Finite Group Databasebeta

← #626903

Subgroups — #626903

Order 512.

2610 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic
2C2×#2316C211yesmaximal, characteristic
3#15811C214yesmaximal, complemented
4#15807C214yesmaximal, complemented
5#15766C214yesmaximal, complemented
6#15698C214yesmaximal, complemented
7#14766C214yesmaximal
8#14702C211yesmaximal, characteristic
9#14686C211yesmaximal, characteristic
10#14353C211yesmaximal, characteristic
11#14281C211yesmaximal, characteristic
12#13878C214yesmaximal
13C2×#1436C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17C23×C42C2211yescharacteristic
18#2627—28no—
19C2×#400C2212yes—
20C2×#398C2214yes—
21C2×#398C2214yes—
22C2×#395C2212yes—
23#2316—28no—
24C2×#391C2212yes—
25C2×#389C2214yes—
26C2×#389C2214yes—
27C2×#385C2212yes—
28C2×#384C2212yes—
29C2×#381C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2211yescharacteristic
34C2×#380C2214yes—
35C2×#380C2214yes—
36C22×#95C2211yescharacteristic
37C22×#95C2212yes—
38C22×#95C2212yes—
39#1446C2212yes—
40#1445C2214yes—
41#1445C2212yes—
42#1444C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1436C2214yes—
56#1436C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2212yes—
61#1436C2212yes—
62#1436C2212yes—
63#1436C2212yes—
64C4⋊#95C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C24×C4C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C24×C4C2311yescharacteristic
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C2×#117—28no—
79C2×#117—48no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—24no—
82C22×C4⋊C4D418yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C42—24no—
102C22×C42—44no—
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2314yes—
107#400—28no—
108#398—216no—
109#398—48no—
110#395—28no—
111#391—44no—
112#389—216no—
113#389—216no—
114#385—28no—
115#384—44no—
116#381—28no—
117#380—28no—
118#380—24no—
119#380—24no—
120#380—24no—
121#380—48no—
122#380—48no—
123C2×#95—24no—
124C2×#95—44no—
125C2×#95Q8116yescomplemented
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C25C2411yescharacteristic
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—42no—
156C22×Q8—42no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4C2×D414yes—
179C23×C4C2×D414yes—
180C23×C4C2×D414yes—
181C23×C4C2×Q818yes—
182C23×C4C2×Q818yes—
183C23×C4C2×Q818yes—
184C23×C4#4114yes—
185C23×C4#4114yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C22⋊Q8—416no—
199#117—432no—
200C2×C4⋊C4—232no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4C2×D418yes—
242C2×C4⋊C4C2×D418yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C42—22no—
253C2×C42—24no—
254C2×C42—28no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—416no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42—42no—
262C2×C42—42no—
263#95—28no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—28no—
268#95—28no—
269#95—28no—
270#95—24no—
271#95—24no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95C2×Q8116yes—
287#95C2×Q8116yes—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24C22×D412yes—
293C24C22×Q814yes—
294C24C2×#4112yes—
295C24C2×#4112yes—
296C24H(5,𝔽2)14yes—
297C24C2511yescharacteristic
298C2×Q8—48no—
299C2×Q8—44no—
300C2×Q8—82no—
301C2×Q8—82no—
302C2×Q8—84no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—216no—
306C22×C4—216no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—216no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—28no—
336C22×C4—216no—
337C22×C4—216no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
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—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—44no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—44no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—82no—
382C22×C4—82no—
383C22×C4—84no—
384C22×C4—84no—
385C22×C4—84no—
386C22×C4—84no—
387C22×C4—84no—
388C22×C4#123116yes—
389C22×C4C4⋊Q8116yes—
390C22×C4C22×D414yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D414yes—
394C22×C4C22×Q814yes—
395C22×C4C22×Q818yes—
396C22×C4C22×Q818yes—
397C22×C4C22×Q814yes—
398C22×C4C2×#4112yes—
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4112yes—
402C22×C4H(5,𝔽2)14yes—
403C22×C42−1+414yes—
404C4⋊C4—216no—
405C4⋊C4—216no—
406C4⋊C4—416no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C4⋊C4—88no—
413C4⋊C4—88no—
414C4⋊C4—88no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C4⋊C4—88no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—48no—
421C22⋊C4—48no—
422C42—28no—
423C42—48no—
424C42—44no—
425C42—416no—
426C23—24no—
427C23—24no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23C2×#12314yes—
443C23C2×C4⋊Q814yes—
444C23#53514yes—
445C23#55418yes—
446C23#55418yes—
447C23C23×D411yescharacteristic
448C23C23×Q812yes—
449C23C22×#4111yescharacteristic
450C23C22×#4111yescharacteristic
451C23C2×H(5,𝔽2)12yes—
452C23C2×H(5,𝔽2)12yes—
453C23C2×2−1+412yes—
454C23#58512yes—
455C23#58512yes—
456Q8—88no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—232no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—28no—
480C2×C4—24no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—24no—
484C2×C4—416no—
485C2×C4—432no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
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—44no—
514C2×C4—42no—
515C2×C4—42no—
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—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—88no—
529C2×C4—816no—
530C2×C4C2×#12314yes—
531C2×C4C2×#12314yes—
532C2×C4C2×C4⋊Q818yes—
533C2×C4#54418yes—
534C22—24no—
535C22—28no—
536C22—28no—
537C22—24no—
538C22—28no—
539C22—28no—
540C22—28no—
541C22—28no—
542C22#250218yes—
543C22C22×#12311yescharacteristic
544C22C22×C4⋊Q811yescharacteristic
545C22C2×#53511yescharacteristic
546C22C2×#54412yes—
547C22#345812yes—
548C22C2×#55412yes—
549C22C2×#55412yes—
550C22#349011yescharacteristic
551C22#349111yescharacteristic
552C22#349512yes—
553C22#349612yes—
554C22#349712yes—
555C22#350112yes—
556C22#350212yes—
557C22#350512yes—
558C22#350911yescharacteristic
559C22#351011yescharacteristic
560C22#353412yes—
561C22#353612yes—
562C22#355912yes—
563C22#356012yes—
564C4—24no—
565C4—24no—
566C4—28no—
567C4—28no—
568C4—44no—
569C4—48no—
570C4—44no—
571C4—82no—
572C4—82no—
573C4—84no—
574C4—84no—
575C4—84no—
576C2—28no—
577C2C2×#250211yescharacteristic
578C2#3516712yes—
579C2#3536011yescharacteristic
580C2#3567311yescharacteristic
581C2#3674112yes—
582C2#3787312yes—
583C2#3787712yes—
584C2#3949212yes—
585C2#3949612yes—
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
3. #15811C2semidirect
4. #15807C2semidirect
5. #15766C2semidirect
6. #15698C2semidirect
125. C2×#95Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C2×#23162
#158112
#158072
#157662
#156982
#147662
#147022
#146862
#143532
#142812
#138782
C2×#14362