Finite Group Databasebeta

← #6436969

Subgroups — #6436969

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35259C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#29949C212yesmaximal, complemented
4#29944C212yesmaximal, complemented
5#29929C212yesmaximal, complemented
6#29921C212yesmaximal, complemented
7#29910C212yesmaximal, complemented
8#29907C212yesmaximal, complemented
9#29899C214yesmaximal, complemented
10#23771C214yesmaximal, complemented
11#23760C212yesmaximal, complemented
12#23755C212yesmaximal, complemented
13#21399C212yesmaximal, complemented
14#21369C212yesmaximal, complemented
15#20720C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C2×#433C2212yes—
19#2856C2211yescharacteristic
20#2854C2212yes—
21#2851C2211yescharacteristic
22#2629C2211yescharacteristic
23#2619C2211yescharacteristic
24#2615C2212yes—
25#2464—22no—
26#2458—22no—
27#2456—22no—
28#2456—22no—
29#2355—22no—
30#2354—22no—
31#2352—22no—
32#2352—22no—
33#2324C2212yes—
34#2314C2211yescharacteristic
35#2314C2212yes—
36#2314C2212yes—
37#2311C2212yes—
38C4×#121C2212yes—
39#2285C2212yes—
40#2285C2212yes—
41#2285C2211yescharacteristic
42C2×#384C2211yescharacteristic
43#2274C2212yes—
44#2101C2214yes—
45#2100C2212yes—
46#2100C2212yes—
47#2094C2212yes—
48#2094C2212yes—
49#2092C2214yes—
50#2084C2212yes—
51#2084C2214yes—
52#2083C2212yes—
53#2077C2214yes—
54#2077C2214yes—
55#2077C2214yes—
56#2075C2214yes—
57#2071C2212yes—
58#2071C2212yes—
59#2071C2214yes—
60#2034C2214yes—
61#2034C2214yes—
62#2034C2214yes—
63#2033C2214yes—
64#1989C2214yes—
65#1982C2214yes—
66#1949C2214yes—
67#1948C2214yes—
68#1875C2214yes—
69#1874C2214yes—
70#1854C2214yes—
71#1853C2214yes—
72#1817C2212yes—
73#1798C2212yes—
74#1797C2212yes—
75#1780C2212yes—
76#1776C2214yes—
77C4×#106C2214yes—
78#1762C2214yes—
79C4×#102C2214yes—
80#1750C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C22×#41C2311yescharacteristic
84#540—28no—
85C2×#127C2311yescharacteristic
86C2×#125C2311yescharacteristic
87C2×#123—22no—
88C2×#123—22no—
89C2×#121—22no—
90C2×#121—22no—
91C2×#121C2312yes—
92C2×#121C2312yes—
93C2×#121C2312yes—
94C4×#41—28no—
95C2×C4×Q8D412yes—
96C2×C4×D4—22no—
97C2×C4×D4D412yes—
98C2×C4×D4D412yes—
99C2×C4×D4D412yes—
100C2×C4×D4C2312yes—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117C2312yes—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C4⋊C4—22no—
108C22×C4⋊C4C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#433—28no—
113C2×#107C2314yes—
114C2×#106C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×C22⋊C8C2314yes—
123C2×C22⋊C8C2314yes—
124C2×C22⋊C8C2312yes—
125C2×C22⋊C8C2312yes—
126C2×#97C2312yes—
127C2×C4×C8C2312yes—
128#398—22no—
129#396—22no—
130#394—22no—
131#393—22no—
132#392—22no—
133#392—22no—
134#392—22no—
135#392—22no—
136#391—22no—
137#390—22no—
138#390—22no—
139#390—22no—
140#390C2312yes—
141#390C2312yes—
142#389C2312yes—
143#389C2312yes—
144#388—22no—
145#388—24no—
146#388—24no—
147#388—22no—
148#388—22no—
149#388—22no—
150#388—22no—
151#387—24no—
152#387—24no—
153#387—22no—
154#387—22no—
155#386—22no—
156#386—22no—
157#386—22no—
158#386—22no—
159#386—22no—
160#385—22no—
161#385—22no—
162#384—22no—
163#384C2312yes—
164#384C2311yescharacteristic
165#384C2312yes—
166#384C2312yes—
167#382—22no—
168#382—22no—
169#382C2312yes—
170#382C2311yescharacteristic
171#380—22no—
172#380—22no—
173#380—22no—
174#380C2314yes—
175#380C2314yes—
176#380C2312yes—
177#380C2312yes—
178#380C2312yes—
179C4×C4⋊C4—24no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
186C4×C22⋊C4—24no—
187C4×C22⋊C4—22no—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2312yes—
197#336C2312yes—
198#336C2314yes—
199C2×#41—24no—
200C2×#41—42no—
201C2×#41—42no—
202C22×Q8C2×D411yescharacteristic
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4C2×D411yescharacteristic
208C22×D4C2×D411yescharacteristic
209C22×D4C2×D411yescharacteristic
210C22×D4C2412yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4C2×D412yes—
