Finite Group Databasebeta

← #360807

Subgroups — #360807

Order 512.

1410 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2351C211yesmaximal, characteristic
2C2×#1835C211yesmaximal, characteristic
3C2×#1805C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#13955C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11292C212yesmaximal
8#10850C211yesmaximal, characteristic
9#10834C211yesmaximal, characteristic
10#10789C212yesmaximal
11#10789C212yesmaximal
12C22×C4×Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2351—24no—
20#2351—24no—
21C2×#389—22no—
22C2×#389C2212yes—
23C2×#382—21no—
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1835—24no—
30#1805—24no—
31C4×#103C418yescomplemented
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1442—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—21no—
45#1434—21no—
46C23×Q8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×Q8—24no—
51C2×C4×Q8—24no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—24no—
57C2×C4×Q8—42no—
58C2×C4×Q8—42no—
59C2×C4×Q8C2×C414yes—
60C2×C4×Q8D414yes—
61C2×C4×Q8Q814yes—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D412yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C42—21no—
72C22×C42—22no—
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103C2×C414yes—
87C2×#103C2×C414yes—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90C2×C4×C8C2×C414yes—
91#389—216no—
92#389—24no—
93#389—24no—
94#382—28no—
95#382—22no—
96#382—22no—
97#382—22no—
98#382D414yes—
99C4×C4⋊C4—28no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4C2×C414yes—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340—24no—
117#340—24no—
118#340—24no—
119#340—42no—
120#340—42no—
121#340C2×C414yes—
122#340C2×C414yes—
123#336—42no—
124#336—42no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8C22×C412yes—
133C22×Q8C2×D412yes—
134C22×Q8C2×Q812yes—
135C22×Q8#4112yes—
136C22×Q8#4112yes—
137C22×Q8#4112yes—
138C22×Q8#4112yes—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8C22×C412yes—
170C22×C8C22×C412yes—
171C4×Q8—28no—
172C4×Q8—28no—
173C4×Q8—28no—
174C4×Q8—48no—
175C4×Q8—48no—
176C4×Q8—48no—
177C4×Q8—48no—
178C4×Q8C4⋊C4116yes—
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—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×Q812yes—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—28no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—24no—
232C2×C42—44no—
233C2×C42—44no—
234C2×C42—82no—
235C2×C42C22×C412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×Q812yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241C4⋊C8—44no—
242#103—28no—
243#103—28no—
244#103—48no—
245C4×C8—28no—
246#95—28no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—22no—
251#95—22no—
252#95—44no—
253#95—42no—
254#95—42no—
255#95—44no—
256#95—82no—
257#95—82no—
258C24C22×D411yescharacteristic
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8C2×C4⋊C414yes—
281C2×Q8#11714yes—
282C2×Q8#11714yes—
283C2×Q8#12414yes—
284C2×Q8#12414yes—
285C2×Q8#12514yes—
286C2×Q8#12514yes—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4C4×D412yes—
341C22×C4C4×D412yes—
342C22×C4#12012yes—
343C22×C4#12012yes—
344C22×C4C22⋊Q812yes—
345C22×C4C22⋊Q812yes—
346C22×C4#13514yes—
347C22×C4C23×C411yescharacteristic
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×Q811yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4H(5,𝔽2)11yescharacteristic
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C42−1+411yescharacteristic
360C22×C42−1+411yescharacteristic
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—216no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—48no—
393C4⋊C4—416no—
394C4⋊C4—416no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4C2×C4⋊C414yes—
406C42—216no—
407C42—416no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—84no—
413C42C2×C4⋊C414yes—
414C42C4×D414yes—
415C42C22⋊Q814yes—
416C42#12314yes—
417C23C2×C4×D411yescharacteristic
418C23C2×#12011yescharacteristic
419C23C2×C22⋊Q811yescharacteristic
420C23#54611yescharacteristic
421C23#54611yescharacteristic
422C23#54811yescharacteristic
423C23#54811yescharacteristic
424C23C2×#13512yes—
425C23#57512yes—
426C23#57712yes—
427C23#57812yes—
428Q8—24no—
429Q8—24no—
430Q8—24no—
431Q8—24no—
432Q8—24no—
433Q8—24no—
434Q8—24no—
435Q8#38218yes—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4#38514yes—
495C2×C4#38514yes—
496C2×C4C22×C4⋊C411yescharacteristic
497C2×C4C2×#11711yescharacteristic
498C2×C4C2×#11711yescharacteristic
499C2×C4C2×C4×D411yescharacteristic
500C2×C4#52011yescharacteristic
501C2×C4#52011yescharacteristic
502C2×C4C2×C22⋊Q811yescharacteristic
503C2×C4C2×#12311yescharacteristic
504C2×C4C2×#12411yescharacteristic
505C2×C4C2×#12411yescharacteristic
506C2×C4C2×#12511yescharacteristic
507C2×C4C2×#12511yescharacteristic
508C2×C4#53511yescharacteristic
509C2×C4#53611yescharacteristic
510C2×C4#54411yescharacteristic
511C2×C4#54411yescharacteristic
512C2×C4#54611yescharacteristic
513C2×C4#54611yescharacteristic
514C2×C4#54811yescharacteristic
515C2×C4#54811yescharacteristic
516C2×C4#55211yescharacteristic
517C2×C4#55211yescharacteristic
518C2×C4#55211yescharacteristic
519C2×C4#55211yescharacteristic
520C2×C4#55311yescharacteristic
521C2×C4#55311yescharacteristic
522C2×C4#55311yescharacteristic
523C2×C4#55311yescharacteristic
524C8—42no—
525C8—42no—
526C8—82no—
527C22C2×#38511yescharacteristic
528C22#235211yescharacteristic
529C22#235211yescharacteristic
530C22#243511yescharacteristic
531C22#243611yescharacteristic
532C22#244211yescharacteristic
533C22#244211yescharacteristic
534C22C4×#13511yescharacteristic
535C22#294311yescharacteristic
536C22#294911yescharacteristic
537C22#295011yescharacteristic
538C22#300212yes—
539C22#301312yes—
540C22#307612yes—
541C22#308212yes—
542C22#330012yes—
543C22#330112yes—
544C22#332612yes—
545C22#332712yes—
546C22#336412yes—
547C22#336512yes—
548C22#337112yes—
549C22#337712yes—
550C4—28no—
551C4—24no—
552C4—24no—
553C4—44no—
554C4—44no—
555C4—82no—
556C4—84no—
557C4—82no—
558C4—82no—
559C4—162no—
560C4C2×#38211yescharacteristic
561C4#231711yescharacteristic
562C4#235211yescharacteristic
563C4#235211yescharacteristic
564C4#244211yescharacteristic
565C4#244211yescharacteristic
566C4#250311yescharacteristic
567C4#250411yescharacteristic
568C2#1388111yescharacteristic
569C2#2087211yescharacteristic
570C2#2088011yescharacteristic
571C2#2276111yescharacteristic
572C2#2276511yescharacteristic
573C2#2276911yescharacteristic
574C2#2277311yescharacteristic
575C2#2588912yes—
576C2#2593712yes—
577C2#2611412yes—
578C2#2611612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
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
31. C4×#103C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23512
C2×#18352
C2×#18052
C2×C4×#1032
#139552
#112932
#112922
#108502
#108342
#107892