Finite Group Databasebeta

← #369000

Subgroups — #369000

Order 512.

1584 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2584C211yesmaximal, characteristic
2C2×Q8×C4⋊C4C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic
4C2×#1989C211yesmaximal, characteristic
5C2×#1985C211yesmaximal, characteristic
6#11207C212yesmaximal
7#11207C212yesmaximal
8#10885C212yesmaximal
9#10884C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8—21no—
14C22×C4×Q8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17#2584—24no—
18#2584C418yescomplemented
19C2×#398—21no—
20C2×#395—21no—
21Q8×C4⋊C4—24no—
22Q8×C4⋊C4—24no—
23C2×#389—21no—
24C2×#389—22no—
25C2×#389—22no—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#384C2212yes—
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1992—24no—
34#1989—24no—
35#1985—24no—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48C23×Q8D411yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125C2×C414yes—
58C2×C4×Q8—28no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—48no—
64C2×C4×Q8C2×C414yes—
65C2×C4×Q8D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D412yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C42—22no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×#97—22no—
91C2×#97—22no—
92C2×#97—22no—
93#398—28no—
94#395—28no—
95#389—28no—
96#389—216no—
97#389—216no—
98#389—24no—
99#389—24no—
100#389C2×C418yes—
101#389C2×C418yes—
102#384—24no—
103#384—24no—
104#384—22no—
105#384D414yes—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C414yes—
111C4×C4⋊C4D414yes—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117#340—22no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340Q818yescomplemented
127#336—24no—
128#336—24no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—24no—
132C22×Q8—24no—
133C22×Q8C22⋊C414yes—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2411yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8C2×Q814yes—
162#125—28no—
163#125—28no—
164#125—44no—
165#125—44no—
166C4×Q8—28no—
167C4×Q8—28no—
168C4×Q8—28no—
169C4×Q8—28no—
170C4×Q8—432no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4C22⋊C418yes—
222C2×C4⋊C4C22⋊C414yes—
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C414yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×Q814yes—
230C2×C4⋊C4C2×Q814yes—
231C2×C4⋊C4#4114yes—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—28no—
237C2×C42—24no—
238C2×C42—24no—
239C2×C42—44no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—81no—
243C2×C42—81no—
244C2×C42C22×C414yes—
245C2×C42C22×C412yes—
246C2×C42C2×D414yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252C4⋊C8—44no—
253#97—44no—
254#95—28no—
255#95—24no—
256#95—28no—
257#95—28no—
258#95—24no—
259C24C22×D411yescharacteristic
260C2×Q8—28no—
261C2×Q8—216no—
262C2×Q8—28no—
263C2×Q8—28no—
264C2×Q8—216no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4C2×C22⋊C414yes—
318C22×C4C2×C22⋊C412yes—
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C414yes—
321C22×C4#12012yes—
322C22×C4#12012yes—
323C22×C4#12312yes—
324C22×C4#12312yes—
325C22×C4AGL(1,ℤ/8)14yes—
326C22×C4#13714yes—
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×Q812yes—
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4112yes—
335C22×C42−1+412yes—
336C22×C42−1+412yes—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—216no—
360C4⋊C4—28no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—48no—
370C4⋊C4—416no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—416no—
381C4⋊C4—84no—
382C4⋊C4—88no—
383C4⋊C4—88no—
384C4⋊C4C4×Q818yescomplemented
385C4⋊C4C4⋊Q814yes—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—24no—
390C42—48no—
391C42—416no—
392C42—84no—
393C42C4×D414yes—
394C42C4×D414yes—
395C42#12114yes—
396C42#12114yes—
397C23C22×C22⋊C411yescharacteristic
398C23C2×#12011yescharacteristic
399C23C2×#12311yescharacteristic
400C23#54812yes—
401C23D4×Q812yes—
402C23C2×AGL(1,ℤ/8)12yes—
403C23C2×#13712yes—
404C23#57611yescharacteristic
405C23#57711yescharacteristic
406C23#57711yescharacteristic
407C23#57811yescharacteristic
408Q8—232no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—28no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4#38614yes—
469C2×C4#38614yes—
470C2×C4#41718yes—
471C2×C4C2×C4×D411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C2×C4×Q812yes—
474C2×C4#51912yes—
475C2×C4C2×#12111yescharacteristic
476C2×C4C2×#12111yescharacteristic
477C2×C4C2×C4⋊Q811yescharacteristic
478C2×C4#53211yescharacteristic
479C2×C4#53312yes—
480C2×C4#53611yescharacteristic
481C2×C4#53611yescharacteristic
482C2×C4#54112yes—
483C2×C4#54712yes—
484C2×C4#54812yes—
485C2×C4D4×Q812yes—
486C2×C4D4×Q812yes—
487C2×C4#55612yes—
488C2×C4#55712yes—
489C8—44no—
490C8—81no—
491C8—81no—
492C22C2×#38611yescharacteristic
493C22Q8×C22⋊C412yes—
494C22#243211yescharacteristic
495C22#243411yescharacteristic
496C22#244612yes—
497C22C2×#41712yes—
498C22#289711yescharacteristic
499C22#289811yescharacteristic
500C22#300311yescharacteristic
501C22#300411yescharacteristic
502C22#301211yescharacteristic
503C22#301311yescharacteristic
504C22#309212yes—
505C22#309811yescharacteristic
506C22#309911yescharacteristic
507C22#336612yes—
508C22#336812yes—
509C22#337212yes—
510C22#337412yes—
511C22#338412yes—
512C22#338512yes—
513C22#338612yes—
514C22#338712yes—
515C4—24no—
516C4—28no—
517C4—24no—
518C4—44no—
519C4—44no—
520C4—48no—
521C4—82no—
522C4—84no—
523C4—82no—
524C4—82no—
525C4C4×C4⋊Q811yescharacteristic
526C4#232711yescharacteristic
527C4#236512yes—
528C4#242912yes—
529C4#254511yescharacteristic
530C4#254611yescharacteristic
531C2#1402411yescharacteristic
532C2#2090911yescharacteristic
533C2#2091511yescharacteristic
534C2#2252212yes—
535C2#2252412yes—
536C2#2565511yescharacteristic
537C2#2565711yescharacteristic
538C2#2584411yescharacteristic
539C2#2584711yescharacteristic
540C2#2628812yes—
541C2#2629012yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
18. #2584C4semidirect
126. #340Q8semidirect
384. C4⋊C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25842
C2×Q8×C4⋊C42
C2×#19922
C2×#19892
C2×#19852
#112072
#108852
#108842
#108492
#107032