Finite Group Databasebeta

← #4309179

Subgroups — #4309179

Order 512.

2298 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic
3#16598C214yesmaximal, complemented
4#16554C214yesmaximal, complemented
5#16324C214yesmaximal
6#16322C214yesmaximal
7#15221C212yesmaximal, complemented
8#15218C212yesmaximal, complemented
9#15152C212yesmaximal, complemented
10#15133C212yesmaximal, complemented
11#14697C212yesmaximal
12#14343C212yesmaximal
13#14340C211yesmaximal, characteristic
14C22×#125C2211yescharacteristic
15C22×#117C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C23×C42C2211yescharacteristic
18#2629—28no—
19C2×#400C2214yes—
20C2×#398C2214yes—
21C2×#398C2214yes—
22C2×#396C2214yes—
23#2296—28no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#385C2214yes—
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C4⋊C4C2212yes—
39#1445C2214yes—
40#1445C2214yes—
41#1445C2214yes—
42#1445C2214yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1440C2214yes—
57#1440C2214yes—
58#1439C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C22⋊#95C2214yes—
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C2×#125—24no—
72C2×#125—42no—
73C2×#125—42no—
74C2×#117—24no—
75C2×#117—24no—
76C2×#117—42no—
77C2×#117—42no—
78C2×#117D418yes—
79C2×#117Q818yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C42—24no—
102C22×C42—42no—
103C22×C42—42no—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#400—48no—
111#398—216no—
112#398—48no—
113#396—216no—
114#389—28no—
115#389—28no—
116#389—28no—
117#389—44no—
118#389—44no—
119#385—28no—
120#385—28no—
121#385—48no—
122#384—24no—
123#384—42no—
124#382—42no—
125#380—28no—
126#380—28no—
127C4×C4⋊C4—28no—
128C4×C4⋊C4—28no—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
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—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C25C2411yescharacteristic
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4C2×D414yes—
166C23×C4C2×Q814yes—
167C23×C4#4114yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C23×C4C2411yescharacteristic
180#125—416no—
181#117—232no—
182#117—416no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C2×D418yes—
230C2×C4⋊C4C2×Q818yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4C2×D418yes—
244C2×C22⋊C4C2×Q814yes—
245C2×C22⋊C4C2×Q814yes—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—24no—
257C2×C42—41no—
258C2×C42—41no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42—416no—
262C2×C42C2×D414yes—
263C2×C42C2×D414yes—
264C2×C42C2×Q818yes—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—44no—
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#95—44no—
287#95—44no—
288#95—44no—
289C24—22no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24C22×D412yes—
298C24C22×Q812yes—
299C24C2×#4112yes—
300C24C2511yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—28no—
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—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
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—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—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—48no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4C22⋊Q8116yes—
383C22×C4C22×D414yes—
384C22×C4C22×D414yes—
385C22×C4C22×D412yes—
386C22×C4C22×D412yes—
387C22×C4C22×Q814yes—
388C22×C4C22×Q814yes—
389C22×C4C22×Q812yes—
390C22×C4C22×Q812yes—
391C22×C4C2×#4112yes—
392C22×C4C2×#4112yes—
393C22×C4H(5,𝔽2)14yes—
394C22×C4H(5,𝔽2)12yes—
395C22×C4H(5,𝔽2)12yes—
396C22×C42−1+414yes—
397C22×C42−1+412yes—
398C22×C42−1+412yes—
399C4⋊C4—216no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—48no—
405C4⋊C4—416no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—88no—
411C4⋊C4—88no—
412C4⋊C4—88no—
413C4⋊C4—88no—
414C4⋊C4—88no—
415C22⋊C4—28no—
416C22⋊C4—28no—
417C22⋊C4—48no—
418C22⋊C4—416no—
419C22⋊C4—88no—
420C42—216no—
421C42—28no—
422C42—28no—
423C42—44no—
424C42—44no—
425C42—48no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—42no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23C2×C22⋊Q814yes—
449C23C23×D411yescharacteristic
450C23C23×Q811yescharacteristic
451C23C22×#4111yescharacteristic
452C23C2×H(5,𝔽2)11yescharacteristic
453C23C2×H(5,𝔽2)11yescharacteristic
454C23C2×H(5,𝔽2)12yes—
455C23C2×2−1+411yescharacteristic
456C23C2×2−1+411yescharacteristic
457C23C2×2−1+412yes—
458C23#58512yes—
459C23#58512yes—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—232no—
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—24no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—432no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
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—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4C2×C22⋊Q814yes—
539C2×C4C2×C22⋊Q814yes—
540C2×C4#53414yes—
541C2×C4#53614yes—
542C2×C4#53914yes—
543C2×C4#53914yes—
544C2×C4#54414yes—
545C2×C4#54414yes—
546C22—28no—
547C22—24no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—44no—
553C22—44no—
554C22—42no—
555C22—42no—
556C22—44no—
557C22C22×C22⋊Q811yescharacteristic
558C22C2×#53411yescharacteristic
559C22C2×#53611yescharacteristic
560C22C2×#53911yescharacteristic
561C22C2×#53911yescharacteristic
562C22C2×#54411yescharacteristic
563C22C2×#54411yescharacteristic
564C22#346012yes—
565C22#346112yes—
566C22#348812yes—
567C22#348912yes—
568C22#350312yes—
569C22#350412yes—
570C22#353612yes—
571C22#353912yes—
572C22#354112yes—
573C22#354212yes—
574C22#354811yescharacteristic
575C22#355412yes—
576C22#355512yes—
577C22#355611yescharacteristic
578C22#355711yescharacteristic
579C22#355711yescharacteristic
580C4—24no—
581C4—44no—
582C4—44no—
583C4—44no—
584C4—44no—
585C4—44no—
586C4—44no—
587C4—84no—
588C4—84no—
589C4—84no—
590C4—84no—
591C4#262718yes—
592C2—44no—
593C2C2×#262711yescharacteristic
594C2#3525712yes—
595C2#3726812yes—
596C2#3727411yescharacteristic
597C2#3730911yescharacteristic
598C2#3731112yes—
599C2#3737711yescharacteristic
600C2#3742711yescharacteristic
601C2#3752011yescharacteristic
602C2#3752211yescharacteristic
603C2#3752412yes—
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. #16598C2semidirect
4. #16554C2semidirect
7. #15221C2semidirect
8. #15218C2semidirect
9. #15152C2semidirect
10. #15133C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#22962
#165982
#165542
#163242
#163222
#152212
#152182
#151522
#151332
#146972
#143432
#143402