Finite Group Databasebeta

← #4598752

Subgroups — #4598752

Order 512.

2232 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
1C2×#2822C211yesmaximal, characteristic, complemented
2C2×#2504C211yesmaximal, characteristic
3#17130C214yesmaximal, complemented
4#16748C214yesmaximal, complemented
5#16633C212yesmaximal, complemented
6#16596C212yesmaximal, complemented
7#16381C214yesmaximal, complemented
8#15945C212yesmaximal, complemented
9#15340C212yesmaximal
10#15326C214yesmaximal, complemented
11#14916C212yesmaximal, complemented
12#14821C212yesmaximal, complemented
13#14494C211yesmaximal, characteristic, complemented
14C22×#125C2212yes—
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2822—28no—
18#2504—28no—
19C2×#400C2214yes—
20C2×#399C2212yes—
21C2×#398C2212yes—
22C2×#398C2212yes—
23C2×#398C2214yes—
24C2×#398C2212yes—
25C2×#397C2214yes—
26C2×#394C2214yes—
27C2×#390C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2212yes—
30C2×#386C2214yes—
31C2×#385C2214yes—
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#1444C2212yes—
37#1443C2214yes—
38#1443C2214yes—
39#1443C2214yes—
40#1442C2212yes—
41#1442C2214yes—
42#1441C2214yes—
43#1441C2214yes—
44#1441C2212yes—
45#1441C2214yes—
46#1441C2212yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2212yes—
50#1439C2214yes—
51#1439C2212yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1438C2214yes—
60#1438C2212yes—
61#1437C2212yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2214yes—
65#1434C2212yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C2×#125—28no—
74C2×#125—48no—
75C2×#121—24no—
76C2×#121—42no—
77C2×#121—42no—
78C22×C4⋊C4—24no—
79C22×C4⋊C4—42no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106#400—216no—
107#399—44no—
108#398—28no—
109#398—28no—
110#398—216no—
111#398—44no—
112#397—216no—
113#394—48no—
114#390—24no—
115#390—24no—
116#390—44no—
117#386—216no—
118#385—48no—
119#382—24no—
120#382D418yes—
121#380—44no—
122C4×C22⋊C4—24no—
123C2×#95C2314yes—
124C2×#95C2312yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C25C2412yes—
148C25C2411yescharacteristic
149C22×D4—24no—
150C22×D4—28no—
151C22×D4—24no—
152C22×D4—22no—
153C22×D4—24no—
154C22×D4—41no—
155C22×D4—41no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—44no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—44no—
168C23×C4—44no—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2414yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180#125—432no—
181#121—416no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—416no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—44no—
251C2×C42C2×D414yes—
252#95—24no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—24no—
257#95—24no—
258#95—28no—
259#95—42no—
260#95—42no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—42no—
273#95—44no—
274#95—42no—
275#95C2×D4116yes—
276C24—21no—
277C24—22no—
278C24—21no—
279C24—22no—
280C24—24no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—22no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24H(5,𝔽2)12yes—
292C242−1+414yes—
293C24C2511yescharacteristic
294C2×D4—48no—
295C2×D4—416no—
296C2×D4—44no—
297C2×D4—44no—
298C2×D4—48no—
299C2×D4—44no—
300C2×D4—84no—
301C2×D4—84no—
302C2×D4—84no—
303C2×D4—88no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—28no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—44no—
365C22×C4—48no—
366C22×C4—44no—
367C22×C4—48no—
368C22×C4—88no—
369C22×C4—84no—
370C22×C4—88no—
371C22×C4C22×D414yes—
372C22×C4C22×D414yes—
373C22×C4C22×D414yes—
374C22×C4C22×D412yes—
375C22×C4H(5,𝔽2)12yes—
376C22×C4H(5,𝔽2)14yes—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—48no—
380C4⋊C4—44no—
381C4⋊C4—48no—
382C4⋊C4—416no—
383C4⋊C4—48no—
384C4⋊C4—416no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—88no—
388C4⋊C4—84no—
389C22⋊C4—48no—
390C22⋊C4—48no—
391C22⋊C4—416no—
392C22⋊C4—88no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C22⋊C4—88no—
396C22⋊C4—88no—
397C22⋊C4—88no—
398C22⋊C4—88no—
399C42—28no—
400C42—44no—
401C42—84no—
402C23—22no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—28no—
407C23—22no—
408C23—28no—
409C23—24no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—42no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—42no—
429C23—44no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23#56314yes—
439C23C23×D411yescharacteristic
440C23C2×H(5,𝔽2)12yes—
441C23C2×H(5,𝔽2)11yescharacteristic
442C23C2×H(5,𝔽2)11yescharacteristic
443C23C2×2−1+412yes—
444C23#58512yes—
445C23#58512yes—
446C23#58514yes—
447D4—88no—
448D4—816no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—416no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—816no—
513C2×C4#53514yes—
514C22—24no—
515C22—28no—
516C22—42no—
517C22—44no—
518C22—48no—
519C22—44no—
520C22—42no—
521C22—44no—
522C22—82no—
523C22—84no—
524C22—84no—
525C22—84no—
526C22—82no—
527C22—84no—
528C22—82no—
529C22—84no—
530C22—84no—
531C22C2×#53511yescharacteristic
532C22#348411yescharacteristic
533C22#348711yescharacteristic
534C22#348912yes—
535C22#350012yes—
536C22C2×#56311yescharacteristic
537C22#353312yes—
538C22#353412yes—
539C22#353412yes—
540C22#354114yes—
541C22#354511yescharacteristic
542C22#355112yes—
543C22#355114yes—
544C22#355314yes—
545C22#356112yes—
546C22#356714yes—
547C4—24no—
548C4—44no—
549C4—44no—
550C4—42no—
551C4—44no—
552C4—82no—
553C4—82no—
554C4—84no—
555C4—84no—
556C4—84no—
557C4—84no—
558C2—44no—
559C2—162no—
560C2#3671011yescharacteristic
561C2#4291012yes—
562C2#4613611yescharacteristic
563C2#5121112yes—
564C2#5197811yescharacteristic
565C2#5243112yes—
566C2#5739914yes—
567C2#5795612yes—
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×#2822C2semidirect
3. #17130C2semidirect
4. #16748C2semidirect
5. #16633C2semidirect
6. #16596C2semidirect
7. #16381C2semidirect
8. #15945C2semidirect
10. #15326C2semidirect
11. #14916C2semidirect
12. #14821C2semidirect
13. #14494C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28222
C2×#25042
#171302
#167482
#166332
#165962
#163812
#159452
#153402
#153262
#149162
#148212
#144942