Finite Group Databasebeta

← #4380256

Subgroups — #4380256

Order 512.

2656 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2591C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic, complemented
3C2×#2624C211yesmaximal, characteristic
4C2×#2316C211yesmaximal, characteristic, complemented
5#16397C214yesmaximal, complemented
6#15888C214yesmaximal, complemented
7#15675C214yesmaximal, complemented
8#15621C214yesmaximal, complemented
9#15458C214yesmaximal, complemented
10#14890C214yesmaximal, complemented
11#14659C211yesmaximal, characteristic
12#14297C211yesmaximal, characteristic, complemented
13#13939C211yesmaximal, characteristic, complemented
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18#2624—28no—
19#2591—28no—
20#2484—28no—
21C2×#400C2212yes—
22C2×#400C2212yes—
23C2×#399C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#396C2214yes—
27C2×#394C2214yes—
28C2×#394C2212yes—
29#2316—28no—
30C2×#390C2212yes—
31C2×#389C2214yes—
32C2×#389C2214yes—
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#387C2214yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#385C2212yes—
40C2×#381C2212yes—
41C2×#381C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46#1445C2214yes—
47#1445C2212yes—
48#1443C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1438C2214yes—
63#1438C2212yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2212yes—
70#1434C2214yes—
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×#121—24no—
77C2×#121—24no—
78C2×#121—24no—
79C2×#117—24no—
80C2×#117—24no—
81C2×#117—24no—
82C2×#117—28no—
83C2×#117—44no—
84C2×#117—44no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4D418yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C42C2312yes—
113C22×C42C2314yes—
114C22×C42C2312yes—
115#400—28no—
116#400—28no—
117#399—28no—
118#398—28no—
119#397—28no—
120#396—48no—
121#394—28no—
122#394—216no—
123#390—28no—
124#389—216no—
125#389—28no—
126#389—216no—
127#388—216no—
128#388—48no—
129#387—216no—
130#386—28no—
131#386—28no—
132#385—28no—
133#381—24no—
134#381—28no—
135#380—24no—
136#380—28no—
137#380D418yes—
138C4×C22⋊C4—24no—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C25C2412yes—
161C25C2411yescharacteristic
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—42no—
169C22×D4—42no—
170C22×D4—44no—
171C22×D4—44no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4C2×D414yes—
183C23×C4C2×D414yes—
184C23×C4C2×D414yes—
185C23×C4C2×D414yes—
186C23×C4#4114yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2411yescharacteristic
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198#121—416no—
199#117—432no—
200#117—416no—
201C2×C4⋊C4—232no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
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—44no—
235C2×C4⋊C4C2×D418yes—
236C2×C4⋊C4C2×D418yes—
237C2×C4⋊C4C2×D418yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4C2×D418yes—
273C2×C42—24no—
274C2×C42—28no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—28no—
278C2×C42—24no—
279C2×C42—44no—
280C2×C42—42no—
281C2×C42—42no—
282#95—24no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95C2×D418yes—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—24no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24—24no—
317C24C22×D412yes—
318C24C22×D412yes—
319C24C2×#4112yes—
320C24C2511yescharacteristic
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—44no—
326C2×D4—84no—
327C2×D4—82no—
328C2×D4—82no—
329C2×D4—88no—
330C2×D4—88no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—216no—
334C22×C4—28no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—216no—
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—28no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
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—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—48no—
396C22×C4—84no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4#121116yes—
400C22×C4C22×D414yes—
401C22×C4C22×D418yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D414yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4H(5,𝔽2)14yes—
410C22×C42−1+414yes—
411C4⋊C4—216no—
412C4⋊C4—216no—
413C4⋊C4—48no—
414C4⋊C4—416no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—416no—
419C4⋊C4—84no—
420C4⋊C4—88no—
421C4⋊C4—88no—
422C4⋊C4—88no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—416no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—416no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C42—416no—
436C42—48no—
437C42—84no—
438C23—24no—
439C23—22no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—42no—
454C23—44no—
455C23—42no—
456C23—44no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23C2×#12114yes—
475C23C23×D411yescharacteristic
476C23C23×D411yescharacteristic
477C23C22×#4111yescharacteristic
478C23C2×H(5,𝔽2)12yes—
479C23C2×2−1+412yes—
480C23#58512yes—
481C23#58512yes—
482C23#58514yes—
483D4—88no—
484D4—816no—
485C2×C4—232no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—416no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—88no—
550C2×C4C2×#12114yes—
551C2×C4C2×#12114yes—
552C2×C4#53718yes—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—48no—
561C22—84no—
562C22—84no—
563C22—82no—
564C22—84no—
565C22—84no—
566C22—82no—
567C22—82no—
568C22—84no—
569C22—84no—
570C22C22×#12111yescharacteristic
571C22C2×#53712yes—
572C22#348411yescharacteristic
573C22#348512yes—
574C22#348512yes—
575C22#348611yescharacteristic
576C22#348712yes—
577C22#349111yescharacteristic
578C22#349111yescharacteristic
579C22#349212yes—
580C22#350012yes—
581C22#351012yes—
582C22#353914yes—
583C22#355114yes—
584C22#356514yes—
585C22#356514yes—
586C4—24no—
587C4—24no—
588C4—28no—
589C4—48no—
590C4—44no—
591C4—82no—
592C4—82no—
593C4—84no—
594C4—82no—
595C4—84no—
596C4—84no—
597C2—44no—
598C2—162no—
599C2#3522311yescharacteristic
600C2#3544711yescharacteristic
601C2#3567711yescharacteristic
602C2#4267414yes—
603C2#4721612yes—
604C2#4721712yes—
605C2#5018212yes—
606C2#5018512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2484C2semidirect
4. C2×#2316C2semidirect
5. #16397C2semidirect
6. #15888C2semidirect
7. #15675C2semidirect
8. #15621C2semidirect
9. #15458C2semidirect
10. #14890C2semidirect
12. #14297C2semidirect
13. #13939C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25912
C2×#24842
C2×#26242
C2×#23162
#163972
#158882
#156752
#156212
#154582
#148902
#146592
#142972
#139392