Finite Group Databasebeta

← #2034634

Subgroups — #2034634

Order 512.

2466 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2734C211yesmaximal, characteristic, complemented
2C2×#2857C211yesmaximal, characteristic
3#17078C212yesmaximal, complemented
4#16303C214yesmaximal
5#16000C214yesmaximal, complemented
6#15790C212yesmaximal, complemented
7#15762C212yesmaximal, complemented
8#15625C214yesmaximal, complemented
9#15430C212yesmaximal
10#15265C212yesmaximal
11#14697C212yesmaximal
12#14471C211yesmaximal, characteristic, complemented
13#14370C214yesmaximal
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C42C2211yescharacteristic
19#2857—28no—
20#2734—28no—
21C2×#399C2212yes—
22C2×#398C2212yes—
23C2×#395C2214yes—
24C2×#395C2212yes—
25C2×#395C2214yes—
26C2×#395C2214yes—
27C2×#393C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#389C2214yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#387C2214yes—
34C2×#385C2214yes—
35C2×#384C2211yescharacteristic
36C2×#384C2212yes—
37C2×#380C2214yes—
38C2×#380C2212yes—
39C2×#380C2212yes—
40#1446C2212yes—
41#1445C2214yes—
42#1445C2212yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1437C2212yes—
58#1437C2212yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2212yes—
64#1436C2214yes—
65#1436C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C23×Q8C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C2×C4⋊Q8—24no—
75C2×C4⋊Q8—28no—
76C2×C4⋊Q8—44no—
77C2×C4⋊Q8—48no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—42no—
81C2×C22⋊Q8—42no—
82C2×C22⋊Q8—42no—
83C2×C22⋊Q8—42no—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C42—24no—
105C22×C42—44no—
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111#399—44no—
112#398—28no—
113#395—216no—
114#395—48no—
115#395—48no—
116#395—44no—
117#393—44no—
118#391—24no—
119#391D418yes—
120#389—28no—
121#389—216no—
122#389—44no—
123#387—216no—
124#385—48no—
125#384—24no—
126#384—44no—
127#380—216no—
128#380—44no—
129#380Q8116yescomplemented
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C25C2411yescharacteristic
154C22×Q8—22no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—41no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—44no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—44no—
168C22×Q8C2×D414yes—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4C2×Q818yes—
181C23×C4C2412yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2411yescharacteristic
190C23×C4C2412yes—
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C4⋊Q8—416no—
194C4⋊Q8—432no—
195C22⋊Q8—416no—
196C22⋊Q8—416no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4C2×D414yes—
243C2×C4⋊C4C2×Q818yes—
244C2×C4⋊C4C2×Q818yes—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4C2×Q8116yes—
257C2×C42—22no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—416no—
263C2×C42—44no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—44no—
268C2×C42C2×D414yes—
269#95—28no—
270#95—24no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—42no—
287#95—44no—
288#95—44no—
289#95—42no—
290#95C2×D4116yes—
291#95C2×Q8116yes—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24C22×Q814yes—
300C24H(5,𝔽2)12yes—
301C24C2511yescharacteristic
302C2×Q8—28no—
303C2×Q8—44no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—48no—
308C2×Q8—44no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×Q8—84no—
312C2×Q8—84no—
313C2×Q8—84no—
314C2×Q8—84no—
315C2×Q8—88no—
316C2×Q8—88no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—28no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—216no—
354C22×C4—216no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4C22×D414yes—
389C22×C4C22×D414yes—
390C22×C4C22×D412yes—
391C22×C4C22×D414yes—
392C22×C4C22×Q818yes—
393C22×C4C22×Q818yes—
394C22×C4C22×Q814yes—
395C22×C4C22×Q814yes—
396C22×C4H(5,𝔽2)12yes—
397C22×C4H(5,𝔽2)14yes—
398C22×C4H(5,𝔽2)14yes—
399C22×C42−1+414yes—
400C22×C42−1+414yes—
401C4⋊C4—216no—
402C4⋊C4—44no—
403C4⋊C4—48no—
404C4⋊C4—416no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—416no—
408C4⋊C4—44no—
409C4⋊C4—416no—
410C4⋊C4—416no—
411C4⋊C4—84no—
412C4⋊C4—84no—
413C4⋊C4—84no—
414C4⋊C4—88no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C22⋊C4—416no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—88no—
421C42—28no—
422C42—44no—
423C42—48no—
424C42—48no—
425C42—48no—
426C23—24no—
427C23—24no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—24no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23Q8214yes—
447C23C23×D411yescharacteristic
448C23C23×Q812yes—
449C23C2×H(5,𝔽2)12yes—
450C23C2×H(5,𝔽2)12yes—
451C23C2×H(5,𝔽2)11yescharacteristic
452C23C2×H(5,𝔽2)11yescharacteristic
453C23C2×2−1+412yes—
454C23C2×2−1+412yes—
455C23#58512yes—
456Q8—88no—
457Q8—88no—
458Q8—816no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—28no—
487C2×C4—24no—
488C2×C4—432no—
489C2×C4—416no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
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—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
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—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4#53514yes—
531C2×C4#54418yes—
532C2×C4D4×Q818yes—
533C22—24no—
534C22—28no—
535C22—44no—
536C22—42no—
537C22—42no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22C2×#53511yescharacteristic
543C22C2×#54412yes—
544C22C2×D4×Q812yes—
545C22C2×Q8211yescharacteristic
546C22#348411yescharacteristic
547C22#348411yescharacteristic
548C22#348912yes—
549C22#350112yes—
550C22#350112yes—
551C22#350312yes—
552C22#350312yes—
553C22#352911yescharacteristic
554C22#353512yes—
555C22#353712yes—
556C22#353712yes—
557C22#353812yes—
558C22#354411yescharacteristic
559C22#355011yescharacteristic
560C22#355012yes—
561C22#355412yes—
562C22#355912yes—
563C4—28no—
564C4—28no—
565C4—24no—
566C4—42no—
567C4—44no—
568C4—44no—
569C4—48no—
570C4—48no—
571C4—84no—
572C4—82no—
573C4—82no—
574C4—84no—
575C2—44no—
576C2#3671711yescharacteristic
577C2#3730812yes—
578C2#3768512yes—
579C2#3959612yes—
580C2#3962312yes—
581C2#4579212yes—
582C2#4622811yescharacteristic
583C2#5318312yes—
584C2#5915111yescharacteristic
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. C2×#2734C2semidirect
3. #17078C2semidirect
5. #16000C2semidirect
6. #15790C2semidirect
7. #15762C2semidirect
8. #15625C2semidirect
12. #14471C2semidirect
129. #380Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#27342
C2×#28572
#170782
#163032
#160002
#157902
#157622
#156252
#154302
#152652
#146972
#144712
#143702