Finite Group Databasebeta

← #364776

Subgroups — #364776

Order 512.

1646 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#474C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic, complemented
3C2×#1993C211yesmaximal, characteristic, complemented
4C2×#1762C211yesmaximal, characteristic
5#13941C211yesmaximal, characteristic
6#11219C212yesmaximal, complemented
7#11066C212yesmaximal
8#10931C212yesmaximal
9#10907C212yesmaximal, complemented
10#10850C211yesmaximal, characteristic, complemented
11#10835C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#474—24no—
15C2×#474—24no—
16C2×#474C418yescomplemented
17C22×#107C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×#102C2212yes—
21C22×#102C2211yescharacteristic
22C22×#97C2211yescharacteristic
23#2369—24no—
24#2369—24no—
25C2×#391—22no—
26C2×#390C2212yes—
27C2×#389—21no—
28C2×#389C2212yes—
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1993—24no—
35#1762—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1439—21no—
45#1439—21no—
46C4⋊#95C2212yes—
47C22⋊#95—22no—
48C23×Q8—21no—
49C23×D4C2311yescharacteristic
50C24×C4—21no—
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8C2×C414yes—
58C2×C4⋊Q8D414yes—
59C2×C4⋊Q8D414yes—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4C2×C414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C42—21no—
79C22×C42—22no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82#474—232no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#107C2×C418yes—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8C2×C414yes—
89C2×#102—22no—
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—22no—
96C2×#102C2×C418yes—
97C2×#102Q814yescomplemented
98C2×#97—22no—
99C2×#97—22no—
100C2×#97—22no—
101#391—216no—
102#390—24no—
103#390—42no—
104#389—28no—
105#389—24no—
106#389—24no—
107#382—22no—
108#382—22no—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4D414yes—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D412yes—
124#340—24no—
125#340—24no—
126#340—22no—
127#340—22no—
128#340—24no—
129#340—42no—
130#336—24no—
131#336—42no—
132C25—21no—
133C22×Q8—24no—
134C22×Q8—24no—
135C22×Q8—24no—
136C22×Q8—82no—
137C22×Q8—82no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4C22×C412yes—
150C22×D4C2×Q812yes—
151C22×D4#4112yes—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—24no—
157C23×C4—22no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—82no—
161C23×C4—82no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8C22×C414yes—
186C22×C8C2×Q812yes—
187C22×C8#4112yes—
188C4⋊Q8—28no—
189C4⋊Q8—28no—
190C4⋊Q8—28no—
191C4⋊Q8C22⋊C4116yescomplemented
192C4×D4—28no—
193C4×D4—28no—
194C4×D4—44no—
195C4×D4—44no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×Q812yes—
235C2×C4⋊C4#4112yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—82no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—28no—
253C2×C42—44no—
254C2×C42—82no—
255C2×C42—82no—
256C2×C42C22×C412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42#4112yes—
261C2×C42#4112yes—
262#107—216no—
263C4⋊C8—28no—
264#102—28no—
265#102—216no—
266#102—28no—
267#97—44no—
268#95—28no—
269#95—28no—
270#95—24no—
271#95—24no—
272#95—42no—
273#95—42no—
274#95—82no—
275#95—82no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24C22×D411yescharacteristic
288C2×Q8—216no—
289C2×Q8—88no—
290C2×Q8—84no—
291C2×Q8—84no—
292C2×Q8—88no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4C4×Q814yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
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—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—48no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—88no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—88no—
372C22×C4C4×D412yes—
373C22×C4C4×D412yes—
374C22×C4#12012yes—
375C22×C4#12012yes—
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4#13714yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×Q811yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4H(5,𝔽2)11yescharacteristic
390C22×C42−1+411yescharacteristic
391C22×C42−1+411yescharacteristic
392C22×C42−1+411yescharacteristic
