Finite Group Databasebeta

← #96912

Subgroups — #96912

Order 512.

1204 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×#2293C211yesmaximal, characteristic
2#10795C211yesmaximal, characteristic, complemented
3#10795C211yesmaximal, characteristic, complemented
4C2×#1271C212yesmaximal, complemented
5C2×#1270C212yesmaximal
6C22×C4×D4C411yescharacteristic
7C22×#117—21no—
8#2293—24no—
9#2293—24no—
10C2×#390C411yescharacteristic
11C2×#390C2211yescharacteristic
12C2×#389C411yescharacteristic
13C2×#389C2211yescharacteristic
14C2×#388—21no—
15C2×#388—21no—
16C2×C4×C4⋊C4C411yescharacteristic
17C2×C4×C22⋊C4—21no—
18C2×C43C2211yescharacteristic
19C2×#336—21no—
20C2×#336—21no—
21C2×#336—21no—
22C2×#336—21no—
23C2×#336C2212yes—
24C2×#336C2212yes—
25#1271—24no—
26#1270—24no—
27C23×D4C2×C411yescharacteristic
28C24×C4—21no—
29C23×C8—22no—
30C23×C8—22no—
31C2×C4×D4—22no—
32C2×C4×D4—22no—
33C2×C4×D4—22no—
34C2×C4×D4—22no—
35C2×C4×D4—22no—
36C2×C4×D4C814yescomplemented
37C2×#117—28no—
38C2×#117—44no—
39C2×#117—42no—
40C2×#117—42no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C4⋊C4C2×C411yescharacteristic
46C22×C22⋊C4—21no—
47C22×C22⋊C4—21no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C42—21no—
52C22×C42C2×C411yescharacteristic
53C22×C42C2×C411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42C2311yescharacteristic
59#390—22no—
60#390—22no—
61#390—22no—
62#390D414yes—
63#389—22no—
64#389—22no—
65#389—22no—
66#389D414yes—
67#388—28no—
68#388—28no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C22⋊C4—28no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—21no—
75C43—22no—
76C43—22no—
77#336—24no—
78#336—24no—
79#336—22no—
80#336—22no—
81#336—22no—
82#336—22no—
83#336—42no—
84#336—42no—
85C25—21no—
86C22×D4—21no—
87C22×D4—21no—
88C22×D4—21no—
89C22×D4—41no—
90C22×D4—41no—
91C22×D4—41no—
92C22×D4—41no—
93C22×D4C22⋊C412yes—
94C22×D4C2×C812yes—
95C22×D4M1612yes—
96C23×C4—24no—
97C23×C4—24no—
98C23×C4—21no—
99C23×C4—24no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—42no—
103C23×C4—42no—
104C23×C4—42no—
105C23×C4—42no—
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C22×C8—24no—
114C22×C8—24no—
115C22×C8—42no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8—42no—
119C22×C8—42no—
120C22×C8—42no—
121C22×C8—42no—
122C22×C8—42no—
123C22×C8—42no—
124C22×C8—42no—
125C22×C8—42no—
126C22×C8—42no—
127C4×D4—28no—
128C4×D4—28no—
129C4×D4—28no—
130C4×D4—28no—
131#117—416no—
132#117—416no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C2×C812yes—
153C2×C4⋊C4M1612yes—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—42no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—42no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—42no—
170C2×C22⋊C4—41no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—24no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—41no—
189C2×C42—41no—
190C2×C42—41no—
191C2×C42—41no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42C2×C812yes—
197C2×C42M1612yes—
198C2×C42D812yes—
199C2×C42SD1612yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202#95—24no—
203#95—24no—
204#95—24no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—41no—
209C24—41no—
210C24—41no—
211C24—41no—
212C24—41no—
213C24—41no—
214C24—41no—
215C24—41no—
216C24—41no—
217C24—41no—
218C24—41no—
219C24—41no—
220C24C2×C22⋊C411yescharacteristic
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4C22⋊C814yes—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—216no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—48no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—48no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4C22⋊C812yes—
300C22×C4C22⋊C812yes—
301C22×C4#10212yes—
302C22×C4#10212yes—
303C22×C4#10412yes—
304C22×C4#10412yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4#12012yes—
308C22×C4#12012yes—
309C22×C4C22×C811yescharacteristic
