Finite Group Databasebeta

← #216039

Subgroups — #216039

Order 512.

729 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic
3#20812C211yesmaximal, characteristic
4#9907C211yesmaximal, characteristic, complemented
5#9906C211yesmaximal, characteristic, complemented
6#9901C211yesmaximal, characteristic, complemented
7#9900C211yesmaximal, characteristic, complemented
8#9900C211yesmaximal, characteristic, complemented
9#9900C211yesmaximal, characteristic, complemented
10#9898C211yesmaximal, characteristic, complemented
11#9896C211yesmaximal, characteristic, complemented
12#8729C211yesmaximal, characteristic
13#8728C211yesmaximal, characteristic
14#8727C211yesmaximal, characteristic
15#8726C211yesmaximal, characteristic
16C2×#433—21no—
17#2868C2211yescharacteristic
18#2859C2211yescharacteristic
19#2859C2211yescharacteristic
20#2859C2211yescharacteristic
21#2629—21no—
22#2628—21no—
23#2570—21no—
24#2570C412yes—
25#2550—21no—
26#2550C412yes—
27#2441—21no—
28#2441C412yes—
29#2439—21no—
30#2439C412yes—
31#2331—21no—
32#2330—21no—
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35C4×#117C2211yescharacteristic
36#1923—21no—
37#1923—21no—
38#1916—21no—
39#1916—21no—
40#1842C2211yescharacteristic
41#1842C2211yescharacteristic
42#1842C2211yescharacteristic
43#1832—21no—
44#1831C2211yescharacteristic
45#1798—21no—
46#1797C2211yescharacteristic
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1768—21no—
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55#1296C2211yescharacteristic
56#1296C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C2×C4⋊Q8—21no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8C2×C412yes—
74C2×#125—21no—
75C2×#125—21no—
76C2×#125C2×C412yes—
77C2×#117—21no—
78C2×#117C2×C412yes—
79C2×#117D411yescharacteristic
80C2×#117D411yescharacteristic
81C22×C42C2311yescharacteristic
82#433—44no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105#400—22no—
106#399—22no—
107#398—22no—
108#398—22no—
109#396—22no—
110#396—22no—
111#395—22no—
112#393—22no—
113#391—22no—
114#391—22no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—22no—
124#389—22no—
125#389C2×C412yes—
126#389C2×C412yes—
127#389C2×C412yes—
128#389C2×C412yes—
129#387—22no—
130#387—22no—
131#387—22no—
132#387—22no—
133#384—21no—
134#384—21no—
135#384C2×C412yes—
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2311yescharacteristic
140#382—21no—
141#382—21no—
142#382—21no—
143#382C2×C412yes—
144#382C2×C412yes—
145#380—21no—
146#380—21no—
147#380—21no—
148#380C2×C412yes—
149C4×C4⋊C4—21no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4C2×C412yes—
153C4×C4⋊C4C2×C411yescharacteristic
154C4×C4⋊C4C2×C411yescharacteristic
155C4×C4⋊C4D411yescharacteristic
156C4×C4⋊C4D411yescharacteristic
157C4×C22⋊C4C2×C411yescharacteristic
158C4×C22⋊C4C2×C411yescharacteristic
159C4×C22⋊C4D411yescharacteristic
160C4×C22⋊C4D411yescharacteristic
161#376D411yescharacteristic
162#376D411yescharacteristic
163C43C2311yescharacteristic
164C43C2311yescharacteristic
165#336—21no—
166#336—21no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175C22×Q8—22no—
176C23×C4C22×C411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C4⋊Q8—42no—
188C4⋊Q8—42no—
189C4⋊Q8—81no—
190C4⋊Q8—81no—
191C4⋊Q8—82no—
192#125—42no—
193#125—42no—
194#125—81no—
195#125—81no—
196#125—82no—
197#117—22no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—81no—
202#117—81no—
203#117—82no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22⋊C411yescharacteristic
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4C22⋊C411yescharacteristic
