Finite Group Databasebeta

← #900742

Subgroups — #900742

Order 512.

2358 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
1#17076C212yesmaximal, complemented
2#17003C212yesmaximal, complemented
3#16057C214yesmaximal, complemented
4#15440C214yesmaximal, complemented
5#15108C212yesmaximal, complemented
6#14819C214yesmaximal, complemented
7#14707C212yesmaximal, complemented
8#14513C212yesmaximal
9#14440C211yesmaximal, characteristic
10#14440C211yesmaximal, characteristic
11#14043C214yesmaximal, complemented
12#14015C212yesmaximal
13C4×#390C211yesmaximal, characteristic
14C22×C4×D4C2211yescharacteristic
15C23×C4⋊C4C2212yes—
16C2×#400C2214yes—
17C2×#390C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#386C2212yes—
23C2×#386C2212yes—
24C2×#385C2214yes—
25C2×#385C2212yes—
26C2×#385C2214yes—
27C2×#385C2214yes—
28C2×#382C2212yes—
29C2×#382C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4C2214yes—
32C2×C43C2211yescharacteristic
33#1445C2214yes—
34#1444C2214yes—
35#1444C2214yes—
36#1444C2214yes—
37#1443C2214yes—
38#1442C2212yes—
39#1442C2214yes—
40#1442C2212yes—
41#1442C2214yes—
42#1441C2212yes—
43#1441C2212yes—
44#1441C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1437C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62#1434C2212yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C22⋊#95C2212yes—
67C23×D4C2311yescharacteristic
68C24×C4C2312yes—
69C24×C4C2312yes—
70C24×C4C2312yes—
71C2×C4×D4—24no—
72C2×C4×D4—42no—
73C2×C4×D4—42no—
74C22×C4⋊C4—48no—
75C22×C4⋊C4—44no—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C42C2311yescharacteristic
97C22×C42C2312yes—
98C22×C42C2312yes—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102#400—48no—
103#390—24no—
104#390Q818yes—
105#389—44no—
106#389—44no—
107#389—44no—
108#386—28no—
109#386—28no—
110#385—216no—
111#385—44no—
112#385—48no—
113#385—48no—
114#382—28no—
115#382—44no—
116C4×C4⋊C4—42no—
117C4×C22⋊C4—48no—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C43—42no—
142C25C2412yes—
143C22×D4—41no—
144C22×D4—41no—
145C22×D4—41no—
146C22×D4—41no—
147C22×D4C2×Q814yes—
148C22×D4#4114yes—
149C22×D4#4114yes—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—28no—
154C23×C4—24no—
155C23×C4—28no—
156C23×C4—44no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—44no—
161C23×C4—42no—
162C23×C4—44no—
163C23×C4—42no—
164C23×C4—44no—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2414yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2414yes—
172C23×C4C2411yescharacteristic
173C23×C4C2412yes—
174C23×C4C2411yescharacteristic
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C4×D4—416no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—432no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4C2×Q814yes—
215C2×C4⋊C4#4118yes—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4C2×Q8116yes—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—22no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—41no—
248C2×C42—42no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42—41no—
252C2×C42—44no—
253C2×C42C2×Q814yes—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—42no—
265#95—42no—
266#95—44no—
267#95—44no—
268#95—42no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277C24—24no—
278C24—24no—
279C24—22no—
280C24—24no—
281C24—22no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24C22×Q814yes—
286C24C2×#4114yes—
287C24C2×#4114yes—
288C24C2511yescharacteristic
289C2×D4—44no—
290C2×D4—44no—
291C2×D4—44no—
292C2×D4—44no—
293C2×D4—44no—
294C2×D4—44no—
295C2×D4#125116yes—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—48no—
359C22×C4—416no—
360C22×C4—416no—
361C22×C4—48no—
362C22×C4—416no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—88no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—88no—
371C22×C4—88no—
372C22×C4—84no—
373C22×C4C22×Q814yes—
374C22×C4C22×Q812yes—
375C22×C4C22×Q814yes—
376C22×C4C2×#4118yes—
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4C2×#4114yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C4H(5,𝔽2)14yes—
382C22×C4H(5,𝔽2)14yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C42−1+414yes—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—416no—
392C4⋊C4—416no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—88no—
396C4⋊C4—84no—
397C4⋊C4—88no—
398C22⋊C4—232no—
399C22⋊C4—416no—
400C22⋊C4—416no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C22⋊C4—84no—
404C22⋊C4—88no—
405C42—216no—
406C42—28no—
407C42—44no—
408C42—48no—
409C42—44no—
410C42—84no—
411C23—24no—
412C23—24no—
413C23—28no—
414C23—28no—
415C23—28no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—28no—
421C23—28no—
422C23—28no—
423C23—44no—
424C23—42no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—44no—
429C23—42no—
430C23—42no—
431C23—44no—
432C23—44no—
433C23C2×#12518yes—
434C23C23×Q811yescharacteristic
435C23C22×#4111yescharacteristic
436C23C22×#4111yescharacteristic
437C23C22×#4112yes—
438C23C2×H(5,𝔽2)11yescharacteristic
439C23C2×H(5,𝔽2)12yes—
440C23C2×H(5,𝔽2)11yescharacteristic
441C23C2×H(5,𝔽2)12yes—
442C23C2×2−1+412yes—
443C23#58511yescharacteristic
444C23#58511yescharacteristic
445D4—416no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—432no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
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—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—816no—
519C2×C4—816no—
520C2×C4C2×#12514yes—
521C2×C4#55418yes—
522C2×C4H(3,𝔽4)14yes—
523C22—28no—
524C22—216no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—48no—
529C22—44no—
530C22—44no—
531C22—48no—
532C22—44no—
533C22—48no—
534C22C22×#12511yescharacteristic
535C22#345911yescharacteristic
536C22#345912yes—
537C22#345911yescharacteristic
538C22C2×#55412yes—
539C22#348111yescharacteristic
540C22#348112yes—
541C22#348212yes—
542C22#349011yescharacteristic
543C22#349111yescharacteristic
544C22#349212yes—
545C22#350111yescharacteristic
546C22#350111yescharacteristic
547C22#350312yes—
548C22#351712yes—
549C22#351912yes—
550C22#352012yes—
551C22C2×H(3,𝔽4)11yescharacteristic
552C22#353712yes—
553C22#354311yescharacteristic
554C22#355011yescharacteristic
555C22#355912yes—
556C22#356112yes—
557C4—24no—
558C4—24no—
559C4—28no—
560C4—44no—
561C4—48no—
562C4—44no—
563C4—42no—
564C4—82no—
565C4—84no—
566C4—84no—
567C4—82no—
568C4—84no—
569C2—48no—
570C2#3499411yescharacteristic
571C2#3629012yes—
572C2#3725411yescharacteristic
573C2#3876112yes—
574C2#3881012yes—
575C2#3966912yes—
576C2#4600411yescharacteristic
577C2#4608911yescharacteristic
578C2#5230111yescharacteristic
579C2#5946611yescharacteristic
580C2#6004611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17076C2semidirect
2. #17003C2semidirect
3. #16057C2semidirect
4. #15440C2semidirect
5. #15108C2semidirect
6. #14819C2semidirect
7. #14707C2semidirect
11. #14043C2semidirect

Maximal subgroups

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

subgroupindex
#170762
#170032
#160572
#154402
#151082
#148192
#147072
#145132
#144402
#140432
#140152
C4×#3902