310C22×C4C2×M1611yescharacteristic
311C22×C4C2×M1611yescharacteristic
312C22×C4C2×M1611yescharacteristic
313C22×C4C2×D811yescharacteristic
314C22×C4C2×SD1611yescharacteristic
315C22×C4#13512yes—
316C22×C4AGL(1,ℤ/8)12yes—
317C22×C4AGL(1,ℤ/8)11yescharacteristic
318C22×C4#13711yescharacteristic
319C2×C8—44no—
320C2×C8—44no—
321C2×C8—44no—
322C2×C8—44no—
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—82no—
326C2×C8—82no—
327C2×C8—82no—
328C2×C8—82no—
329C2×C8—82no—
330C2×C8—82no—
331C2×C8—82no—
332C2×C8—82no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—48no—
341C4⋊C4—44no—
342C4⋊C4—44no—
343C4⋊C4C22⋊C814yes—
344C22⋊C4—216no—
345C22⋊C4—48no—
346C22⋊C4—48no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—24no—
356C42—24no—
357C42—42no—
358C42—42no—
359C42—44no—
360C42—44no—
361C42—42no—
362C42—42no—
363C42—48no—
364C42—42no—
365C42—42no—
366C23—24no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—81no—
401C23#37911yescharacteristic
402C23C2×C22⋊C811yescharacteristic
403C23#40711yescharacteristic
404C23#41011yescharacteristic
405C23#41011yescharacteristic
406C23#41311yescharacteristic
407C23#41311yescharacteristic
408C23C2×#10211yescharacteristic
409C23#41611yescharacteristic
410C23#41711yescharacteristic
411C23#41811yescharacteristic
412C23C2×#10411yescharacteristic
413C23C2×#10411yescharacteristic
414C23#42111yescharacteristic
415C23#42111yescharacteristic
416D4—44no—
417D4—44no—
418D4—44no—
419D4—44no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—432no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4D4⋊C814yes—
498C2×C4C2×C22⋊C811yescharacteristic
499C2×C4C2×C22⋊C811yescharacteristic
500C2×C4#40711yescharacteristic
501C2×C4#40711yescharacteristic
502C2×C4#44712yes—
503C2×C4#44912yes—
504C2×C4#45012yes—
505C2×C4#45212yes—
506C8—84no—
507C8—84no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22#145811yescharacteristic
524C22#145811yescharacteristic
525C22#146411yescharacteristic
526C22#146411yescharacteristic
527C22C2×D4⋊C811yescharacteristic
528C22#147611yescharacteristic
529C22#147711yescharacteristic
530C22#147811yescharacteristic
531C22#148611yescharacteristic
532C22#148811yescharacteristic
533C22#148911yescharacteristic
534C22#149011yescharacteristic
535C22#150011yescharacteristic
536C22#150211yescharacteristic
537C22#150511yescharacteristic
538C22#150911yescharacteristic
539C22#151211yescharacteristic
540C22#151411yescharacteristic
541C22#151711yescharacteristic
542C22#152111yescharacteristic
543C22#152411yescharacteristic
544C22#152911yescharacteristic
545C22#153211yescharacteristic
546C22#153511yescharacteristic
547C22#154011yescharacteristic
548C22#154711yescharacteristic
549C22#155311yescharacteristic
550C22#155711yescharacteristic
551C22#177911yescharacteristic
552C22#178211yescharacteristic
553C22#178511yescharacteristic
554C22#178611yescharacteristic
555C22#178811yescharacteristic
556C22#178911yescharacteristic
557C22#178911yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—24no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—22no—
567C4—44no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—82no—
572C4—82no—
573C2—84no—
574C2#756011yescharacteristic
575C2#757411yescharacteristic
576C2#758011yescharacteristic
577C2#759411yescharacteristic
578C2#760811yescharacteristic
579C2#761411yescharacteristic
580C2#762811yescharacteristic
581C2#764211yescharacteristic
582C2#830011yescharacteristic
583C2#831011yescharacteristic
584C2#831211yescharacteristic
585C2#833011yescharacteristic
586C2#833211yescharacteristic
587C2#833611yescharacteristic
588C2#834111yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #10795C2semidirect
3. #10795C2semidirect
4. C2×#1271C2semidirect
36. C2×C4×D4C8semidirect

Maximal subgroups

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

subgroupindex
C2×#22932
#107952
C2×#12712
C2×#12702