Finite Group Databasebeta

← #641233

Subgroups — #641233

Order 512.

3398 conjugacy classes of subgroups.

The subgroup lattice has 657 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C212yesmaximal, complemented
2C2×C4×#121C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic, complemented
4#17074C212yesmaximal, complemented
5#17071C212yesmaximal, complemented
6#16899C214yesmaximal, complemented
7#15194C212yesmaximal, complemented
8#15181C212yesmaximal, complemented
9#15131C214yesmaximal, complemented
10#14729C214yesmaximal, complemented
11#14686C211yesmaximal, characteristic, complemented
12#14275C212yesmaximal, complemented
13#14271C212yesmaximal
14#13872C212yesmaximal, complemented
15C22×#121C2211yescharacteristic
16C22×C4×D4C2212yes—
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C42C2211yescharacteristic
22#2487—216no—
23C2×#400C2214yes—
24C2×#399C2214yes—
25C2×#396C2218yes—
26C4×#121—28no—
27C2×#390C2211yescharacteristic
28C2×#389—28no—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×#388C2212yes—
36C2×#386C2212yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#385C2214yes—
40C2×#384C2212yes—
41C2×#380C2214yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2214yes—
46C2×#380C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C22×#95C2212yes—
50#1445C2214yes—
51#1444C2214yes—
52#1442C2214yes—
53#1441C2218yes—
54#1441C2214yes—
55#1441C2218yes—
56#1441C2218yes—
57#1439C2218yes—
58#1439C2214yes—
59#1439C2218yes—
60#1438C2214yes—
61C4⋊#95C2212yes—
62C4⋊#95C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C22⋊#95C2214yes—
66C23×D4C2312yes—
67C23×D4C2311yescharacteristic
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2312yes—
73C24×C4C2311yescharacteristic
74C2×#121—24no—
75C2×#121—24no—
76C2×#121Q818yes—
77C2×C4×D4—28no—
78C2×C4×D4—28no—
79C2×C4×D4—28no—
80C2×C4×D4—24no—
81C2×C4×D4—44no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—44no—
87C22×C4⋊C4—48no—
88C22×C4⋊C4D418yescomplemented
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2318yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2318yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2318yes—
111C22×C42—44no—
112C22×C42D418yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118#400—216no—
119#399—216no—
120#396—416no—
121#390—24no—
122#389—216no—
123#389—28no—
124#389—28no—
125#389—28no—
126#389—28no—
127#389—24no—
128#388—28no—
129#386—216no—
130#386—28no—
131#386—48no—
132#385—216no—
133#384—44no—
134#380—28no—
135#380—24no—
136#380—28no—
137#380—216no—
138#380—44no—
139#380—48no—
140C4×C4⋊C4—24no—
141C4×C22⋊C4—28no—
142C2×#95—28no—
143C2×#95C2314yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2318yes—
153C2×#95C2318yes—
154C2×#95C2314yes—
155C2×#95C2318yes—
156C25C2412yes—
157C25C2411yescharacteristic
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—24no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4C2×Q818yes—
169C22×D4C2×Q814yes—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—28no—
184C23×C4—28no—
185C23×C4—44no—
186C23×C4—42no—
187C23×C4—44no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4C2×D414yes—
193C23×C4C2×D414yes—
194C23×C4C2×D414yes—
195C23×C4C2×D418yes—
196C23×C4C2×D414yes—
197C23×C4C2×Q814yes—
198C23×C4#4114yes—
199C23×C4#4114yes—
200C23×C4C2412yes—
201C23×C4C2418yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212#121—232no—
213C4×D4—416no—
214C4×D4—432no—
215C2×C4⋊C4—232no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—216no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—216no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—48no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—432no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4C2×D418yes—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×Q814yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—48no—
271C2×C22⋊C4—48no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4C2×Q818yes—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—416no—
282C2×C42—42no—
283C2×C42—42no—
284C2×C42C2×D414yes—
285C2×C42C2×D414yes—
286C2×C42C2×D414yes—
287C2×C42C2×D414yes—
288#95—28no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—216no—
294#95—28no—
295#95—216no—
296#95—28no—
297#95—44no—
298#95—48no—
299#95—48no—
300#95—48no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—24no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24—24no—
315C24C22×D412yes—
316C24C22×D412yes—
317C24C22×Q814yes—
318C24C22×Q812yes—
319C24C2×#4112yes—
320C24C2×#4112yes—
321C24H(5,𝔽2)12yes—
322C242−1+412yes—
323C24C2511yescharacteristic
