Finite Group Databasebeta

← #404158

Subgroups — #404158

Order 512.

1382 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2295C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic
3C2×#2024C211yesmaximal, characteristic
4#14714C211yesmaximal, characteristic
5#11293C212yesmaximal
6#11207C211yesmaximal, characteristic
7#11124C212yesmaximal
8#11050C211yesmaximal, characteristic
9#10907C212yesmaximal
10#10795C211yesmaximal, characteristic
11#10789C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17#2295—24no—
18#2295—24no—
19C2×#391—21no—
20C2×#391—21no—
21C2×#391C2211yescharacteristic
22C2×#391C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#387—21no—
26C2×#387—21no—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×#376—21no—
31C2×C43C2211yescharacteristic
32#2057—24no—
33#2024—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41C2×#336C2211yescharacteristic
42C2×#336C2211yescharacteristic
43C4⋊#95—21no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8—21no—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8Q814yescomplemented
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—21no—
73C22×C42—21no—
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—42no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—42no—
89C2×#103—42no—
90#391—22no—
91#391—22no—
92#391—22no—
93#391—44no—
94#391—44no—
95#391D414yes—
96#389—22no—
97#389—22no—
98#389—22no—
99#389D414yes—
100#387—28no—
101#387—28no—
102#384—42no—
103C4×C4⋊C4—28no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106#376—28no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C43—22no—
119C43—22no—
120#340—24no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—42no—
125#336—24no—
126#336—22no—
127#336—22no—
128#336—24no—
129C22×Q8—24no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8C2×D412yes—
144C22×Q8C2×Q812yes—
145C22×Q8#4112yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
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×C8—22no—
171C22×C8—22no—
172C22×C8#4114yes—
173C4⋊Q8—48no—
174C4⋊Q8—48no—
175C4×Q8—28no—
176C4×Q8—28no—
177C4×Q8—28no—
178C4×Q8—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
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—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
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⋊C4—42no—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×Q812yes—
221C2×C4⋊C4#4114yes—
222C2×C4⋊C4#4112yes—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—41no—
247C2×C42—41no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42Q1614yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×Q812yes—
254C2×C42#4112yes—
255#107—48no—
256#103—48no—
257#103—48no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269C24C22×D411yescharacteristic
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—48no—
280C2×Q8—48no—
281C2×Q8—48no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—84no—
289C2×Q8—84no—
290C2×Q8—84no—
291C2×Q8—84no—
292C2×Q8C22⋊Q814yes—
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—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
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—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4#12012yes—
366C22×C4#12012yes—
367C22×C4C22⋊Q812yes—
368C22×C4C22⋊Q812yes—
369C22×C4C2×Q1612yes—
370C22×C4C2×Q1612yes—
371C22×C4C2×Q1612yes—
372C22×C4#13512yes—
373C22×C4AGL(1,ℤ/8)12yes—
374C22×C4AGL(1,ℤ/8)12yes—
375C22×C4#13712yes—
376C22×C4#13712yes—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×Q811yescharacteristic
380C22×C4C2×#4111yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4112yes—
384C22×C4C2×#4112yes—
385C22×C4H(5,𝔽2)12yes—
386C22×C42−1+412yes—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
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—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—48no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—84no—
421C4⋊C4—84no—
422C4⋊C4C22⋊Q814yes—
423C42—216no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—24no—
429C42—24no—
430C42—28no—
431C42—44no—
432C42—44no—
433C42—48no—
434C42—42no—
435C42—42no—
436C42—44no—
437C23C2×#12011yescharacteristic
438C23C2×C22⋊Q811yescharacteristic
439C23#52511yescharacteristic
440C23#52511yescharacteristic
441C23#53211yescharacteristic
442C23#53511yescharacteristic
443C23#53611yescharacteristic
444C23C22×Q1611yescharacteristic
445C23C2×#13511yescharacteristic
446C23C2×AGL(1,ℤ/8)11yescharacteristic
447C23C2×AGL(1,ℤ/8)11yescharacteristic
448C23C2×#13711yescharacteristic
449C23C2×#13711yescharacteristic
450C23#57511yescharacteristic
451C23#57511yescharacteristic
452Q8—28no—
453Q8—44no—
454Q8—44no—
455Q8—44no—
456Q8—816no—
457C2×C4—22no—
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—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—82no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4#44914yes—
539C2×C4#45114yes—
540C2×C4#47714yes—
541C2×C4C2×C22⋊Q811yescharacteristic
542C2×C4C2×C22⋊Q811yescharacteristic
543C2×C4#52511yescharacteristic
544C2×C4#52511yescharacteristic
545C2×C4#52912yes—
546C2×C4#53312yes—
547C2×C4#54212yes—
548C2×C4#54412yes—
549C2×C4#54612yes—
550C2×C4#54612yes—
551C2×C4#54712yes—
552C2×C4#54812yes—
553C2×C4#55112yes—
554C2×C4#55212yes—
555C2×C4#55512yes—
556C2×C4#55612yes—
557C8—44no—
558C8—82no—
559C22#239611yescharacteristic
560C22#240411yescharacteristic
561C22#243211yescharacteristic
562C22#259911yescharacteristic
563C22#261011yescharacteristic
564C22#262811yescharacteristic
565C22#284811yescharacteristic
566C22C2×#45111yescharacteristic
567C22C2×#44911yescharacteristic
568C22#300611yescharacteristic
569C22#300711yescharacteristic
570C22C2×#47711yescharacteristic
571C22#307611yescharacteristic
572C22#307811yescharacteristic
573C22#307911yescharacteristic
574C22#310411yescharacteristic
575C22#310411yescharacteristic
576C22#310511yescharacteristic
577C22#310711yescharacteristic
578C22#311211yescharacteristic
579C22#311311yescharacteristic
580C22#311411yescharacteristic
581C22#311511yescharacteristic
582C22#317011yescharacteristic
583C22#317111yescharacteristic
584C22#318111yescharacteristic
585C22#318211yescharacteristic
586C22#320911yescharacteristic
587C22#321111yescharacteristic
588C22#321511yescharacteristic
589C22#321711yescharacteristic
590C22#323011yescharacteristic
591C22#323111yescharacteristic
592C22#323911yescharacteristic
593C22#324011yescharacteristic
594C4—24no—
595C4—24no—
596C4—24no—
597C4—44no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—44no—
602C4—82no—
603C4—82no—
604C4—84no—
605C4—162no—
606C4#243812yes—
607C4#243912yes—
608C4#247412yes—
609C4#247712yes—
610C2#1488711yescharacteristic
611C2#2377711yescharacteristic
612C2#2378311yescharacteristic
613C2#2393411yescharacteristic
614C2#2393611yescharacteristic
615C2#2567411yescharacteristic
616C2#2567811yescharacteristic
617C2#2939111yescharacteristic
618C2#2939411yescharacteristic
619C2#2955311yescharacteristic
620C2#2956111yescharacteristic
621C2#2998411yescharacteristic
622C2#2998511yescharacteristic
623C2#3295511yescharacteristic
624C2#3295711yescharacteristic
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

Complements

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

NH ≅ G/Ntype
60. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22952
C2×#20572
C2×#20242
#147142
#112932
#112072
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