Finite Group Databasebeta

← #360622

Subgroups — #360622

Order 512.

1506 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2347C211yesmaximal, characteristic
2C2×#1825C211yesmaximal, characteristic
3C2×#1808C211yesmaximal, characteristic
4C2×#1735C211yesmaximal, characteristic
5#13951C211yesmaximal, characteristic
6#11328C212yesmaximal, complemented
7#11293C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17#2347—24no—
18#2347—24no—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#388—22no—
22C2×#383—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#1825—24no—
29#1808—24no—
30#1735C418yescomplemented
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1439—22no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45#1434—21no—
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—24no—
50C23×C8—42no—
51C23×C8C2×C414yes—
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—24no—
60C2×#117—42no—
61C2×#117C2×C414yes—
62C2×#117C2×C414yes—
63C2×#117D414yes—
64C2×#117Q814yes—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D412yes—
78C22×C42—21no—
79C22×C42—21no—
80C22×C42—22no—
81C22×C42C2312yes—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88#389—24no—
89#389—24no—
90#389—24no—
91#389—42no—
92#388—216no—
93#383—28no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380D414yes—
98C4×C4⋊C4—28no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95D411yescharacteristic
111C2×#95D411yescharacteristic
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—42no—
117#340C2×C414yes—
118#340C2×C414yes—
119#340C2×C414yes—
120#340C2×C414yes—
121#336—24no—
122#336—42no—
123C25C2×D411yescharacteristic
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4C22×C412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2×Q812yes—
141C23×C4#4112yes—
142C23×C4#4112yes—
143C23×C4C2411yescharacteristic
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—28no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—28no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—44no—
161C22×C8—48no—
162C22×C8—48no—
163C22×C8C22×C412yes—
164C22×C8C22×C412yes—
165C22×C8#4114yes—
166#117—28no—
167#117—28no—
168#117—28no—
169#117—28no—
170#117—44no—
171#117—48no—
172#117C4⋊C4116yescomplemented
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
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—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×Q812yes—
209C2×C4⋊C4C2×Q812yes—
210C2×C4⋊C4#4114yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—81no—
222C2×C22⋊C4—81no—
223C2×C22⋊C4—81no—
224C2×C22⋊C4—81no—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—28no—
236C2×C42—44no—
237C2×C42—82no—
238C2×C42—82no—
239#107—48no—
240C22⋊C8—44no—
241#95—28no—
242#95—28no—
243#95—24no—
244#95—22no—
245#95—22no—
246#95—44no—
247#95—44no—
248#95—42no—
249#95—82no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24C4×D412yes—
259C24#12012yes—
260C24C22⋊Q812yes—
261C24C22×D411yescharacteristic
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4C2×C4⋊C414yes—
328C22×C4C4×D414yes—
329C22×C4C4×D412yes—
330C22×C4#12012yes—
331C22×C4C22⋊Q814yes—
332C22×C4C22⋊Q812yes—
333C22×C4#12314yes—
334C22×C4#13514yes—
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C22×Q811yescharacteristic
339C22×C4C2×#4112yes—
340C22×C4C2×#4112yes—
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4H(5,𝔽2)12yes—
344C22×C42−1+412yes—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—416no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8#11714yes—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—416no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—88no—
384C4⋊C4C2×C4⋊C414yes—
385C4⋊C4C2×C4⋊C414yes—
386C4⋊C4#11714yes—
387C22⋊C4—216no—
388C22⋊C4—48no—
389C22⋊C4—48no—
390C22⋊C4—84no—
391C22⋊C4—84no—
392C42—216no—
393C42—48no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—88no—
399C23—24no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—22no—
407C23—24no—
408C23—22no—
409C23—22no—
410C23—24no—
411C23—22no—
412C23—22no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23#38514yes—
422C23C2×C4×D411yescharacteristic
423C23C2×#12011yescharacteristic
424C23C2×C22⋊Q811yescharacteristic
425C23#54612yes—
426C23#54712yes—
427C23C2×#13512yes—
428C23#57512yes—
429C23#57611yescharacteristic
430C23#57611yescharacteristic
431C23#57811yescharacteristic
432C23#57811yescharacteristic
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—28no—
441C2×C4—28no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
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#38514yes—
494C2×C4C22×C4⋊C411yescharacteristic
495C2×C4C2×#11711yescharacteristic
496C2×C4C2×#11711yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4#51811yescharacteristic
499C2×C4#51911yescharacteristic
500C2×C4C2×C22⋊Q811yescharacteristic
501C2×C4C2×#12311yescharacteristic
502C2×C4#53212yes—
503C2×C4#53312yes—
504C2×C4#53712yes—
505C2×C4#54412yes—
506C2×C4#54612yes—
507C2×C4#54712yes—
508C2×C4#55112yes—
509C2×C4#55112yes—
510C2×C4#55212yes—
511C2×C4#55212yes—
512C8—24no—
513C8—42no—
514C8—82no—
515C22—24no—
516C22—24no—
517C22—24no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22C2×#38511yescharacteristic
527C22#234211yescharacteristic
528C22#234411yescharacteristic
529C22#243012yes—
530C22#243812yes—
531C22C4×#13511yescharacteristic
532C22#294311yescharacteristic
533C22#295111yescharacteristic
534C22#295111yescharacteristic
535C22#300212yes—
536C22#300811yescharacteristic
537C22#301111yescharacteristic
538C22#307612yes—
539C22#308311yescharacteristic
540C22#308311yescharacteristic
541C22#329712yes—
542C22#330212yes—
543C22#330912yes—
544C22#331312yes—
545C22#333912yes—
546C22#334212yes—
547C22#334712yes—
548C22#334912yes—
549C4—24no—
550C4—44no—
551C4—44no—
552C4—44no—
553C4—44no—
554C4—82no—
555C4—84no—
556C4—82no—
557C4—162no—
558C4#228511yescharacteristic
559C4#231611yescharacteristic
560C4#234211yescharacteristic
561C4#234411yescharacteristic
562C4#243812yes—
563C4#249812yes—
564C2—44no—
565C2#1387811yescharacteristic
566C2#2087211yescharacteristic
567C2#2088111yescharacteristic
568C2#2256911yescharacteristic
569C2#2257611yescharacteristic
570C2#2261411yescharacteristic
571C2#2261611yescharacteristic
572C2#2553412yes—
573C2#2554212yes—
574C2#2598912yes—
575C2#2599712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11328C2semidirect
7. #11293C2semidirect
8. #10907C2semidirect
9. #10884C2semidirect
30. #1735C4semidirect
172. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23472
C2×#18252
C2×#18082
C2×#17352
#139512
#113282
#112932
#109072
#108842
#107312
#107142