324C2×D4—216no—
325C2×D4—216no—
326C2×D4—216no—
327C2×D4—28no—
328C2×D4—48no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—48no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—216no—
348C22×C4—216no—
349C22×C4—216no—
350C22×C4—28no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—22no—
367C22×C4—28no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—28no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—28no—
387C22×C4—24no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—416no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4—416no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—88no—
408C22×C4—88no—
409C22×C4#121116yes—
410C22×C4C22⋊Q8116yes—
411C22×C4C22×D418yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×Q814yes—
421C22×C4C22×Q812yes—
422C22×C4C22×Q812yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4H(5,𝔽2)14yes—
428C22×C4H(5,𝔽2)14yes—
429C22×C42−1+414yes—
430C4⋊C4—216no—
431C4⋊C4—216no—
432C4⋊C4—28no—
433C4⋊C4—44no—
434C4⋊C4—48no—
435C4⋊C4—416no—
436C4⋊C4—44no—
437C4⋊C4—48no—
438C4⋊C4—48no—
439C4⋊C4—416no—
440C4⋊C4—416no—
441C4⋊C4—88no—
442C4⋊C4—88no—
443C4⋊C4—816no—
444C22⋊C4—216no—
445C22⋊C4—48no—
446C22⋊C4—48no—
447C22⋊C4—416no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—816no—
451C22⋊C4—816no—
452C42—28no—
453C42—28no—
454C42—44no—
455C42—44no—
456C42—48no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—24no—
461C23—22no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—28no—
477C23—28no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—42no—
484C23—42no—
485C23—44no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23C2×#12114yes—
497C23C2×C22⋊Q814yes—
498C23#53714yes—
499C23D4×Q814yes—
500C23#55014yes—
501C23#55414yes—
502C23#55614yes—
503C23C23×D411yescharacteristic
504C23C23×D411yescharacteristic
505C23C23×Q811yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4111yescharacteristic
508C23C2×H(5,𝔽2)12yes—
509C23C2×H(5,𝔽2)12yes—
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×2−1+412yes—
512C23C2×2−1+411yescharacteristic
513C23#58512yes—
514D4—416no—
515D4—432no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—232no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—216no—
541C2×C4—416no—
542C2×C4—416no—
543C2×C4—432no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—48no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4—44no—
568C2×C4—48no—
569C2×C4—44no—
570C2×C4—48no—
571C2×C4—48no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—816no—
576C2×C4—816no—
577C2×C4C2×#12114yes—
578C2×C4C2×#12114yes—
579C2×C4C2×C22⋊Q814yes—
580C2×C4C2×C22⋊Q814yes—
581C2×C4#53414yes—
582C2×C4#53514yes—
583C2×C4#53614yes—
584C2×C4#53718yes—
585C2×C4#53918yes—
586C22—28no—
587C22—24no—
588C22—28no—
589C22—24no—
590C22—24no—
591C22—28no—
592C22—24no—
593C22—24no—
594C22—28no—
595C22—24no—
596C22—44no—
597C22—48no—
598C22—44no—
599C22—44no—
600C22—48no—
601C22—84no—
602C22—82no—
603C22—84no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22#253818yes—
608C22C22×#12111yescharacteristic
609C22C22×C22⋊Q811yescharacteristic
610C22C2×#53411yescharacteristic
611C22C2×#53511yescharacteristic
612C22C2×#53611yescharacteristic
613C22C2×#53712yes—
614C22C2×#53711yescharacteristic
615C22#344911yescharacteristic
616C22C2×#53912yes—
617C22C2×D4×Q811yescharacteristic
618C22C2×#55011yescharacteristic
619C22C2×#55411yescharacteristic
620C22C2×#55611yescharacteristic
621C22#349612yes—
622C22#350112yes—
623C22#350312yes—
624C22#350512yes—
625C22#350612yes—
626C22#351112yes—
627C22#353612yes—
628C22#353712yes—
629C22#355912yes—
630C22#356312yes—
631C4—24no—
632C4—24no—
633C4—24no—
634C4—24no—
635C4—42no—
636C4—48no—
637C4—44no—
638C4—88no—
639C4—84no—
640C4—82no—
641C4—82no—
642C4#261818yes—
643C2—28no—
644C2—84no—
645C2C2×#261811yescharacteristic
646C2C2×#253811yescharacteristic
647C2#3501011yescharacteristic
648C2#3516511yescharacteristic
649C2#3527911yescharacteristic
650C2#3743911yescharacteristic
651C2#3754911yescharacteristic
652C2#3961412yes—
653C2#3963112yes—
654C2#3974212yes—
655C2#3975912yes—
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
1. C2×#2487C2semidirect
3. C22×#389C2semidirect
4. #17074C2semidirect
5. #17071C2semidirect
6. #16899C2semidirect
7. #15194C2semidirect
8. #15181C2semidirect
9. #15131C2semidirect
10. #14729C2semidirect
11. #14686C2semidirect
12. #14275C2semidirect
14. #13872C2semidirect
88. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C2×C4×#1212
C22×#3892
#170742
#170712
#168992
#151942
#151812
#151312
#147292
#146862
#142752
#142712
#138722