Finite Group Databasebeta

← #402853

Subgroups — #402853

Order 512.

1428 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2C2×#2099C211yesmaximal, characteristic
3C2×#2043C211yesmaximal, characteristic
4#14699C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic
6#11190C212yesmaximal
7#11067C212yesmaximal
8#11010C211yesmaximal, characteristic
9#10795C211yesmaximal, characteristic
10#10789C212yesmaximal
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C4×C4⋊Q8—24no—
19C4×C4⋊Q8—24no—
20C2×#391C2212yes—
21C2×#389—21no—
22C2×#389—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#387—21no—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4—21no—
29C2×C43C2211yescharacteristic
30#2099—24no—
31#2043—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40#1436—21no—
41#1436—21no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46#1434—21no—
47C23×Q8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8D414yes—
54C2×C4×Q8—24no—
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—24no—
58C2×C4×Q8—42no—
59C2×C4×Q8—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—22no—
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92#391—24no—
93#391—24no—
94#389—28no—
95#389—28no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#387—44no—
101#384—22no—
102#384D414yes—
103C4×C4⋊C4—28no—
104C4×C4⋊C4—28no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C43—22no—
118C43—22no—
119#340—24no—
120#340—24no—
121#340—42no—
122#340—42no—
123#336—22no—
124#336—22no—
125#336—24no—
126#336—42no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C2×D414yes—
139C22×Q8#4114yes—
140C22×Q8#4114yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C4⋊Q8—216no—
169C4⋊Q8—216no—
170C4×Q8—48no—
171C4×Q8—48no—
172C4×Q8—48no—
173C4×Q8—48no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C2×D414yes—
217C2×C4⋊C4#4112yes—
218C2×C4⋊C4#4112yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—28no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42#4114yes—
248#103—48no—
249#103—48no—
250#103—48no—
251#103—48no—
252#95—28no—
253#95—28no—
254#95—24no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95—42no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264C24C22×D411yescharacteristic
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8#12318yes—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4#12012yes—
366C22×C4#12012yes—
367C22×C4#12412yes—
368C22×C4#12412yes—
369C22×C4#13512yes—
370C22×C4#13512yes—
371C22×C4AGL(1,ℤ/8)12yes—
372C22×C4#13712yes—
373C22×C4#13712yes—
374C22×C4#13712yes—
375C22×C4#13712yes—
376C22×C4#13712yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4C2×#4112yes—
381C22×C4C2×#4112yes—
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4H(5,𝔽2)11yescharacteristic
385C22×C4H(5,𝔽2)12yes—
386C22×C42−1+411yescharacteristic
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C4⋊C4—216no—
402C4⋊C4—216no—
403C4⋊C4—22no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—48no—
409C4⋊C4—416no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—84no—
422C42—28no—
423C42—28no—
424C42—24no—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—416no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—44no—
434C42—44no—
435C23C2×#12011yescharacteristic
436C23#52511yescharacteristic
437C23#52511yescharacteristic
438C23#52511yescharacteristic
439C23C2×#12411yescharacteristic
440C23#53511yescharacteristic
441C23#53711yescharacteristic
442C23C2×#13511yescharacteristic
443C23C2×#13511yescharacteristic
444C23C2×AGL(1,ℤ/8)11yescharacteristic
445C23C2×#13711yescharacteristic
446C23C2×#13711yescharacteristic
447C23C2×#13711yescharacteristic
448C23C2×#13711yescharacteristic
449C23C2×#13711yescharacteristic
450Q8—28no—
451Q8—28no—
452Q8—44no—
453Q8—44no—
454Q8—44no—
455Q8—44no—
456Q8—44no—
457Q8—44no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—28no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—44no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—81no—
519C2×C4—82no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4#44914yes—
540C2×C4#45214yes—
541C2×C4#49114yes—
542C2×C4C2×#12312yes—
543C2×C4#52512yes—
544C2×C4#52912yes—
545C2×C4#52912yes—
546C2×C4#53812yes—
547C2×C4#53912yes—
548C2×C4#54612yes—
549C2×C4#54611yescharacteristic
550C2×C4#54612yes—
551C2×C4#54711yescharacteristic
552C2×C4#54811yescharacteristic
553C2×C4#55011yescharacteristic
554C2×C4#55112yes—
555C2×C4#55212yes—
556C2×C4#55312yes—
557C2×C4#55312yes—
558C8—82no—
559C8—82no—
560C22#239811yescharacteristic
561C22#240711yescharacteristic
562C22#243511yescharacteristic
563C22#256211yescharacteristic
564C22#259811yescharacteristic
565C22#263611yescharacteristic
566C22#284911yescharacteristic
567C22C2×#44911yescharacteristic
568C22C2×#45211yescharacteristic
569C22#300411yescharacteristic
570C22#300411yescharacteristic
571C22#310711yescharacteristic
572C22#310711yescharacteristic
573C22#310811yescharacteristic
574C22#310811yescharacteristic
575C22#310811yescharacteristic
576C22#310811yescharacteristic
577C22#311011yescharacteristic
578C22#311211yescharacteristic
579C22#311211yescharacteristic
580C22#311311yescharacteristic
581C22#311311yescharacteristic
582C22#311311yescharacteristic
583C22#313111yescharacteristic
584C22C2×#49111yescharacteristic
585C22#313911yescharacteristic
586C22#314111yescharacteristic
587C22#320711yescharacteristic
588C22#321011yescharacteristic
589C22#321811yescharacteristic
590C22#321811yescharacteristic
591C22#324811yescharacteristic
592C22#324811yescharacteristic
593C22#325411yescharacteristic
594C22#325511yescharacteristic
595C4—24no—
596C4—24no—
597C4—24no—
598C4—42no—
599C4—48no—
600C4—42no—
601C4—42no—
602C4—84no—
603C4—82no—
604C4—82no—
605C4—161no—
606C4—161no—
607C4#241912yes—
608C4#244412yes—
609C4#246012yes—
610C4#246712yes—
611C2#1486011yescharacteristic
612C2#2382011yescharacteristic
613C2#2382311yescharacteristic
614C2#2415311yescharacteristic
615C2#2415711yescharacteristic
616C2#2585711yescharacteristic
617C2#2586111yescharacteristic
618C2#2912611yescharacteristic
619C2#2913211yescharacteristic
620C2#2936911yescharacteristic
621C2#2937111yescharacteristic
622C2#3013011yescharacteristic
623C2#3013211yescharacteristic
624C2#3300311yescharacteristic
625C2#3300811yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#20992
C2×#20432
#146992
#113732
#111902
#110672
#110102
#107952
#107892