232C2×C22⋊C4C22⋊C411yescharacteristic
233C2×C22⋊C4C22⋊C411yescharacteristic
234C2×C22⋊C4C22⋊C411yescharacteristic
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2411yescharacteristic
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—42no—
274C4⋊C8—42no—
275C4⋊C8—42no—
276C4⋊C8—42no—
277C4⋊C8—81no—
278C4⋊C8—81no—
279C22⋊C8—42no—
280C22⋊C8—42no—
281C22⋊C8—81no—
282C22⋊C8—81no—
283#97—42no—
284C4×C8—42no—
285C4×C8—81no—
286C4×C8—81no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95C22⋊C411yescharacteristic
298#95C22⋊C411yescharacteristic
299#95C22⋊C411yescharacteristic
300#95C22⋊C411yescharacteristic
301#95C22⋊C411yescharacteristic
302#95C22⋊C411yescharacteristic
303#95C22⋊C411yescharacteristic
304#95C22⋊C411yescharacteristic
305C24C2×C22⋊C411yescharacteristic
306C2×Q8—44no—
307C2×Q8—82no—
308C2×Q8—82no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C2×C22⋊C411yescharacteristic
335C22×C4C2×C22⋊C411yescharacteristic
336C22×C4C2×C22⋊C411yescharacteristic
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C411yescharacteristic
340C22×C4C23×C411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—82no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—41no—
428C42—82no—
429C42—82no—
430C42#13512yes—
431C42#13512yes—
432C42AGL(1,ℤ/8)12yes—
433C42#13712yes—
434C422−1+411yescharacteristic
435C422−1+411yescharacteristic
436C422−1+411yescharacteristic
437C422−1+411yescharacteristic
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23C22×C22⋊C411yescharacteristic
446Q8—164no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4#41612yes—
501C2×C4#41612yes—
502C2×C4#41712yes—
503C2×C4#41712yes—
504C2×C4#51911yescharacteristic
505C2×C4#51911yescharacteristic
506C2×C4#53611yescharacteristic
507C2×C4#53611yescharacteristic
508C2×C4#53611yescharacteristic
509C2×C4#53611yescharacteristic
510C2×C4C2×#13511yescharacteristic
511C2×C4C2×#13511yescharacteristic
512C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
513C2×C4C2×#13711yescharacteristic
514C2×C4#57611yescharacteristic
515C2×C4#57711yescharacteristic
516C2×C4#57711yescharacteristic
517C2×C4#57811yescharacteristic
518C8—81no—
519C8—81no—
520C8—81no—
521C8—81no—
522C8—81no—
523C8—81no—
524C8—81no—
525C8—81no—
526C22—41no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22#231011yescharacteristic
534C22#288811yescharacteristic
535C22#288911yescharacteristic
536C22C2×#41611yescharacteristic
537C22C2×#41711yescharacteristic
538C22#289711yescharacteristic
539C22#289811yescharacteristic
540C4—22no—
541C4—22no—
542C4—21no—
543C4—21no—
544C4—22no—
545C4—22no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4#322811yescharacteristic
558C4#322811yescharacteristic
559C4#323111yescharacteristic
560C4#323111yescharacteristic
561C4#324111yescharacteristic
562C4#324211yescharacteristic
563C4#324311yescharacteristic
564C4#324411yescharacteristic
565C2—81no—
566C2#1772411yescharacteristic
567C2#1772511yescharacteristic
568C2#1776411yescharacteristic
569C2#1776511yescharacteristic
570C2#2183211yescharacteristic
571C2#2183511yescharacteristic
572C2#2184311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9907C2semidirect
5. #9906C2semidirect
6. #9901C2semidirect
7. #9900C2semidirect
8. #9900C2semidirect
9. #9900C2semidirect
10. #9898C2semidirect
11. #9896C2semidirect

Maximal subgroups

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

subgroupindex
#352942
#219222
#208122
#99072
#99062
#99012
#99002
#98982
#98962
#87292
#87282
#87272
#87262