Finite Group Databasebeta

← #415066

Subgroups — #415066

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic
2C2×#1985C211yesmaximal, characteristic
3C2×#1847C211yesmaximal, characteristic
4C2×#1805C211yesmaximal, characteristic
5C2×#1763C211yesmaximal, characteristic
6#13925C211yesmaximal, characteristic
7#11220C212yesmaximal
8#11219C212yesmaximal
9#11124C212yesmaximal
10#11122C212yesmaximal
11#10849C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#97C2211yescharacteristic
21#2582—24no—
22#2582—24no—
23C2×#398—22no—
24C2×#391—21no—
25C2×#391C2212yes—
26C2×#389—21no—
27C2×#389C2212yes—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1985—24no—
32#1847—24no—
33#1805—24no—
34#1763—24no—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1446—21no—
43#1445—21no—
44#1436—22no—
45C4⋊#95C2212yes—
46C4⋊#95C2212yes—
47C23×Q8D411yescharacteristic
48C23×Q8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—42no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—42no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—22no—
91C2×#103—22no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103Q818yescomplemented
95C2×#97—22no—
96C2×#97—22no—
97C2×#97D414yes—
98#398—216no—
99#391—28no—
100#391—24no—
101#391—24no—
102#389—28no—
103#389—24no—
104#389—24no—
105#384—22no—
106#384D414yes—
107#382—22no—
108#382—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121#340—24no—
122#340—22no—
123#340—22no—
124#340—42no—
125#340—42no—
126#336—24no—
127#336—24no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—24no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—42no—
141C22×Q8C2×D412yes—
142C22×Q8C2×Q814yes—
143C23×C4—21no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8C2×D414yes—
169C22×C8C2×Q814yes—
170C22×C8#4114yes—
171C4×Q8—28no—
172C4×Q8—28no—
173C4×Q8—44no—
174C4×Q8—44no—
175C4×Q8—44no—
176C4×Q8—44no—
177C4×Q8—44no—
178C4×Q8—44no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4C2×Q814yes—
222C2×C42—24no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—42no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—81no—
233C2×C42—81no—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42#4112yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240#107—48no—
241#106—48no—
242C4⋊C8—44no—
243#103—216no—
244#103—44no—
245#103—44no—
246#97—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—44no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—44no—
255#95—82no—
256#95—82no—
257C24C22×D411yescharacteristic
258C2×Q8—28no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—22no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—48no—
270C2×Q8—48no—
271C2×Q8—44no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—44no—
280C2×Q8C4⋊Q814yes—
281C22×C4—22no—
282C22×C4—22no—
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—22no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4#12012yes—
337C22×C4#12012yes—
338C22×C4#12112yes—
339C22×C4#12112yes—
340C22×C4#12312yes—
341C22×C4#12312yes—
342C22×C4#13714yes—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×Q812yes—
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4112yes—
350C22×C4C2×#4111yescharacteristic
351C22×C4H(5,𝔽2)12yes—
352C22×C42−1+412yes—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8C22⋊Q818yes—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—48no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—48no—
384C4⋊C4—44no—
385C4⋊C4—48no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—84no—
393C4⋊C4—88no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—48no—
399C42—84no—
400C42#12114yes—
401C42#12114yes—
402C42#12314yes—
403C42#12414yes—
404C23C2×#12011yescharacteristic
405C23C2×#12111yescharacteristic
406C23C2×#12311yescharacteristic
407C23D4×Q812yes—
408C23#55012yes—
409C23C2×#13712yes—
410C23#57512yes—
411C23#57611yescharacteristic
412C23#57711yescharacteristic
413C23#57711yescharacteristic
414C23#57811yescharacteristic
415Q8—24no—
416Q8—24no—
417Q8—416no—
418Q8—42no—
419Q8—44no—
420Q8—44no—
421Q8—42no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4#39414yes—
483C2×C4#39414yes—
484C2×C4C2×#12111yescharacteristic
485C2×C4C2×#12111yescharacteristic
486C2×C4C2×C22⋊Q812yes—
487C2×C4C2×#12311yescharacteristic
488C2×C4C2×#12411yescharacteristic
489C2×C4C2×C4⋊Q811yescharacteristic
490C2×C4#53211yescharacteristic
491C2×C4#53312yes—
492C2×C4#53511yescharacteristic
493C2×C4#53611yescharacteristic
494C2×C4#53712yes—
495C2×C4#54212yes—
496C2×C4#54612yes—
497C2×C4D4×Q812yes—
498C2×C4#55212yes—
499C2×C4#55412yes—
500C2×C4#55412yes—
501C2×C4#55612yes—
502C8—28no—
503C8—81no—
504C8—81no—
505C22C2×#39411yescharacteristic
506C22#242312yes—
507C22#243111yescharacteristic
508C22#243411yescharacteristic
509C22#253812yes—
510C22#300411yescharacteristic
511C22#300511yescharacteristic
512C22#301211yescharacteristic
513C22#301311yescharacteristic
514C22#305612yes—
515C22#306211yescharacteristic
516C22#306311yescharacteristic
517C22#309512yes—
518C22#309811yescharacteristic
519C22#309911yescharacteristic
520C22#338512yes—
521C22#338712yes—
522C22#338812yes—
523C22#339012yes—
524C22#339812yes—
525C22#340012yes—
526C22#340112yes—
527C22#340312yes—
528C4—24no—
529C4—24no—
530C4—44no—
531C4—44no—
532C4—84no—
533C4—84no—
534C4—82no—
535C4—82no—
536C4—161no—
537C4—161no—
538C4#241712yes—
539C4#249212yes—
540C4#249911yescharacteristic
541C4#250211yescharacteristic
542C4#262011yescharacteristic
543C4#285711yescharacteristic
544C2#1518211yescharacteristic
545C2#2348111yescharacteristic
546C2#2348511yescharacteristic
547C2#2507412yes—
548C2#2507512yes—
549C2#2563911yescharacteristic
550C2#2564011yescharacteristic
551C2#2584611yescharacteristic
552C2#2584711yescharacteristic
553C2#2827712yes—
554C2#2827812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
94. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#19852
C2×#18472
C2×#18052
C2×#17632
#139252
#112202
#112192
#111242
#111222
#108492