Finite Group Databasebeta

← #229335

Subgroups — #229335

Order 512.

765 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23182C211yesmaximal, characteristic, complemented
2#23093C211yesmaximal, characteristic
3#22962C211yesmaximal, characteristic, complemented
4#22665C211yesmaximal, characteristic, complemented
5C2×#1685C211yesmaximal, characteristic
6#9842C211yesmaximal, characteristic, complemented
7#9614C211yesmaximal, characteristic, complemented
8#9565C211yesmaximal, characteristic, complemented
9#9563C211yesmaximal, characteristic, complemented
10#8984C211yesmaximal, characteristic, complemented
11#8977C211yesmaximal, characteristic
12#8922C211yesmaximal, characteristic, complemented
13#8921C211yesmaximal, characteristic, complemented
14#8285C211yesmaximal, characteristic, complemented
15#8210C211yesmaximal, characteristic, complemented
16C2×#492C2211yescharacteristic
17C2×#474C2211yescharacteristic
18C2×#459C2211yescharacteristic
19C2×C4×SD16—21no—
20C2×#436C2211yescharacteristic
21C2×#428—21no—
22#2366C2211yescharacteristic
23#2365C2211yescharacteristic
24#2354C2211yescharacteristic
25D4×C4⋊C4C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1982C2211yescharacteristic
29#1981C2211yescharacteristic
30#1970—21no—
31#1968—21no—
32#1957C2211yescharacteristic
33#1957C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1942—21no—
39#1925—21no—
40#1918—21no—
41#1880—21no—
42#1879—21no—
43#1877—21no—
44#1862—21no—
45#1854—21no—
46#1852—21no—
47#1845C2211yescharacteristic
48#1809—21no—
49C4×#106C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×#102—21no—
54C4×#102—21no—
55C4×#102C2211yescharacteristic
56#1685C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×SD16—21no—
70C22×D8—21no—
71C22×M16—21no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×#127C2311yescharacteristic
74C2×C4×Q8—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4C2311yescharacteristic
79C2×#117—21no—
80C22×C4⋊C4—21no—
81#492—22no—
82#492C2×C414yescomplemented
83#474—22no—
84#474C2×C414yescomplemented
85#459—22no—
86#459C2×C414yes—
87C4×SD16—44no—
88#436—22no—
89#436C2×C414yescomplemented
90#428—44no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#107—21no—
95C2×#107—21no—
96C2×#107—21no—
97C2×#106—21no—
98C2×#106D411yescharacteristic
99C2×#106D411yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#103—21no—
105C2×#103—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—41no—
113C2×#102D411yescharacteristic
114C2×#102D411yescharacteristic
115C2×C22⋊C8—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122#391—21no—
123#390—21no—
124#390—21no—
125#390—21no—
126#389—21no—
127#389—21no—
128#388—41no—
129#387—41no—
130#386—41no—
131#385—41no—
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134#380—21no—
135#380—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2311yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—41no—
148#340D411yescharacteristic
149#340D411yescharacteristic
150#340D411yescharacteristic
151#340D411yescharacteristic
152#336—21no—
153#336—21no—
154#336—21no—
155#335C2×C414yescomplemented
156#332C2×C414yescomplemented
157#331C2×C414yescomplemented
158D4⋊C8—22no—
159#322—22no—
160C22×Q8—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C2×D411yescharacteristic
165C23×C4—21no—
166C23×C4—41no—
167C23×C4—41no—
168C2×SD16—24no—
169C2×SD16—24no—
170C2×SD16—42no—
171C2×SD16—42no—
172C2×SD16—81no—
173C2×SD16—81no—
174C2×D8—24no—
175C2×D8—81no—
176C2×D8—81no—
177C2×M16—24no—
178C2×M16—42no—
179C2×M16—42no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8C22×C412yes—
193#127—21no—
194#127—21no—
195#127—21no—
196#127C22×C412yes—
197C4×Q8—42no—
198C4×Q8—81no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—42no—
204C4×D4—81no—
205C4×D4—81no—
206C4×D4—81no—
207C4×D4C22×C412yes—
208C4×D4#4112yes—
209#117—42no—
210#117—81no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—81no—
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C22⋊C4—21no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42C2411yescharacteristic
243#107—24no—
244#107—42no—
245#107—42no—
246#107—81no—
247#107—81no—
248#107—81no—
249#106—22no—
250#106—42no—
251#106D814yescomplemented
