Finite Group Databasebeta

← #642353

Subgroups — #642353

Order 512.

2480 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2642C211yesmaximal, characteristic, complemented
2C2×#2621C211yesmaximal, characteristic
3C2×#2629C211yesmaximal, characteristic
4C2×C4×#125C211yesmaximal, characteristic, complemented
5#16472C214yesmaximal
6#16464C214yesmaximal, complemented
7#15189C212yesmaximal, complemented
8#15179C212yesmaximal
9#14871C214yesmaximal, complemented
10#14854C214yesmaximal
11#14714C211yesmaximal, characteristic, complemented
12#14277C211yesmaximal, characteristic, complemented
13#14142C212yesmaximal, complemented
14#13997C212yesmaximal
15#13761C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×#117C2211yescharacteristic
20#2642—28no—
21#2629—28no—
22#2621—28no—
23C2×#400C2214yes—
24C2×#400C2214yes—
25C2×#398C2214yes—
26C2×#398C2214yes—
27C2×#397C2214yes—
28C2×#396C2214yes—
29C4×#125—28no—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#387C2214yes—
39C2×#384C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2214yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×#376C2211yescharacteristic
47C2×C43C2211yescharacteristic
48#1445C2214yes—
49#1444C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1437C2214yes—
63#1437C2212yes—
64#1437C2212yes—
65#1436C2214yes—
66#1436C2212yes—
67#1436C2212yes—
68C4⋊#95C2214yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72#1434C2214yes—
73C4×#95C2211yescharacteristic
74C4×#95C2211yescharacteristic
75C23×Q8C2312yes—
76C24×C4C2311yescharacteristic
77C2×C4⋊Q8—24no—
78C2×C4⋊Q8—24no—
79C2×C4⋊Q8—24no—
80C2×#125—24no—
81C2×#125—24no—
82C2×#125D418yes—
83C2×C22⋊Q8—28no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—44no—
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—24no—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2314yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#400—216no—
116#400—216no—
117#398—216no—
118#398—216no—
119#397—216no—
120#396—216no—
121#391—28no—
122#391—28no—
123#391—28no—
124#391—28no—
125#391—44no—
126#389—28no—
127#389—28no—
128#389—44no—
129#387—216no—
130#384—24no—
131#382D418yes—
132#380—28no—
133C4×C4⋊C4—28no—
134C4×C4⋊C4—216no—
135C4×C22⋊C4—24no—
136C4×C22⋊C4—24no—
137#376—24no—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2311yescharacteristic
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C43—24no—
166C25C2411yescharacteristic
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4#4114yes—
181C23×C4C2414yes—
182C23×C4C2411yescharacteristic
183C23×C4C2414yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C4⋊Q8—416no—
196#125—232no—
197C22⋊Q8—432no—
198#117—416no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
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×D418yes—
231C2×C4⋊C4C2×D4116yes—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—28no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—28no—
267C2×C42—22no—
268C2×C42—42no—
269C2×C42—44no—
270C2×C42—42no—
271C2×C42C2×D414yes—
272C2×C42C2×D414yes—
273C2×C42#4114yes—
274C2×C42#4114yes—
275#95—24no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—22no—
288#95—22no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—24no—
293#95—24no—
294#95—42no—
295#95—42no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95C2×D4116yes—
302C24—21no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24—21no—
310C24—22no—
311C24—22no—
312C24C2×#4112yes—
313C24C2511yescharacteristic
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—48no—
317C2×Q8—48no—
318C2×Q8—48no—
319C2×Q8—48no—
320C2×Q8—48no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4C22×D418yes—
402C22×C4C22×D418yes—
403C22×C4C22×D414yes—
404C22×C4C22×D414yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4114yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C42−1+414yes—
416C22×C42−1+414yes—
417C22×C42−1+412yes—
418C4⋊C4—232no—
419C4⋊C4—216no—
420C4⋊C4—216no—
421C4⋊C4—416no—
422C4⋊C4—416no—
423C4⋊C4—48no—
424C4⋊C4—416no—
425C4⋊C4—48no—
426C4⋊C4—88no—
427C4⋊C4—88no—
428C4⋊C4—88no—
429C22⋊C4—48no—
430C22⋊C4—416no—
431C22⋊C4—44no—
432C22⋊C4—48no—
433C22⋊C4—44no—
434C22⋊C4—88no—
435C42—28no—
436C42—48no—
437C42—416no—
438C42—48no—
439C42—44no—
440C42—44no—
441C42—44no—
442C42—44no—
443C42#121116yes—
444C23—22no—
445C23—24no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23C23×D411yescharacteristic
468C23C23×D411yescharacteristic
469C23C22×#4111yescharacteristic
470C23C22×#4111yescharacteristic
471C23C22×#4111yescharacteristic
472C23C2×H(5,𝔽2)11yescharacteristic
473C23C2×2−1+412yes—
474C23C2×2−1+411yescharacteristic
475C23C2×2−1+412yes—
476C23#58512yes—
477C23#58512yes—
478Q8—816no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—28no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—416no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—44no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4C2×#12118yes—
562C2×C4C2×#12114yes—
563C2×C4#52914yes—
564C2×C4#54114yes—
565C2×C4#54214yes—
566C2×C4#54614yes—
567C2×C4#54814yes—
568C2×C4#54814yes—
569C2×C4#55014yes—
570C22—44no—
571C22—42no—
572C22—44no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22—82no—
580C22—82no—
581C22C22×#12111yescharacteristic
582C22C2×#52911yescharacteristic
583C22C2×#54111yescharacteristic
584C22C2×#54211yescharacteristic
585C22#345511yescharacteristic
586C22#346112yes—
587C22#346112yes—
588C22#346112yes—
589C22C2×#54611yescharacteristic
590C22C2×#54811yescharacteristic
591C22C2×#54811yescharacteristic
592C22C2×#55011yescharacteristic
593C22#348912yes—
594C22#348912yes—
595C22#349512yes—
596C22#349912yes—
597C22#350012yes—
598C22#350911yescharacteristic
599C22#351211yescharacteristic
600C22#355212yes—
601C22#355512yes—
602C22#356212yes—
603C22#356312yes—
604C4—24no—
605C4—28no—
606C4—42no—
607C4—48no—
608C4—44no—
609C4—42no—
610C4—48no—
611C4—82no—
612C4—84no—
613C4—81no—
614C4—84no—
615C4—81no—
616C4—82no—
617C4#242618yes—
618C2—82no—
619C2#3462911yescharacteristic
620C2C2×#242611yescharacteristic
621C2#3530812yes—
622C2#3532111yescharacteristic
623C2#3759812yes—
624C2#3770511yescharacteristic
625C2#3770911yescharacteristic
626C2#3775012yes—
627C2#3938211yescharacteristic
628C2#3938511yescharacteristic
629C2#3942712yes—
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
1. C2×#2642C2semidirect
4. C2×C4×#125C2semidirect
6. #16464C2semidirect
7. #15189C2semidirect
9. #14871C2semidirect
11. #14714C2semidirect
12. #14277C2semidirect
13. #14142C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26422
C2×#26212
C2×#26292
C2×C4×#1252
#164722
#164642
#151892
#151792
#148712
#148542
#147142
#142772
#141422
#139972
#137612