no—
170#385—22no—
171#385—22no—
172#384—21no—
173#384C2311yescharacteristic
174#384C2311yescharacteristic
175#384C2312yes—
176#384C2312yes—
177#382—21no—
178#382—22no—
179#382C2312yes—
180#382C2312yes—
181#382C2312yes—
182#380—21no—
183#380—21no—
184#380—24no—
185#380—22no—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—21no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4D411yescharacteristic
195C4×C4⋊C4D411yescharacteristic
196C4×C4⋊C4D411yescharacteristic
197C4×C4⋊C4D411yescharacteristic
198C4×C4⋊C4C2314yes—
199C4×C4⋊C4C2314yes—
200C4×C22⋊C4—24no—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203#376C2312yes—
204C2×#95—21no—
205C43C2312yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2314yes—
213#340C2314yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#340C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2312yes—
222#336C2312yes—
223#336C2312yes—
224#336C2311yescharacteristic
225#336C2311yescharacteristic
226#336C2312yes—
227#336C2314yes—
228#336C2314yes—
229#336C2314yes—
230#336C2314yes—
231C22×Q8—28no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—82no—
235C22×Q8—82no—
236C23×C4—22no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2411yescharacteristic
242C2×M16—24no—
243C2×M16—24no—
244C2×M16—24no—
245C22×C8—24no—
246C22×C8—24no—
247C22×C8—24no—
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C4⋊Q8—28no—
252C4⋊Q8—28no—
253C4⋊Q8—28no—
254C4⋊Q8—28no—
255#125—84no—
256C4×Q8—84no—
257C4×Q8—84no—
258#117—22no—
259#117—22no—
260#117—24no—
261#117—24no—
262#117—22no—
263#117—22no—
264#117—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—82no—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4#4118yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2414yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D412yes—
326C2×C42C2×D412yes—
327C2×C42C2412yes—
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331#107—28no—
332#106—28no—
333#106—28no—
334#106—28no—
335#95—22no—
336#95—22no—
337#95—24no—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95C2×D412yes—
356#95C2×D412yes—
357#95C2×D412yes—
358#95C2×D412yes—
359C24C22×D411yescharacteristic
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—82no—
365C2×Q8—82no—
366C2×Q8—816no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—28no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D412yes—
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C2×#4114yes—
401C22×C42−1+412yes—
402C22×C42−1+412yes—
403C22×C42−1+414yes—
404C22×C4C2511yescharacteristic
405M16—48no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—24no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—48no—
416C4⋊C4—28no—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—82no—
445C4⋊C4—82no—
446C4⋊C4—81no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—81no—
450C4⋊C4—82no—
451C4⋊C4—84no—
452C4⋊C4—84no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—82no—
456C4⋊C4—82no—
457C4⋊C4C2×#4118yes—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C22⋊C4—42no—
461C22⋊C4—44no—
462C22⋊C4—42no—
463C42—22no—
464C42—22no—
465C42—24no—
466C42—24no—
467C42—22no—
468C42—22no—
469C42—22no—
470C42—22no—
471C42—24no—
472C42—42no—
473C42—42no—
474C42—42no—
475C42—44no—
476C42—42no—
477C42—84no—
478C42—84no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23#52512yes—
484C23#53612yes—
485C23#53612yes—
486C23C23×D411yescharacteristic
487Q8—416no—
488Q8—88no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—88no—
541C2×C4—84no—
542C2×C4—162no—
543C2×C4#52512yes—
544C2×C4#53612yes—
545C2×C4#53612yes—
546C2×C4#54118yes—
547C2×C4#54118yes—
548C2×C4#57712yes—
549C2×C4#57712yes—
550C2×C4C22×#4112yes—
551C2×C4C2×2−1+412yes—
552C2×C4C2×2−1+411yescharacteristic
553C2×C4C2×2−1+411yescharacteristic
554C8—42no—
555C8—42no—
556C8—44no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22#263414yes—
564C22C2×#52511yescharacteristic
565C22C2×#53611yescharacteristic
566C22C2×#53611yescharacteristic
567C22C2×#57711yescharacteristic
568C22C2×#57711yescharacteristic
569C22#358511yescharacteristic
570C22#358611yescharacteristic
571C4—22no—
572C4—22no—
573C4—41no—
574C4—48no—
575C4—41no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—81no—
581C4—81no—
582C4—82no—
583C4—82no—
584C4—82no—
585C4—81no—
586C4—81no—
587C4—82no—
588C4#263414yes—
589C4C2×#54112yes—
590C4C2×#54112yes—
591C2—24no—
592C2C2×#263411yescharacteristic
593C2#6057911yescharacteristic
594C2#6058211yescharacteristic
595C2#6088111yescharacteristic
596C2#6088611yescharacteristic
597C2#6088911yescharacteristic
598C2#6089211yescharacteristic
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. #30102C2semidirect
4. #30100C2semidirect
5. #30096C2semidirect
6. #30094C2semidirect
7. #30093C2semidirect
8. #30092C2semidirect

Maximal subgroups

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

subgroupindex
#352652
#342162
#301022
#301002
#300962
#300942
#300932
#300922
#237492
#237472
#237442
#236612
#235522
C2×#20582
#210382
#210352
#207742