Finite Group Databasebeta

← #4380255

Subgroups — #4380255

Order 512.

2482 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic
3C2×#2625C211yesmaximal, characteristic
4C2×#2316C211yesmaximal, characteristic
5#16396C214yesmaximal
6#15860C214yesmaximal, complemented
7#15678C214yesmaximal, complemented
8#15618C214yesmaximal, complemented
9#15347C214yesmaximal, complemented
10#14907C214yesmaximal
11#14660C211yesmaximal, characteristic
12#14297C211yesmaximal, characteristic
13#13939C211yesmaximal, characteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18#2625—28no—
19#2583—28no—
20#2484—28no—
21C2×#400C2212yes—
22C2×#400C2212yes—
23C2×#398C2212yes—
24C2×#398C2212yes—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#397C2214yes—
28C2×#395C2212yes—
29#2316—28no—
30C2×#391C2212yes—
31C2×#389C2214yes—
32C2×#389C2214yes—
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#389C2214yes—
36C2×#387C2214yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#381C2212yes—
41C2×#381C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#1446C2212yes—
47#1445C2212yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69#1434C2214yes—
70#1434C2214yes—
71C23×Q8C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C2×C22⋊Q8—24no—
76C2×C22⋊Q8—24no—
77C2×C22⋊Q8—24no—
78C2×#117—24no—
79C2×#117—24no—
80C2×#117—24no—
81C2×#117—28no—
82C2×#117—44no—
83C2×#117—44no—
84C22×C4⋊C4—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4D418yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C42C2312yes—
109C22×C42C2314yes—
110C22×C42C2312yes—
111#400—28no—
112#400—28no—
113#398—28no—
114#398—28no—
115#398—28no—
116#398—48no—
117#397—216no—
118#395—28no—
119#391—28no—
120#389—216no—
121#389—216no—
122#389—28no—
123#389—216no—
124#389—48no—
125#387—216no—
126#385—28no—
127#385—28no—
128#385—28no—
129#381—24no—
130#381—28no—
131#380—24no—
132#380—28no—
133#380D418yes—
134C4×C22⋊C4—24no—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C25C2411yescharacteristic
161C22×Q8—22no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—42no—
165C22×Q8—42no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4C2×D414yes—
177C23×C4C2×D414yes—
178C23×C4C2×D414yes—
179C23×C4C2×D414yes—
180C23×C4#4114yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C22⋊Q8—416no—
194#117—432no—
195#117—416no—
196C2×C4⋊C4—232no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C2×D418yes—
240C2×C4⋊C4C2×D418yes—
241C2×C4⋊C4C2×D418yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4C2×D418yes—
259C2×C42—24no—
260C2×C42—28no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—28no—
264C2×C42—24no—
265C2×C42—44no—
266C2×C42—42no—
267C2×C42—42no—
268#95—24no—
269#95—28no—
270#95—24no—
271#95—24no—
272#95—28no—
273#95—28no—
274#95—24no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95C2×D418yes—
293C24—22no—
294C24—24no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24C22×D412yes—
299C24C22×D412yes—
300C24C2×#4112yes—
301C24C2511yescharacteristic
302C2×Q8—48no—
303C2×Q8—44no—
304C2×Q8—84no—
305C2×Q8—82no—
306C2×Q8—82no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—216no—
310C22×C4—28no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—216no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—28no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
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—24no—
370C22×C4—24no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—48no—
376C22×C4—84no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4#121116yes—
380C22×C4C22×D414yes—
381C22×C4C22×D418yes—
382C22×C4C22×D412yes—
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D414yes—
387C22×C4C2×#4112yes—
388C22×C4C2×#4112yes—
389C22×C4H(5,𝔽2)14yes—
390C22×C42−1+418yes—
391C22×C42−1+414yes—
392C4⋊C4—216no—
393C4⋊C4—216no—
394C4⋊C4—48no—
395C4⋊C4—416no—
396C4⋊C4—416no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—48no—
401C4⋊C4—416no—
402C4⋊C4—416no—
403C4⋊C4—84no—
404C4⋊C4—88no—
405C4⋊C4—88no—
406C4⋊C4—88no—
407C22⋊C4—216no—
408C22⋊C4—48no—
409C22⋊C4—48no—
410C22⋊C4—48no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C42—416no—
414C42—48no—
415C42—84no—
416C23—24no—
417C23—22no—
418C23—24no—
419C23—24no—
420C23—24no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23C2×#12114yes—
432C23C23×D411yescharacteristic
433C23C23×D411yescharacteristic
434C23C22×#4111yescharacteristic
435C23C2×H(5,𝔽2)12yes—
436C23C2×2−1+412yes—
437C23C2×2−1+414yes—
438C23#58512yes—
439C23#58512yes—
440Q8—88no—
441C2×C4—232no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—416no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
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—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—88no—
511C2×C4C2×#12114yes—
512C2×C4C2×#12114yes—
513C2×C4#53718yes—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—48no—
522C22C22×#12111yescharacteristic
523C22C2×#53712yes—
524C22#348411yescharacteristic
525C22#348512yes—
526C22#348512yes—
527C22#348612yes—
528C22#348611yescharacteristic
529C22#349111yescharacteristic
530C22#349111yescharacteristic
531C22#349212yes—
532C22#349912yes—
533C22#350912yes—
534C22#354114yes—
535C22#354714yes—
536C22#355514yes—
537C22#356314yes—
538C4—24no—
539C4—24no—
540C4—28no—
541C4—48no—
542C4—44no—
543C4—48no—
544C4—82no—
545C4—84no—
546C4—82no—
547C4—82no—
548C4—82no—
549C4—84no—
550C2—44no—
551C2#3522311yescharacteristic
552C2#3544711yescharacteristic
553C2#3566711yescharacteristic
554C2#4267514yes—
555C2#4721112yes—
556C2#4721412yes—
557C2#5018312yes—
558C2#5018412yes—
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
6. #15860C2semidirect
7. #15678C2semidirect
8. #15618C2semidirect
9. #15347C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#24842
C2×#26252
C2×#23162
#163962
#158602
#156782
#156182
#153472
#149072
#146602
#142972
#139392