Finite Group Databasebeta

← #620508

Subgroups — #620508

Order 512.

2446 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#15942C214yesmaximal, complemented
4#15766C214yesmaximal, complemented
5#15488C214yesmaximal
6#15305C214yesmaximal
7#15218C211yesmaximal, characteristic, complemented
8#15218C211yesmaximal, characteristic, complemented
9#15196C211yesmaximal, characteristic, complemented
10#15151C211yesmaximal, characteristic, complemented
11#14697C212yesmaximal
12#14264C212yesmaximal
13#14046C212yesmaximal, complemented
14#14045C212yesmaximal, complemented
15#13777C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2629—28no—
21C2×#398C2214yes—
22C2×#398C2214yes—
23C2×#396C2214yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#389C2214yes—
30C2×#385C2214yes—
31#2285—28no—
32C2×#384C2211yescharacteristic
33C2×#384C2212yes—
34C2×#382C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44#1445C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1444C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1437C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65C4⋊#95C2211yescharacteristic
66C4⋊#95C2212yes—
67C4⋊#95C2214yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C22⋊#95C2214yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×#125—24no—
77C2×#125—44no—
78C2×#117—24no—
79C2×#117—44no—
80C2×#117—44no—
81C2×#117D418yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C42—24no—
104C22×C42—24no—
105C22×C42—24no—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2314yes—
115#398—216no—
116#398—48no—
117#396—216no—
118#389—28no—
119#389—28no—
120#389—28no—
121#389—48no—
122#389—44no—
123#389—44no—
124#385—48no—
125#384—28no—
126#384—24no—
127#382—28no—
128#380—28no—
129#380—28no—
130#380—28no—
131#380Q8116yes—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—44no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C25C2411yescharacteristic
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—24no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4C2×D414yes—
174C23×C4C2×Q818yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2411yescharacteristic
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189#125—416no—
190#117—416no—
191#117—416no—
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—28no—
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—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
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—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4C2×D418yes—
241C2×C4⋊C4C2×Q818yes—
242C2×C4⋊C4C2×Q818yes—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4C2×D418yes—
254C2×C22⋊C4C2×Q8116yes—
255C2×C42—232no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42C2×D414yes—
276C2×C42C2×D414yes—
277C2×C42#4114yes—
278C2×C42#4114yes—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95C2×Q8116yes—
301C24—21no—
302C24—21no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24C22×D412yes—
309C24C22×Q814yes—
310C24C2511yescharacteristic
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—216no—
323C22×C4—216no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—48no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4C4⋊Q8116yes—
395C22×C4C22×D414yes—
396C22×C4C22×D414yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×Q818yes—
400C22×C4C22×Q818yes—
401C22×C4C22×Q814yes—
402C22×C4C22×Q814yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4H(5,𝔽2)14yes—
410C22×C4H(5,𝔽2)14yes—
411C22×C42−1+414yes—
412C4⋊C4—216no—
413C4⋊C4—216no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—416no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—416no—
424C4⋊C4—88no—
425C4⋊C4—88no—
426C4⋊C4—88no—
427C4⋊C4—88no—
428C4⋊C4—88no—
429C22⋊C4—48no—
430C22⋊C4—416no—
431C22⋊C4—48no—
432C22⋊C4—88no—
433C42—28no—
434C42—28no—
435C42—28no—
436C42—28no—
437C42—28no—
438C42—28no—
439C42—44no—
440C42—44no—
441C42—88no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23C2×C4⋊Q814yes—
463C23C23×D411yescharacteristic
464C23C23×Q812yes—
465C23C22×#4111yescharacteristic
466C23C22×#4111yescharacteristic
467C23C2×H(5,𝔽2)12yes—
468C23C2×H(5,𝔽2)12yes—
469C23C2×2−1+412yes—
470C23#58512yes—
471C23#58512yes—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—232no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—416no—
502C2×C4—416no—
503C2×C4—44no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4#52514yes—
554C2×C4C2×C4⋊Q814yes—
555C2×C4C2×C4⋊Q814yes—
556C2×C4#53414yes—
557C2×C4#54014yes—
558C2×C4#54114yes—
559C2×C4#55418yes—
560C2×C4#55418yes—
561C22—24no—
562C22—28no—
563C22—44no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—42no—
569C22—42no—
570C22C2×#52511yescharacteristic
571C22C22×C4⋊Q811yescharacteristic
572C22C2×#53411yescharacteristic
573C22C2×#54011yescharacteristic
574C22C2×#54111yescharacteristic
575C22#345511yescharacteristic
576C22#345511yescharacteristic
577C22C2×#55412yes—
578C22C2×#55412yes—
579C22#348812yes—
580C22#348912yes—
581C22#349412yes—
582C22#349512yes—
583C22#350112yes—
584C22#350212yes—
585C22#350312yes—
586C22#350412yes—
587C22#354612yes—
588C22#354712yes—
589C22#354812yes—
590C22#355112yes—
591C4—24no—
592C4—24no—
593C4—24no—
594C4—44no—
595C4—44no—
596C4—48no—
597C4—48no—
598C4—42no—
599C4—42no—
600C4—82no—
601C4—82no—
602C4—84no—
603C4—84no—
604C4#258318yes—
605C2—44no—
606C2C2×#258311yescharacteristic
607C2#3526111yescharacteristic
608C2#3531011yescharacteristic
609C2#3556612yes—
610C2#3727212yes—
611C2#3960212yes—
612C2#3960312yes—
613C2#3960612yes—
614C2#3960712yes—
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. #15942C2semidirect
4. #15766C2semidirect
7. #15218C2semidirect
8. #15218C2semidirect
9. #15196C2semidirect
10. #15151C2semidirect
13. #14046C2semidirect
14. #14045C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#22852
#159422
#157662
#154882
#153052
#152182
#151962
#151512
#146972
#142642
#140462
#140452
#137772