Finite Group Databasebeta

← #878984

Subgroups — #878984

Order 512.

2134 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17076C211yesmaximal, characteristic, complemented
2#17070C211yesmaximal, characteristic, complemented
3#17003C212yesmaximal, complemented
4#15786C214yesmaximal
5#15292C214yesmaximal
6#14940C214yesmaximal, complemented
7#14893C212yesmaximal
8#14697C212yesmaximal
9#14604C212yesmaximal, complemented
10#14441C212yesmaximal, complemented
11#14211C214yesmaximal, complemented
12#13938C212yesmaximal
13#13756C211yesmaximal, characteristic
14C22×#117C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C2×#400C2212yes—
18C2×#399C2212yes—
19C2×#398C2214yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2214yes—
24C2×#387C2214yes—
25C2×#385C2214yes—
26C2×#383C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2212yes—
30#1445C2214yes—
31#1444C2214yes—
32#1444C2214yes—
33#1444C2214yes—
34#1442C2214yes—
35#1442C2214yes—
36#1442C2214yes—
37#1441C2212yes—
38#1441C2214yes—
39#1441C2214yes—
40#1441C2214yes—
41#1441C2214yes—
42#1441C2212yes—
43#1441C2212yes—
44#1441C2214yes—
45#1439C2212yes—
46#1439C2214yes—
47#1439C2212yes—
48#1439C2212yes—
49#1437C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55#1436C2214yes—
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2212yes—
59C4⋊#95C2212yes—
60C4⋊#95C2211yescharacteristic
61C4⋊#95C2211yescharacteristic
62#1434C2212yes—
63#1434C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C4×#95C2214yes—
67C24×C4C2311yescharacteristic
68C24×C4C2312yes—
69C2×#117—24no—
70C2×#117—42no—
71C2×#117—42no—
72C2×#117—42no—
73C2×#117—42no—
74C2×#117Q818yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C42—24no—
96C22×C42—42no—
97C22×C42—42no—
98C22×C42C2312yes—
99C22×C42C2312yes—
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2314yes—
103C22×C42C2311yescharacteristic
104#400—28no—
105#399—44no—
106#398—216no—
107#389—28no—
108#389—44no—
109#389—44no—
110#389—48no—
111#387—48no—
112#385—48no—
113#383—44no—
114#380—28no—
115#380—44no—
116C4×C4⋊C4—44no—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2312yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C25C2411yescharacteristic
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4C2×Q814yes—
157C23×C4C2414yes—
158C23×C4C2412yes—
159C23×C4C2414yes—
160C23×C4C2414yes—
161C23×C4C2412yes—
162C23×C4C2412yes—
163C23×C4C2414yes—
164C23×C4C2412yes—
165C23×C4C2412yes—
166C23×C4C2411yescharacteristic
167C23×C4C2411yescharacteristic
168C23×C4C2412yes—
169#117—416no—
170#117—416no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4C2×Q814yes—
213C2×C4⋊C4C2×Q814yes—
214C2×C4⋊C4#4118yes—
215C2×C4⋊C4#4114yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4C2×Q818yes—
228C2×C42—24no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—42no—
236C2×C42—41no—
237C2×C42—41no—
238C2×C42—42no—
239C2×C42—41no—
240C2×C42—41no—
241C2×C42—416no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42C2×Q818yes—
246C2×C42#4114yes—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—24no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—42no—
266#95—42no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—42no—
272C24—21no—
273C24—22no—
274C24—21no—
275C24—21no—
276C24—22no—
277C24—21no—
278C24—22no—
279C24—22no—
280C24C22×Q812yes—
281C24H(5,𝔽2)12yes—
282C242−1+412yes—
283C24C2511yescharacteristic
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—28no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—216no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
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—24no—
343C22×C4—24no—
344C22×C4—22no—
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—48no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—48no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4C22×Q814yes—
361C22×C4C22×Q814yes—
362C22×C4C22×Q812yes—
363C22×C4C22×Q812yes—
364C22×C4C2×#4114yes—
365C22×C4C2×#4118yes—
366C22×C4C2×#4112yes—
367C22×C4C2×#4112yes—
368C22×C4C2×#4114yes—
369C22×C4C2×#4114yes—
370C22×C4H(5,𝔽2)12yes—
371C22×C4H(5,𝔽2)12yes—
372C22×C4H(5,𝔽2)14yes—
373C22×C42−1+414yes—
374C4⋊C4—216no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—44no—
382C4⋊C4—48no—
383C4⋊C4—416no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C4⋊C4—88no—
388C4⋊C4—88no—
389C4⋊C4—88no—
390C22⋊C4—48no—
391C22⋊C4—416no—
392C22⋊C4—48no—
393C22⋊C4—88no—
394C42—216no—
395C42—44no—
396C42—48no—
397C42—48no—
398C42—44no—
399C42—88no—
400C23—22no—
401C23—22no—
402C23—24no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—22no—
407C23—22no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—42no—
413C23—42no—
414C23—41no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—41no—
419C23—42no—
420C23—41no—
421C23—41no—
422C23#54414yes—
423C23C23×Q811yescharacteristic
424C23C22×#4111yescharacteristic
425C23C22×#4111yescharacteristic
426C23C22×#4112yes—
427C23C2×H(5,𝔽2)11yescharacteristic
428C23C2×H(5,𝔽2)11yescharacteristic
429C23C2×H(5,𝔽2)11yescharacteristic
430C23C2×H(5,𝔽2)12yes—
431C23C2×2−1+411yescharacteristic
432C23C2×2−1+412yes—
433C23#58512yes—
434C2×C4—232no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—24no—
465C2×C4—432no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
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—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4#53314yes—
510C2×C4#54314yes—
511C2×C4#55318yes—
512C22—28no—
513C22—24no—
514C22—44no—
515C22—42no—
516C22—42no—
517C22—44no—
518C22—42no—
519C22—44no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22C2×#53311yescharacteristic
524C22C2×#54311yescharacteristic
525C22C2×#54411yescharacteristic
526C22#346012yes—
527C22C2×#55312yes—
528C22Q8×#4112yes—
529C22#348111yescharacteristic
530C22#348212yes—
531C22#349011yescharacteristic
532C22#349011yescharacteristic
533C22#349212yes—
534C22#349211yescharacteristic
535C22#349211yescharacteristic
536C22#349512yes—
537C22#350312yes—
538C22#351712yes—
539C22#352012yes—
540C22#352911yescharacteristic
541C22#353412yes—
542C22#353612yes—
543C22#354911yescharacteristic
544C22#355011yescharacteristic
545C22#356312yes—
546C4—28no—
547C4—24no—
548C4—24no—
549C4—44no—
550C4—44no—
551C4—48no—
552C4—44no—
553C4—44no—
554C4—84no—
555C4—84no—
556C4—82no—
557C4—84no—
558C2—44no—
559C2#3505211yescharacteristic
560C2#3648112yes—
561C2#3730611yescharacteristic
562C2#3794112yes—
563C2#3844712yes—
564C2#3880312yes—
565C2#4623511yescharacteristic
566C2#4631811yescharacteristic
567C2#5228811yescharacteristic
568C2#5922011yescharacteristic
569C2#6004611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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. #17070C2semidirect
3. #17003C2semidirect
6. #14940C2semidirect
9. #14604C2semidirect
10. #14441C2semidirect
11. #14211C2semidirect

Maximal subgroups

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

subgroupindex
#170762
#170702
#170032
#157862
#152922
#149402
#148932
#146972
#146042
#144412
#142112
#139382
#137562