Finite Group Databasebeta

← #376218

Subgroups — #376218

Order 512.

1596 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic
2C2×#2043C211yesmaximal, characteristic
3C2×#1928C212yesmaximal
4C2×#1863C212yesmaximal
5C4×#389C211yesmaximal, characteristic
6#11324C211yesmaximal, characteristic
7#11207C211yesmaximal, characteristic
8#11010C211yesmaximal, characteristic
9#10932C212yesmaximal
10#10795C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2212yes—
14C22×#107C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C4×C4⋊Q8—28no—
20C2×#391C2212yes—
21C2×#389—21no—
22C2×#389—21no—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C43C2211yescharacteristic
32#2043C418yescomplemented
33#1928—28no—
34#1863—28no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42#1436—21no—
43#1436—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2211yescharacteristic
46C4×#95—21no—
47C23×Q8C2312yes—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
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×C42—21no—
74C22×C42—21no—
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103C2×C418yes—
96C2×#103C2×C418yes—
97#391—24no—
98#391D418yes—
99#389—28no—
100#389—28no—
101#389—22no—
102#389—22no—
103#389C2×C414yes—
104#389D414yes—
105#384—24no—
106C4×C4⋊C4—28no—
107C4×C4⋊C4—28no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—44no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C43—24no—
123#340—24no—
124#340—24no—
125#340—42no—
126#336—24no—
127#336—24no—
128#336—22no—
129#336C2×C414yes—
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×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8C22×C414yes—
142C22×Q8C2×D414yes—
143C22×Q8#4114yes—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2411yescharacteristic
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—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8C22×C414yes—
171C4⋊Q8—216no—
172C4⋊Q8—216no—
173C4×Q8—48no—
174C4×Q8—48no—
175C4×Q8—48no—
176C4×Q8—48no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4#4112yes—
221C2×C4⋊C4#4112yes—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—28no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—44no—
243C2×C42—42no—
244C2×C42—44no—
245C2×C42—42no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42Q1614yes—
250C2×C42C22×C412yes—
251C2×C42C2×D414yes—
252C2×C42C2×D412yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255#107—48no—
256#103—216no—
257#103—216no—
258#103—48no—
259#103—48no—
260#95—24no—
261#95—24no—
262#95—28no—
263#95—28no—
264#95—24no—
265#95—42no—
266#95—42no—
267#95—42no—
268#95—42no—
269#95—42no—
270C24C22×D411yescharacteristic
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—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—24no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8C4×D418yes—
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—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4C4×D412yes—
370C22×C4C4×D412yes—
371C22×C4#12012yes—
372C22×C4#12012yes—
373C22×C4#12112yes—
374C22×C4#12112yes—
375C22×C4C2×Q1612yes—
376C22×C4C2×Q1612yes—
377C22×C4C2×Q1612yes—
378C22×C4#13512yes—
379C22×C4AGL(1,ℤ/8)12yes—
380C22×C4#13712yes—
381C22×C4#13712yes—
382C22×C4#13712yes—
383C22×C4C23×C411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4112yes—
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4H(5,𝔽2)12yes—
392C22×C42−1+412yes—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C4⋊C4—216no—
408C4⋊C4—216no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—416no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—48no—
427C4⋊C4C4×D414yes—
428C42—28no—
429C42—24no—
430C42—24no—
431C42—24no—
432C42—24no—
433C42—24no—
434C42—24no—
435C42—28no—
436C42—44no—
437C42—44no—
438C42—48no—
439C42—416no—
440C42—84no—
441C23C2×C4×D411yescharacteristic
442C23C2×#12011yescharacteristic
443C23C2×#12111yescharacteristic
444C23#52511yescharacteristic
445C23#52511yescharacteristic
446C23#53511yescharacteristic
447C23#53611yescharacteristic
448C23C22×Q1611yescharacteristic
449C23C2×#13511yescharacteristic
450C23C2×AGL(1,ℤ/8)11yescharacteristic
451C23C2×#13711yescharacteristic
452C23C2×#13711yescharacteristic
453C23C2×#13711yescharacteristic
454C23#57511yescharacteristic
455C23#57511yescharacteristic
456Q8—28no—
457Q8—28no—
458Q8—44no—
459Q8—44no—
460Q8—44no—
461Q8—44no—
462Q8—44no—
463Q8—44no—
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—22no—
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—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—44no—
497C2×C4—41no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4C4×Q1614yes—
545C2×C4#44014yes—
546C2×C4#45114yes—
547C2×C4#46214yes—
548C2×C4C2×C4×D412yes—
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C4×#4111yescharacteristic
551C2×C4#52012yes—
552C2×C4#52512yes—
553C2×C4#52911yescharacteristic
554C2×C4#53211yescharacteristic
555C2×C4#54212yes—
556C2×C4D4212yes—
557C2×C4#54612yes—
558C2×C4#54712yes—
559C2×C4#54712yes—
560C2×C4#54812yes—
561C2×C4#55112yes—
562C2×C4#55212yes—
563C2×C4#55312yes—
564C8—44no—
565C8—82no—
566C22C4×#12011yescharacteristic
567C22C4×#12111yescharacteristic
568C22#232611yescharacteristic
569C22#240611yescharacteristic
570C22#240811yescharacteristic
571C22#253611yescharacteristic
572C22#259811yescharacteristic
573C22C2×C4×Q1611yescharacteristic
574C22C2×#44011yescharacteristic
575C22C4×#13711yescharacteristic
576C22C4×#13711yescharacteristic
577C22C2×#45111yescharacteristic
578C22#300211yescharacteristic
579C22#300411yescharacteristic
580C22#300611yescharacteristic
581C22C2×#46211yescharacteristic
582C22#303511yescharacteristic
583C22#303611yescharacteristic
584C22#303911yescharacteristic
585C22#310411yescharacteristic
586C22#310411yescharacteristic
587C22#310711yescharacteristic
588C22#310811yescharacteristic
589C22#311011yescharacteristic
590C22#311211yescharacteristic
591C22#311611yescharacteristic
592C22#311711yescharacteristic
593C22#320711yescharacteristic
594C22#320911yescharacteristic
595C22#321811yescharacteristic
596C22#321911yescharacteristic
597C22#323011yescharacteristic
598C22#323111yescharacteristic
599C22#323611yescharacteristic
600C22#323711yescharacteristic
601C4—24no—
602C4—24no—
603C4—24no—
604C4—24no—
605C4—44no—
606C4—42no—
607C4—48no—
608C4—42no—
609C4—84no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4#235912yes—
614C4#236112yes—
615C4#246112yes—
616C4#246812yes—
617C2#1417611yescharacteristic
618C2C4×#45111yescharacteristic
619C2#2145411yescharacteristic
620C2C4×#46211yescharacteristic
621C2#2150411yescharacteristic
622C2#2222111yescharacteristic
623C2#2222511yescharacteristic
624C2#2406411yescharacteristic
625C2#2406511yescharacteristic
626C2#2421211yescharacteristic
627C2#2421411yescharacteristic
628C2#2820611yescharacteristic
629C2#2820911yescharacteristic
630C2#2936911yescharacteristic
631C2#2937011yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
32. #2043C4semidirect

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×#20432
C2×#19282
C2×#18632
C4×#3892
#113242
#112072
#110102
#109322
#107952
#107892