Finite Group Databasebeta

← #4308688

Subgroups — #4308688

Order 512.

2490 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2864C211yesmaximal, characteristic, complemented
2C2×C4×#117C211yesmaximal, characteristic
3#15221C212yesmaximal, complemented
4#15220C212yesmaximal
5#15133C212yesmaximal, complemented
6#15131C212yesmaximal
7#15118C214yesmaximal, complemented
8#14891C214yesmaximal
9#14707C211yesmaximal, characteristic, complemented
10#14707C211yesmaximal, characteristic, complemented
11#14687C211yesmaximal, characteristic, complemented
12#14682C211yesmaximal, characteristic
13#14682C211yesmaximal, characteristic
14#14316C214yesmaximal, complemented
15#14315C214yesmaximal
16C22×#117C2212yes—
17C23×C42C2211yescharacteristic
18#2864—28no—
19C2×#398C2214yes—
20C2×#396C2214yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#387C2214yes—
26C2×#387C2214yes—
27C2×#384C2212yes—
28C2×#382C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C4×#117—28no—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2212yes—
42C2×#376C2212yes—
43C2×C43C2211yescharacteristic
44C2×C43C2211yescharacteristic
45#1445C2214yes—
46#1444C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2214yes—
69#1434C2212yes—
70#1434C2212yes—
71#1434C2212yes—
72#1434C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C2×#117—28no—
81C2×#117—28no—
82C2×#117—28no—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42—24no—
100C22×C42—24no—
101C22×C42Q818yes—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2311yescharacteristic
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#398—216no—
115#396—216no—
116#389—28no—
117#389—28no—
118#389—44no—
119#389—44no—
120#387—216no—
121#387—216no—
122#384—44no—
123#382—24no—
124#382—24no—
125#382—24no—
126#382—24no—
127#382—44no—
128#380—44no—
129#380—44no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—28no—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—24no—
135C4×C22⋊C4—28no—
136#376—28no—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C43—24no—
166C43D418yes—
167C25C2411yescharacteristic
168C23×C4—24no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4C2×Q814yes—
183C23×C4C2×Q814yes—
184C23×C4C2×Q814yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199#117—264no—
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—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—44no—
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#4118yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4#4118yes—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—232no—
264C2×C42—24no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42C2×D418yes—
276C2×C42C2×D414yes—
277C2×C42C2×D414yes—
278C2×C42C2×D418yes—
279C2×C42C2×D414yes—
280C2×C42C2×Q814yes—
281C2×C42C2×Q814yes—
282C2×C42C2×Q814yes—
283C2×C42C2×Q814yes—
284C2×C42#4114yes—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313C24—21no—
314C24—22no—
315C24—21no—
316C24—21no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24C22×Q812yes—
323C24C2×#4114yes—
324C24C2511yescharacteristic
325C22×C4—216no—
326C22×C4—216no—
327C22×C4—28no—
328C22×C4—21no—
329C22×C4—21no—
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—28no—
340C22×C4—28no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—28no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—28no—
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—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4C22×D414yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D414yes—
413C22×C4C22×D412yes—
414C22×C4C22×Q812yes—
415C22×C4C22×Q812yes—
416C22×C4C22×Q812yes—
417C22×C4C22×Q812yes—
418C22×C4C22×Q812yes—
419C22×C4C22×Q812yes—
420C22×C4C2×#4114yes—
421C22×C4C2×#4114yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4114yes—
425C22×C4C2×#4114yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4112yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C4H(5,𝔽2)12yes—
430C22×C42−1+412yes—
431C22×C42−1+412yes—
432C4⋊C4—216no—
433C4⋊C4—216no—
434C4⋊C4—416no—
435C4⋊C4—416no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—88no—
441C4⋊C4—88no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C22⋊C4—216no—
445C22⋊C4—216no—
446C22⋊C4—88no—
447C22⋊C4—88no—
448C42—216no—
449C42—28no—
450C42—28no—
451C42—28no—
452C42—216no—
453C42—28no—
454C42—28no—
455C42—48no—
456C42—48no—
457C42—44no—
458C42—44no—
459C42C22⋊Q8116yes—
460C23—22no—
461C23—24no—
462C23—22no—
463C23—24no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—41no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23#53314yes—
483C23C23×D411yescharacteristic
484C23C23×Q811yescharacteristic
485C23C22×#4111yescharacteristic
486C23C22×#4112yes—
487C23C22×#4112yes—
488C23C2×H(5,𝔽2)11yescharacteristic
489C23C2×H(5,𝔽2)11yescharacteristic
490C23C2×2−1+411yescharacteristic
491C23C2×2−1+411yescharacteristic
492C23#58512yes—
493C23#58512yes—
494C2×C4—232no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—416no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—416no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4C2×C22⋊Q814yes—
577C2×C4C2×C22⋊Q814yes—
578C2×C4C2×C22⋊Q814yes—
579C2×C4#52518yes—
580C2×C4#53314yes—
581C2×C4#53714yes—
582C2×C4#53714yes—
583C2×C4#54014yes—
584C2×C4#54114yes—
585C2×C4#54414yes—
586C2×C4#54414yes—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—42no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—42no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22C22×C22⋊Q811yescharacteristic
599C22C2×#52512yes—
600C22C2×#53311yescharacteristic
601C22C2×#53311yescharacteristic
602C22#344412yes—
603C22C2×#53711yescharacteristic
604C22C2×#53711yescharacteristic
605C22#344912yes—
606C22C2×#54011yescharacteristic
607C22C2×#54111yescharacteristic
608C22#345512yes—
609C22C2×#54411yescharacteristic
610C22C2×#54411yescharacteristic
611C22#345811yescharacteristic
612C22#345811yescharacteristic
613C22#351412yes—
614C22#351412yes—
615C22#351412yes—
616C22#351512yes—
617C22#351512yes—
618C22#351512yes—
619C22#351712yes—
620C22#351812yes—
621C4—24no—
622C4—24no—
623C4—28no—
624C4—24no—
625C4—42no—
626C4—44no—
627C4—42no—
628C4—44no—
629C4—44no—
630C4—84no—
631C4—84no—
632C4—84no—
633C4—84no—
634C4#262918yes—
635C2—44no—
636C2C2×#262911yescharacteristic
637C2#3463411yescharacteristic
638C2#3513012yes—
639C2#3513211yescharacteristic
640C2#3513211yescharacteristic
641C2#3513612yes—
642C2#3514212yes—
643C2#3518912yes—
644C2#3519211yescharacteristic
645C2#3520211yescharacteristic
646C2#3662511yescharacteristic
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×#2864C2semidirect
3. #15221C2semidirect
5. #15133C2semidirect
7. #15118C2semidirect
9. #14707C2semidirect
10. #14707C2semidirect
11. #14687C2semidirect
14. #14316C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28642
C2×C4×#1172
#152212
#152202
#151332
#151312
#151182
#148912
#147072
#146872
#146822
#143162
#143152