214C23×C4C2×D412yes—
215C23×C4C2×D412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C22×C8C2414yes—
220C22×C8C2414yes—
221#127—24no—
222#125—24no—
223#123—48no—
224#123—84no—
225#121—28no—
226#121—42no—
227#121—44no—
228#121—42no—
229#121—48no—
230#121—84no—
231C4×Q8—28no—
232C4×D4—28no—
233C4×D4—28no—
234C4×D4—28no—
235C4×D4—28no—
236C4×D4—84no—
237#117—24no—
238#117—24no—
239#117—44no—
240#117—84no—
241#117—84no—
242#117#4118yes—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—84no—
255C2×C4⋊C4C2×D412yes—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2414yes—
259C2×C4⋊C4C2414yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D412yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—22no—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294#107—44no—
295#106—44no—
296#106—44no—
297#106—44no—
298C4⋊C8—28no—
299#102—44no—
300#102—44no—
301#102—44no—
302#102—44no—
303C22⋊C8—28no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306C22⋊C8—44no—
307#97—24no—
308C4×C8—24no—
309#95—24no—
310#95—24no—
311#95—22no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321C24—22no—
322C24C22×D411yescharacteristic
323C24C22×D411yescharacteristic
324C24C22×D411yescharacteristic
325#41—416no—
326C2×Q8—22no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—82no—
347C2×D4—82no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4—82no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—84no—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4C2×#4114yes—
389C22×C4C2511yescharacteristic
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—84no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4#12318yes—
417C22⋊C4—24no—
418C22⋊C4—24no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C22⋊C4—82no—
428C22⋊C4—82no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C22⋊C4—82no—
432C22⋊C4—82no—
433C22⋊C4#12318yes—
434C42—24no—
435C42—24no—
436C42—24no—
437C42—22no—
438C42—22no—
439C42—44no—
440C42—84no—
441C42—84no—
442C42—82no—
443C42C2×#4114yes—
444C42C2×#4114yes—
445C23—22no—
446C23—21no—
447C23—21no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—22no—
458C23—22no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23C2×#12312yes—
467C23C23×D411yescharacteristic
468Q8—44no—
469D4—44no—
470D4—44no—
471D4—44no—
472D4—44no—
473D4—44no—
474D4—88no—
475D4—164no—
476D4—164no—
477D4—164no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—48no—
497C2×C4—48no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—164no—
517C2×C4—164no—
518C2×C4C2×#12312yes—
519C2×C4C2×#12314yes—
520C2×C4C2×#12314yes—
521C2×C4#57714yes—
522C2×C4C22×#4112yes—
523C2×C4#58512yes—
524C2×C4#58512yes—
525C8—44no—
526C8—44no—
527C22—22no—
528C22—24no—
529C22—42no—
530C22—41no—
531C22—41no—
532C22—42no—
533C22—42no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—42no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22C22×#12311yescharacteristic
550C22#344911yescharacteristic
551C22#344911yescharacteristic
552C22C2×#57712yes—
553C22#358512yes—
554C4—22no—
555C4—24no—
556C4—24no—
557C4—24no—
558C4—41no—
559C4—41no—
560C4—42no—
561C4—84no—
562C4—82no—
563C4—82no—
564C4—84no—
565C4—82no—
566C4#345514yes—
567C2—42no—
568C2—81no—
569C2—81no—
570C2—162no—
571C2#3463511yescharacteristic
572C2#6052512yes—
573C2#6098912yes—
574C2#6101211yescharacteristic
575C2#6101911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35259C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34188C2semidirect
3. #29949C2semidirect
4. #29944C2semidirect
5. #29929C2semidirect
6. #29921C2semidirect
7. #29910C2semidirect
8. #29907C2semidirect
9. #29899C2semidirect
10. #23771C2semidirect
11. #23760C2semidirect
12. #23755C2semidirect
13. #21399C2semidirect
14. #21369C2semidirect
15. #20720C2semidirect

Maximal subgroups

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

subgroupindex
#352592
#341882
#299492
#299442
#299292
#299212
#299102
#299072
#298992
#237712
#237602
#237552
#213992
#213692
#207202