Finite Group Databasebeta

← #4466153

Subgroups — #4466153

Order 512.

2420 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2724C211yesmaximal, characteristic, complemented
2C2×#2620C211yesmaximal, characteristic
3#17199C214yesmaximal, complemented
4#16993C212yesmaximal, complemented
5#16856C214yesmaximal, complemented
6#16269C214yesmaximal, complemented
7#15950C212yesmaximal
8#15790C212yesmaximal, complemented
9#15430C212yesmaximal, complemented
10#15392C214yesmaximal, complemented
11#15262C212yesmaximal, complemented
12#14907C212yesmaximal, complemented
13#14503C211yesmaximal, characteristic, complemented
14C22×#124C2211yescharacteristic
15C22×#123C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#2724—28no—
20#2620—28no—
21C2×#399C2212yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#397C2214yes—
25C2×#397C2212yes—
26C2×#396C2214yes—
27C2×#395C2214yes—
28C2×#394C2214yes—
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#389C2212yes—
32C2×#386C2212yes—
33C2×#385C2214yes—
34C2×#385C2212yes—
35C2×#385C2214yes—
36C2×#385C2214yes—
37C2×#384C2211yescharacteristic
38C2×#384C2212yes—
39#1445C2212yes—
40#1444C2214yes—
41#1444C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1441C2214yes—
48#1441C2214yes—
49#1441C2212yes—
50#1441C2214yes—
51#1440C2212yes—
52#1440C2214yes—
53#1440C2214yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2212yes—
59#1439C2214yes—
60#1438C2214yes—
61#1438C2212yes—
62#1438C2212yes—
63#1437C2214yes—
64#1437C2212yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2212yes—
68#1434C2212yes—
69C23×Q8C2311yescharacteristic
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C2×#124—24no—
75C2×#124—44no—
76C2×#123—28no—
77C2×#123—48no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—42no—
80C2×C22⋊Q8—42no—
81C2×#121—24no—
82C2×#121—42no—
83C2×#121—42no—
84C2×#117—24no—
85C2×#117—44no—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113#399—44no—
114#398—28no—
115#398—44no—
116#397—216no—
117#397—44no—
118#396—216no—
119#395—48no—
120#394—216no—
121#391—24no—
122#390D418yes—
123#389—28no—
124#386—44no—
125#385—216no—
126#385—48no—
127#385—48no—
128#385—44no—
129#384—24no—
130#384—44no—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C25C2412yes—
153C25C2411yescharacteristic
154C22×Q8—22no—
155C22×Q8—24no—
156C22×Q8—41no—
157C22×Q8—41no—
158C22×Q8—42no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—41no—
163C22×D4—41no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4C2×D414yes—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4C2412yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2414yes—
189C23×C4C2411yescharacteristic
190#124—416no—
191#123—432no—
192C22⋊Q8—416no—
193#121—416no—
194#117—416no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C2×D414yes—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4C2×D4116yes—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—44no—
268C2×C42C2×D414yes—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—24no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—42no—
288#95—42no—
289#95—42no—
290C24—21no—
291C24—22no—
292C24—24no—
293C24—22no—
294C24—22no—
295C24—24no—
296C24—21no—
297C24—22no—
298C24—24no—
299C24—24no—
300C24—24no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24C22×D414yes—
305C24H(5,𝔽2)12yes—
306C24C2511yescharacteristic
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—84no—
311C2×Q8—84no—
312C2×D4—28no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—84no—
318C2×D4—84no—
319C2×D4—88no—
320C2×D4—84no—
321C2×D4—88no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—28no—
326C22×C4—22no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—48no—
384C22×C4—84no—
385C22×C4—84no—
386C22×C4—88no—
387C22×C4—88no—
388C22×C4C22×D414yes—
389C22×C4C22×D414yes—
390C22×C4C22×D412yes—
391C22×C4H(5,𝔽2)14yes—
392C22×C4H(5,𝔽2)12yes—
393C22×C4H(5,𝔽2)14yes—
394C22×C42−1+418yes—
395C4⋊C4—44no—
396C4⋊C4—48no—
397C4⋊C4—48no—
398C4⋊C4—48no—
399C4⋊C4—44no—
400C4⋊C4—416no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—88no—
404C4⋊C4—88no—
405C4⋊C4—88no—
406C4⋊C4—88no—
407C22⋊C4—416no—
408C22⋊C4—48no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—416no—
413C22⋊C4—416no—
414C22⋊C4—416no—
415C22⋊C4—48no—
416C22⋊C4—416no—
417C22⋊C4—88no—
418C22⋊C4—84no—
419C42—28no—
420C42—44no—
421C42—48no—
422C23—24no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—28no—
427C23—28no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—41no—
432C23—41no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23—44no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23C23×D411yescharacteristic
459C23C2×H(5,𝔽2)12yes—
460C23C2×H(5,𝔽2)11yescharacteristic
461C23C2×H(5,𝔽2)11yescharacteristic
462C23C2×H(5,𝔽2)12yes—
463C23C2×2−1+414yes—
464C23#58514yes—
465Q8—88no—
466D4—88no—
467D4—816no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—416no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—816no—
532C2×C4#53514yes—
533C22—48no—
534C22—42no—
535C22—44no—
536C22—42no—
537C22—44no—
538C22—42no—
539C22—44no—
540C22—48no—
541C22—48no—
542C22—82no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22—82no—
547C22—84no—
548C22—84no—
549C22C2×#53511yescharacteristic
550C22#348411yescharacteristic
551C22#348411yescharacteristic
552C22#348512yes—
553C22#349912yes—
554C22#353312yes—
555C22#353712yes—
556C22#353712yes—
557C22#353812yes—
558C22#354012yes—
559C22#354114yes—
560C22#354114yes—
561C22#354412yes—
562C22#355414yes—
563C22#355914yes—
564C4—24no—
565C4—44no—
566C4—44no—
567C4—44no—
568C4—42no—
569C4—48no—
570C4—82no—
571C4—84no—
572C4—82no—
573C4—84no—
574C4—84no—
575C2—84no—
576C2—82no—
577C2#3671711yescharacteristic
578C2#4264812yes—
579C2#4635812yes—
580C2#5154211yescharacteristic
581C2#5205411yescharacteristic
582C2#5646014yes—
583C2#5809514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2724C2semidirect
3. #17199C2semidirect
4. #16993C2semidirect
5. #16856C2semidirect
6. #16269C2semidirect
8. #15790C2semidirect
9. #15430C2semidirect
10. #15392C2semidirect
11. #15262C2semidirect
12. #14907C2semidirect
13. #14503C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27242
C2×#26202
#171992
#169932
#168562
#162692
#159502
#157902
#154302
#153922
#152622
#149072
#145032