Finite Group Databasebeta

← #360673

Subgroups — #360673

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#1847C211yesmaximal, characteristic, complemented
3C2×#1834C211yesmaximal, characteristic, complemented
4C2×#1750C211yesmaximal, characteristic
5#14340C211yesmaximal, characteristic
6#11096C212yesmaximal
7#11095C212yesmaximal
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10842C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2212yes—
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2314—24no—
19#2314—24no—
20C2×#388—21no—
21C2×#388—21no—
22C2×#387—21no—
23C2×#387—21no—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1847—24no—
34#1834—24no—
35#1750C418yescomplemented
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1441—21no—
44#1441—21no—
45#1439—21no—
46#1439—21no—
47C4⋊#95C2212yes—
48C24×C4C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117C2×C414yes—
56C2×#117Q814yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—42no—
81C2×#106—24no—
82C2×#106—24no—
83C2×#106—42no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8C2×C418yes—
87C2×#97—22no—
88C2×#97—22no—
89C2×#97C2×C414yes—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8C2×C414yes—
93#388—28no—
94#388—28no—
95#387—28no—
96#387—28no—
97#384—22no—
98#384—22no—
99#384—22no—
100#384D414yes—
101#382—22no—
102#382—22no—
103#382—22no—
104#382D414yes—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—42no—
110#380—42no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126#340—24no—
127#340—24no—
128#340—24no—
129#340—24no—
130#340—42no—
131#340—42no—
132#336C2×C418yes—
133C25C2×D411yescharacteristic
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C22×C412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×Q812yes—
155C23×C4C2411yescharacteristic
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×C8C22×C414yes—
167C22×C8C22×C414yes—
168C22×C8#4114yes—
169C22×C8#4114yes—
170#117—28no—
171#117—28no—
172#117—28no—
173#117—28no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
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—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
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—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C4⋊C414yes—
211C2×C4⋊C4C4⋊C414yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—82no—
224C2×C22⋊C4—82no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4C4⋊C414yes—
228C2×C22⋊C4C4⋊C414yes—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—82no—
236C2×C42—82no—
237C2×C42C22×C412yes—
238C2×C42C22×C412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×Q812yes—
242C2×C42C2×Q812yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245#107—48no—
246#106—48no—
247C22⋊C8—216no—
248#97—28no—
249C4×C8—28no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—44no—
257#95—42no—
258#95—42no—
259#95—82no—
260#95—82no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24C2×C4⋊C412yes—
272C24C22×D411yescharacteristic
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
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×C4—42no—
341C22×C4C2×C4⋊C412yes—
342C22×C4C2×C4⋊C412yes—
343C22×C4C2×C4⋊C412yes—
344C22×C4C2×C4⋊C412yes—
345C22×C4C2×C4⋊C412yes—
346C22×C4#12012yes—
347C22×C4#12012yes—
348C22×C4#12312yes—
349C22×C4#12312yes—
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×Q811yescharacteristic
354C22×C4C2×#4112yes—
355C22×C4C2×#4112yes—
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4H(5,𝔽2)12yes—
359C22×C42−1+412yes—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8#11718yes—
374C4⋊C4—28no—
375C4⋊C4—28no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—84no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C22⋊C4—28no—
397C22⋊C4—28no—
398C22⋊C4—84no—
399C22⋊C4—84no—
400C22⋊C4—88no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—84no—
406C42—84no—
407C42C4×D414yes—
408C42C4×D414yes—
409C42C22⋊Q814yes—
410C42C22⋊Q814yes—
411C23—22no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23C22×C4⋊C411yescharacteristic
435C23C2×#12011yescharacteristic
436C23C2×#12311yescharacteristic
437C23#53211yescharacteristic
438C23#53211yescharacteristic
439C23#53511yescharacteristic
440C23#53611yescharacteristic
441C23#57612yes—
442C23#57712yes—
443C23#57712yes—
444C23#57812yes—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
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—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4#38514yes—
509C2×C4#38514yes—
510C2×C4C2×#11712yes—
511C2×C4C2×C4×D411yescharacteristic
512C2×C4C2×C4×D411yescharacteristic
513C2×C4#52012yes—
514C2×C4C2×C22⋊Q811yescharacteristic
515C2×C4C2×C22⋊Q811yescharacteristic
516C2×C4#53312yes—
517C2×C4#54412yes—
518C2×C4#54612yes—
519C2×C4#54712yes—
520C2×C4#54812yes—
521C2×C4#55012yes—
522C2×C4#55212yes—
523C2×C4#55312yes—
524C2×C4#55512yes—
525C2×C4#55612yes—
526C8—28no—
527C8—44no—
528C22—42no—
529C22—44no—
530C22—42no—
531C22—42no—
532C22—44no—
533C22—82no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—82no—
539C22#228511yescharacteristic
540C22C2×#38511yescharacteristic
541C22#231711yescharacteristic
542C22#243111yescharacteristic
543C22#243211yescharacteristic
544C22#249911yescharacteristic
545C22#250011yescharacteristic
546C22#291212yes—
547C22#291312yes—
548C22#300811yescharacteristic
549C22#300911yescharacteristic
550C22#301011yescharacteristic
551C22#301111yescharacteristic
552C22#309812yes—
553C22#309912yes—
554C22#317812yes—
555C22#317812yes—
556C22#317912yes—
557C22#317912yes—
558C22#322412yes—
559C22#322512yes—
560C22#324112yes—
561C22#324212yes—
562C4—24no—
563C4—24no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—82no—
571C4—162no—
572C4—162no—
573C4#235212yes—
574C4#235412yes—
575C4#243912yes—
576C4#244112yes—
577C2—82no—
578C2#1387911yescharacteristic
579C2#2082212yes—
580C2#2084711yescharacteristic
581C2#2084811yescharacteristic
582C2#2199212yes—
583C2#2561311yescharacteristic
584C2#2561411yescharacteristic
585C2#2568511yescharacteristic
586C2#2568611yescharacteristic
587C2#2788812yes—
588C2#2790412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1847C2semidirect
3. C2×#1834C2semidirect
8. #10908C2semidirect
9. #10907C2semidirect
10. #10843C2semidirect
11. #10842C2semidirect
35. #1750C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#18472
C2×#18342
C2×#17502
#143402
#110962
#110952
#109082
#109072
#108432
#108422