Finite Group Databasebeta

← #4598766

Subgroups — #4598766

Order 512.

2368 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2844C211yesmaximal, characteristic, complemented
2C2×#2504C211yesmaximal, characteristic
3#17106C214yesmaximal, complemented
4#16879C214yesmaximal, complemented
5#16692C212yesmaximal, complemented
6#16556C212yesmaximal, complemented
7#16351C214yesmaximal, complemented
8#15946C212yesmaximal, complemented
9#15674C212yesmaximal
10#15325C214yesmaximal, complemented
11#15017C212yesmaximal, complemented
12#14720C212yesmaximal, complemented
13#14494C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2212yes—
15C22×#121C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17#2844—28no—
18#2504—28no—
19C2×#400C2212yes—
20C2×#400C2214yes—
21C2×#399C2214yes—
22C2×#397C2214yes—
23C2×#397C2214yes—
24C2×#397C2212yes—
25C2×#396C2214yes—
26C2×#395C2212yes—
27C2×#395C2214yes—
28C2×#395C2212yes—
29C2×#393C2214yes—
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#386C2214yes—
34C2×#385C2214yes—
35C2×#382C2212yes—
36C2×#382C2211yescharacteristic
37C2×#382C2211yescharacteristic
38C2×C4×C22⋊C4C2211yescharacteristic
39#1444C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1444C2212yes—
43#1443C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2212yes—
48#1441C2212yes—
49#1441C2212yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2212yes—
54#1439C2214yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1438C2212yes—
61#1437C2212yes—
62#1436C2212yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2214yes—
66#1434C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C23×Q8C2312yes—
70C23×D4C2312yes—
71C23×D4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C2×C22⋊Q8—44no—
75C2×C22⋊Q8—48no—
76C2×#121—24no—
77C2×#121—42no—
78C2×#121—42no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107#400—28no—
108#400—216no—
109#399—48no—
110#397—216no—
111#397—48no—
112#397—44no—
113#396—48no—
114#395—28no—
115#395—216no—
116#395—44no—
117#393—48no—
118#390—24no—
119#390—24no—
120#390—44no—
121#386—216no—
122#385—48no—
123#382—24no—
124#382—28no—
125#382D418yes—
126C4×C22⋊C4—24no—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C25C2412yes—
151C25C2411yescharacteristic
152C22×Q8—24no—
153C22×Q8—44no—
154C22×Q8—42no—
155C22×Q8—44no—
156C22×Q8—42no—
157C22×D4—24no—
158C22×D4—28no—
159C22×D4—24no—
160C22×D4—22no—
161C22×D4—24no—
162C22×D4—41no—
163C22×D4—41no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4—44no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4C2414yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2414yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C22⋊Q8—432no—
189#121—416no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—416no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D414yes—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
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×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—44no—
259C2×C42C2×D414yes—
260#95—24no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—24no—
265#95—28no—
266#95—24no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—42no—
274#95—42no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—42no—
280#95—42no—
281#95—44no—
282#95C2×D4116yes—
283C24—21no—
284C24—24no—
285C24—21no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—22no—
293C24—22no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24H(5,𝔽2)12yes—
299C242−1+414yes—
300C24C2511yescharacteristic
301C2×Q8—48no—
302C2×Q8—84no—
303C2×Q8—88no—
304C2×Q8—84no—
305C2×Q8—88no—
306C2×D4—48no—
307C2×D4—416no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—48no—
311C2×D4—44no—
312C2×D4—84no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—88no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—28no—
320C22×C4—21no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
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—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—44no—
376C22×C4—48no—
377C22×C4—44no—
378C22×C4—48no—
379C22×C4—88no—
380C22×C4—84no—
381C22×C4—88no—
382C22×C4C22×D414yes—
383C22×C4C22×D414yes—
384C22×C4C22×D414yes—
385C22×C4C22×D412yes—
386C22×C4H(5,𝔽2)18yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C42−1+414yes—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—44no—
394C4⋊C4—416no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C4⋊C4—88no—
399C4⋊C4—88no—
400C4⋊C4—88no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C22⋊C4—416no—
404C22⋊C4—416no—
405C22⋊C4—88no—
406C22⋊C4—88no—
407C22⋊C4—88no—
408C22⋊C4—88no—
409C22⋊C4—88no—
410C22⋊C4—88no—
411C42—28no—
412C42—44no—
413C42—84no—
414C23—22no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—28no—
419C23—22no—
420C23—28no—
421C23—24no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—44no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23#56314yes—
451C23C23×D411yescharacteristic
452C23C2×H(5,𝔽2)11yescharacteristic
453C23C2×H(5,𝔽2)11yescharacteristic
454C23C2×H(5,𝔽2)14yes—
455C23C2×2−1+412yes—
456C23C2×2−1+412yes—
457C23#58512yes—
458C23#58512yes—
459Q8—816no—
460D4—88no—
461D4—816no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—28no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—28no—
478C2×C4—416no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
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—816no—
526C2×C4#53614yes—
527C22—24no—
528C22—28no—
529C22—44no—
530C22—42no—
531C22—44no—
532C22—44no—
533C22—48no—
534C22—42no—
535C22—84no—
536C22—82no—
537C22—82no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—82no—
543C22—84no—
544C22C2×#53611yescharacteristic
545C22#348411yescharacteristic
546C22#348711yescharacteristic
547C22#348912yes—
548C22#349812yes—
549C22C2×#56311yescharacteristic
550C22#353212yes—
551C22#353512yes—
552C22#353512yes—
553C22#354114yes—
554C22#354511yescharacteristic
555C22#354912yes—
556C22#355214yes—
557C22#355914yes—
558C22#356014yes—
559C22#356112yes—
560C4—24no—
561C4—44no—
562C4—42no—
563C4—48no—
564C4—44no—
565C4—44no—
566C4—82no—
567C4—84no—
568C4—84no—
569C4—82no—
570C4—84no—
571C2—44no—
572C2—162no—
573C2#3671111yescharacteristic
574C2#4291412yes—
575C2#4613611yescharacteristic
576C2#5120712yes—
577C2#5190211yescharacteristic
578C2#5241412yes—
579C2#5751014yes—
580C2#5772512yes—
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×#2844C2semidirect
3. #17106C2semidirect
4. #16879C2semidirect
5. #16692C2semidirect
6. #16556C2semidirect
7. #16351C2semidirect
8. #15946C2semidirect
10. #15325C2semidirect
11. #15017C2semidirect
12. #14720C2semidirect
13. #14494C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28442
C2×#25042
#171062
#168792
#166922
#165562
#163512
#159462
#156742
#153252
#150172
#147202
#144942