Finite Group Databasebeta

← #878985

Subgroups — #878985

Order 512.

2246 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
1#17080C211yesmaximal, characteristic, complemented
2#17072C211yesmaximal, characteristic, complemented
3#16933C212yesmaximal, complemented
4#15995C214yesmaximal
5#15671C214yesmaximal
6#14939C214yesmaximal, complemented
7#14887C212yesmaximal
8#14702C212yesmaximal
9#14538C212yesmaximal, complemented
10#14471C212yesmaximal, complemented
11#14214C214yesmaximal, complemented
12#13938C212yesmaximal
13#13756C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C42C2211yescharacteristic
18C2×#395C2212yes—
19C2×#395C2214yes—
20C2×#393C2212yes—
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#391C2214yes—
24C2×#391C2212yes—
25C2×#387C2214yes—
26C2×#383C2212yes—
27C2×#380C2212yes—
28C2×C4×C4⋊C4C2212yes—
29#1446C2214yes—
30#1446C2214yes—
31#1445C2214yes—
32#1444C2214yes—
33#1443C2214yes—
34#1443C2214yes—
35#1442C2214yes—
36#1442C2214yes—
37#1442C2214yes—
38#1441C2214yes—
39#1441C2212yes—
40#1441C2212yes—
41#1441C2212yes—
42#1441C2214yes—
43#1441C2214yes—
44#1439C2214yes—
45#1439C2212yes—
46#1439C2212yes—
47#1439C2212yes—
48#1437C2214yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1436C2214yes—
53#1436C2214yes—
54#1436C2214yes—
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2212yes—
57C4⋊#95C2212yes—
58C4⋊#95C2211yescharacteristic
59C4⋊#95C2211yescharacteristic
60C4⋊#95C2211yescharacteristic
61#1434C2212yes—
62#1434C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C4×#95C2214yes—
67C23×Q8C2312yes—
68C23×Q8C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—44no—
73C2×C22⋊Q8—44no—
74C2×#117—24no—
75C2×#117—42no—
76C2×#117—42no—
77C2×#117—42no—
78C2×#117—42no—
79C2×#117Q818yescomplemented
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C42—24no—
97C22×C42—42no—
98C22×C42—42no—
99C22×C42C2312yes—
100C22×C42C2312yes—
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2314yes—
104C22×C42C2311yescharacteristic
105#395—28no—
106#395—216no—
107#393—44no—
108#391—28no—
109#391—44no—
110#391—44no—
111#391—48no—
112#387—48no—
113#383—44no—
114#380—44no—
115C4×C4⋊C4—44no—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2312yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C25C2411yescharacteristic
144C22×Q8—24no—
145C22×Q8—24no—
146C22×Q8—28no—
147C22×Q8—24no—
148C22×Q8—24no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8—44no—
155C22×Q8—42no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4C2×Q814yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2414yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2412yes—
182C22⋊Q8—432no—
183#117—416no—
184#117—416no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C2×Q814yes—
215C2×C4⋊C4C2×Q814yes—
216C2×C4⋊C4#4114yes—
217C2×C4⋊C4#4118yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4C2×Q818yes—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—24no—
237C2×C42—42no—
238C2×C42—41no—
239C2×C42—41no—
240C2×C42—42no—
241C2×C42—41no—
242C2×C42—41no—
243C2×C42—416no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—44no—
247C2×C42C2×Q818yes—
248C2×C42#4114yes—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—24no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—42no—
267#95—44no—
268#95—42no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—42no—
275#95—44no—
276C24—21no—
277C24—22no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—21no—
282C24—22no—
283C24—22no—
284C24C22×Q812yes—
285C24H(5,𝔽2)12yes—
286C242−1+412yes—
287C24C2511yescharacteristic
288C2×Q8—48no—
289C2×Q8—416no—
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8—48no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—88no—
299C2×Q8—84no—
300C22×C4—28no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—216no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4C22×Q814yes—
377C22×C4C22×Q814yes—
378C22×C4C22×Q812yes—
379C22×C4C22×Q812yes—
380C22×C4C2×#4114yes—
381C22×C4C2×#4118yes—
382C22×C4C2×#4112yes—
383C22×C4C2×#4112yes—
384C22×C4C2×#4114yes—
385C22×C4C2×#4114yes—
386C22×C4H(5,𝔽2)14yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C4H(5,𝔽2)12yes—
389C22×C42−1+414yes—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—416no—
396C4⋊C4—416no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—416no—
405C22⋊C4—88no—
406C42—216no—
407C42—44no—
408C42—48no—
409C42—48no—
410C42—44no—
411C42—88no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—24no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—42no—
425C23—42no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—42no—
432C23—41no—
433C23—41no—
434C23#54414yes—
435C23C23×Q811yescharacteristic
436C23C22×#4111yescharacteristic
437C23C22×#4111yescharacteristic
438C23C22×#4112yes—
439C23C2×H(5,𝔽2)11yescharacteristic
440C23C2×H(5,𝔽2)11yescharacteristic
441C23C2×H(5,𝔽2)11yescharacteristic
442C23C2×H(5,𝔽2)12yes—
443C23C2×2−1+411yescharacteristic
444C23C2×2−1+412yes—
445C23#58512yes—
446Q8—816no—
447Q8—816no—
448C2×C4—232no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—28no—
478C2×C4—24no—
479C2×C4—432no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
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—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4#53314yes—
524C2×C4#54314yes—
525C2×C4#55618yes—
526C22—28no—
527C22—24no—
528C22—42no—
529C22—44no—
530C22—44no—
531C22—42no—
532C22—44no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22C2×#53311yescharacteristic
538C22C2×#54311yescharacteristic
539C22C2×#54411yescharacteristic
540C22#346012yes—
541C22C2×#55612yes—
542C22Q8×#4112yes—
543C22#348111yescharacteristic
544C22#348112yes—
545C22#349012yes—
546C22#349011yescharacteristic
547C22#349011yescharacteristic
548C22#349211yescharacteristic
549C22#349211yescharacteristic
550C22#349412yes—
551C22#350312yes—
552C22#351812yes—
553C22#352312yes—
554C22#352911yescharacteristic
555C22#353512yes—
556C22#353612yes—
557C22#354911yescharacteristic
558C22#355011yescharacteristic
559C22#356112yes—
560C4—28no—
561C4—24no—
562C4—24no—
563C4—44no—
564C4—44no—
565C4—44no—
566C4—48no—
567C4—44no—
568C4—84no—
569C4—84no—
570C4—84no—
571C4—82no—
572C2—44no—
573C2#3505211yescharacteristic
574C2#3648012yes—
575C2#3730611yescharacteristic
576C2#3795712yes—
577C2#3844612yes—
578C2#3876312yes—
579C2#4623211yescharacteristic
580C2#4631611yescharacteristic
581C2#5228811yescharacteristic
582C2#5924811yescharacteristic
583C2#6004511yescharacteristic
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. #17080C2semidirect
2. #17072C2semidirect
3. #16933C2semidirect
6. #14939C2semidirect
9. #14538C2semidirect
10. #14471C2semidirect
11. #14214C2semidirect
79. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
#170802
#170722
#169332
#159952
#156712
#149392
#148872
#147022
#145382
#144712
#142142
#139382
#137562