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—41no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C4⋊C4—28no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—216no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—44no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—88no—
436C4⋊C4C2×C22⋊C418yes—
437C4⋊C4C22⋊Q814yes—
438C22⋊C4—48no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C42—48no—
442C42—42no—
443C42—42no—
444C42—42no—
445C42—42no—
446C42—88no—
447C42C2×C22⋊C414yes—
448C42C4×D414yes—
449C42#12114yes—
450C42#12314yes—
451C23—24no—
452C23—44no—
453C23—42no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23—44no—
461C23—42no—
462C23—44no—
463C23—44no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—81no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—82no—
473C23—82no—
474C23C2×C4×D411yescharacteristic
475C23C2×#12011yescharacteristic
476C23C2×#12111yescharacteristic
477C23#54611yescharacteristic
478C23#54711yescharacteristic
479C23#54811yescharacteristic
480C23D4×Q811yescharacteristic
481C23C2×#13712yes—
482C23#57512yes—
483C23#57612yes—
484C23#57612yes—
485Q8—816no—
486D4—24no—
487D4—24no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—42no—
492D4—42no—
493D4—42no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—816no—
534C2×C4—82no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4#38614yes—
547C2×C4#38614yes—
548C2×C4C22×C22⋊C411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C2×C4×Q811yescharacteristic
551C2×C4C4×#4111yescharacteristic
552C2×C4#51911yescharacteristic
553C2×C4#52011yescharacteristic
554C2×C4C2×#12111yescharacteristic
555C2×C4C2×C22⋊Q811yescharacteristic
556C2×C4C2×#12311yescharacteristic
557C2×C4#52511yescharacteristic
558C2×C4#52911yescharacteristic
559C2×C4#53311yescharacteristic
560C2×C4#53611yescharacteristic
561C2×C4#53711yescharacteristic
562C2×C4#54111yescharacteristic
563C2×C4#54211yescharacteristic
564C2×C4#54811yescharacteristic
565C2×C4#54811yescharacteristic
566C2×C4#54811yescharacteristic
567C2×C4#54811yescharacteristic
568C2×C4D4×Q811yescharacteristic
569C2×C4D4×Q811yescharacteristic
570C2×C4#55011yescharacteristic
571C2×C4#55011yescharacteristic
572C2×C4#55211yescharacteristic
573C2×C4#55211yescharacteristic
574C2×C4#55411yescharacteristic
575C2×C4#55611yescharacteristic
576C8—44no—
577C8—42no—
578C8—42no—
579C22—44no—
580C22—48no—
581C22—82no—
582C22—82no—
583C22—84no—
584C22—82no—
585C22—82no—
586C22—84no—
587C22—82no—
588C22—84no—
589C22—82no—
590C22C2×#38611yescharacteristic
591C22#236011yescharacteristic
592C22#236211yescharacteristic
593C22#240711yescharacteristic
594C22#240911yescharacteristic
595C22#242611yescharacteristic
596C22#242911yescharacteristic
597C22C4×#13712yes—
598C22#294812yes—
599C22#300411yescharacteristic
600C22#300511yescharacteristic
601C22#300811yescharacteristic
602C22#300811yescharacteristic
603C22#303712yes—
604C22#304112yes—
605C22#331412yes—
606C22#331912yes—
607C22#334612yes—
608C22#334712yes—
609C22#337012yes—
610C22#337512yes—
611C22#338512yes—
612C22#338912yes—
613C4—24no—
614C4—24no—
615C4—44no—
616C4—44no—
617C4—48no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—162no—
622C4C4×C22⋊Q811yescharacteristic
623C4#232311yescharacteristic
624C4Q8×C22⋊C411yescharacteristic
625C4#234111yescharacteristic
626C4#242111yescharacteristic
627C4#242311yescharacteristic
628C4#253611yescharacteristic
629C4#253811yescharacteristic
630C2—84no—
631C2#1396911yescharacteristic
632C2#2092611yescharacteristic
633C2#2093411yescharacteristic
634C2#2297912yes—
635C2#2307912yes—
636C2#2412911yescharacteristic
637C2#2413011yescharacteristic
638C2#2433911yescharacteristic
639C2#2434111yescharacteristic
640C2#2528512yes—
641C2#2546212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2369C2semidirect
3. C2×#1993C2semidirect
6. #11219C2semidirect
9. #10907C2semidirect
10. #10850C2semidirect
11. #10835C2semidirect
16. C2×#474C4semidirect
97. C2×#102Q8semidirect
191. C4⋊Q8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4742
C2×#23692
C2×#19932
C2×#17622
#139412
#112192
#110662
#109312
#109072
#108502
#108352