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8C22×C412yes—
258C4⋊C8C22×C412yes—
259C4⋊C8C22×C412yes—
260#103—42no—
261#103—81no—
262#102—24no—
263#102—24no—
264#102—22no—
265#102—41no—
266#102—42no—
267#102—42no—
268#102—81no—
269#102—81no—
270#102—82no—
271C22⋊C8—24no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—42no—
277C4×C8—81no—
278C4×C8—81no—
279C4×C8C22×C412yes—
280C4×C8#4112yes—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—41no—
285#95—41no—
286#95—41no—
287#95—41no—
288C24—21no—
289C24—41no—
290C2×Q8—22no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—81no—
316C2×D4—81no—
317C2×D4C4×D412yes—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—82no—
337C22×C4—82no—
338C22×C4—161no—
339C22×C4—161no—
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343SD16—48no—
344SD16—84no—
345D8—84no—
346M16—48no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
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—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8C4×D412yes—
371C2×C8C4×D412yes—
372C2×C8C2×D812yes—
373C2×C8#13512yes—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—81no—
399C4⋊C4—81no—
400C4⋊C4C4×D412yes—
401C4⋊C4C2×D812yes—
402C4⋊C4C2×D812yes—
403C4⋊C4#13712yes—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—161no—
411C22⋊C4—161no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—82no—
419C42—82no—
420C42C23×C411yescharacteristic
421C42C2×#4111yescharacteristic
422C42C2×#4111yescharacteristic
423C422−1+411yescharacteristic
424C23—22no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23D4211yescharacteristic
439Q8—42no—
440Q8—42no—
441Q8—81no—
442Q8—81no—
443D4—22no—
444D4—22no—
445D4—41no—
446D4—41no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—42no—
451D4—82no—
452D4—81no—
453D4—81no—
454D4—81no—
455D4—81no—
456D4—161no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—164no—
494C2×C4—164no—
495C2×C4C2×C4×D411yescharacteristic
496C2×C4C2×C4×D411yescharacteristic
497C2×C4#52011yescharacteristic
498C2×C4#54711yescharacteristic
499C2×C4#54811yescharacteristic
500C2×C4#55311yescharacteristic
501C2×C4C22×D811yescharacteristic
502C2×C4C2×#13511yescharacteristic
503C2×C4C2×#13711yescharacteristic
504C2×C4#57511yescharacteristic
505C2×C4#57611yescharacteristic
506C2×C4#57611yescharacteristic
507C2×C4#57711yescharacteristic
508C2×C4#57711yescharacteristic
509C8—22no—
510C8—41no—
511C8—41no—
512C8—42no—
513C8—42no—
514C8—42no—
515C8—81no—
516C8—81no—
517C8C4×D814yes—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—81no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—161no—
529C22—161no—
530C22—161no—
531C22—161no—
532C22#236111yescharacteristic
533C22#326611yescharacteristic
534C22#326811yescharacteristic
535C22D4×D811yescharacteristic
536C22#328411yescharacteristic
537C22#328811yescharacteristic
538C22#328911yescharacteristic
539C4—21no—
540C4—21no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—81no—
546C4—81no—
547C4—81no—
548C4—82no—
549C4—161no—
550C4—161no—
551C4C2×C4×D811yescharacteristic
552C4C4×#13711yescharacteristic
553C4#294611yescharacteristic
554C4#294611yescharacteristic
555C4#335111yescharacteristic
556C4#335711yescharacteristic
557C4#335811yescharacteristic
558C4#336111yescharacteristic
559C2—82no—
560C2—162no—
561C2#1975711yescharacteristic
562C2#1975911yescharacteristic
563C2#1987711yescharacteristic
564C2#1987911yescharacteristic
565C2#2299511yescharacteristic
566C2#2303511yescharacteristic
567C2#2304911yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#459C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #23182C2semidirect
3. #22962C2semidirect
4. #22665C2semidirect
6. #9842C2semidirect
7. #9614C2semidirect
8. #9565C2semidirect
9. #9563C2semidirect
10. #8984C2semidirect
12. #8922C2semidirect
13. #8921C2semidirect
14. #8285C2semidirect
15. #8210C2semidirect
56. #1685C4semidirect
82. #492C2×C4semidirect
84. #474C2×C4semidirect
89. #436C2×C4semidirect
155. #335C2×C4semidirect
156. #332C2×C4semidirect
157. #331C2×C4semidirect
251. #106D8semidirect

Maximal subgroups

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

subgroupindex
#231822
#230932
#229622
#226652
C2×#16852
#98422
#96142
#95652
#95632
#89842
#89772
#89222
#89212
#82852
#82102