Finite Group Databasebeta

← #366471

Subgroups — #366471

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1933C212yesmaximal
2C2×#1805C211yesmaximal, characteristic
3C2×C8×C4⋊C4C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#14150C211yesmaximal, characteristic
6#13942C211yesmaximal, characteristic
7#11292C212yesmaximal
8#11190C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10842C211yesmaximal, characteristic
11#10820C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×C4×C8C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#391C2212yes—
20C2×#389C2212yes—
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1933—28no—
26#1805C418yescomplemented
27C8×C4⋊C4—24no—
28C4×#103—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1444—21no—
38#1442—21no—
39#1441—21no—
40#1441—21no—
41#1436—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—22no—
45C23×Q8C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8C2×C414yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42—21no—
65C22×C42—21no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8C2×C414yes—
71C2×#103—24no—
72C2×#103—22no—
73C2×#103—24no—
74C2×#103—22no—
75C2×#103—22no—
76C2×#103—22no—
77C2×#103—22no—
78C2×#103C2×C418yes—
79C2×#103D414yes—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8—24no—
83C2×C4×C8—24no—
84C2×C4×C8—24no—
85C2×C4×C8D414yes—
86#391—24no—
87#391—42no—
88#389—24no—
89#389—42no—
90#382—22no—
91#382—22no—
92#382—22no—
93#382C2×C414yes—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95D412yes—
111#340—24no—
112#340—22no—
113#340—22no—
114#340—24no—
115#340—24no—
116#340—42no—
117#340C2×C418yes—
118#336—24no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8C22×C412yes—
131C22×Q8C2×D412yes—
132C22×Q8#4112yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8C22×C414yes—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C22×C8#4112yes—
165C4×Q8—28no—
166C4×Q8—28no—
167C4×Q8—44no—
168C4×Q8—44no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
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—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C414yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4#4112yes—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—82no—
220C2×C42C22×C412yes—
221C2×C42C2×D412yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42#4112yes—
225C2×C42#4112yes—
226C4⋊C8—28no—
227#103—28no—
228#103—216no—
229#103—28no—
230C4×C8—28no—
231C4×C8—48no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—42no—
239#95—42no—
240#95—42no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—82no—
245C24C22×D411yescharacteristic
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8C4×D414yes—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
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—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
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—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4C4×D412yes—
327C22×C4C4×D412yes—
328C22×C4#12112yes—
329C22×C4#12112yes—
330C22×C4#12412yes—
331C22×C4#12412yes—
332C22×C4#13514yes—
333C22×C4#13514yes—
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4H(5,𝔽2)11yescharacteristic
345C22×C42−1+411yescharacteristic
346C22×C42−1+411yescharacteristic
347C22×C42−1+411yescharacteristic
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8#12114yes—
369C4⋊C4—28no—
370C4⋊C4—28no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—41no—
387C4⋊C4—48no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—88no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—48no—
398C42—84no—
399C42#11714yes—
400C42C4×D414yes—
401C42#12314yes—
402C42#12614yes—
403C23C2×C4×D411yescharacteristic
404C23C2×#12111yescharacteristic
405C23C2×#12411yescharacteristic
406C23D4211yescharacteristic
407C23#54711yescharacteristic
408C23#54811yescharacteristic
409C23#54811yescharacteristic
410C23C2×#13512yes—
411C23C2×#13512yes—
412C23#57712yes—
413C23#57812yes—
414Q8—24no—
415Q8—24no—
416Q8—42no—
417Q8—42no—
418Q8—42no—
419Q8—42no—
420Q8—42no—
421Q8—42no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4#38814yes—
481C2×C4#38814yes—
482C2×C4C2×#11711yescharacteristic
483C2×C4C2×C4×D411yescharacteristic
484C2×C4C2×C4×D411yescharacteristic
485C2×C4C4×#4111yescharacteristic
486C2×C4#51911yescharacteristic
487C2×C4#52011yescharacteristic
488C2×C4C2×#12111yescharacteristic
489C2×C4C2×#12311yescharacteristic
490C2×C4#52511yescharacteristic
491C2×C4C2×#12611yescharacteristic
492C2×C4#52911yescharacteristic
493C2×C4#53211yescharacteristic
494C2×C4#53611yescharacteristic
495C2×C4#53711yescharacteristic
496C2×C4#54111yescharacteristic
497C2×C4#54211yescharacteristic
498C2×C4#54711yescharacteristic
499C2×C4#54711yescharacteristic
500C2×C4#54711yescharacteristic
501C2×C4#55211yescharacteristic
502C2×C4#55211yescharacteristic
503C2×C4#55211yescharacteristic
504C2×C4#55211yescharacteristic
505C2×C4#55211yescharacteristic
506C2×C4#55311yescharacteristic
507C2×C4#55511yescharacteristic
508C2×C4#55511yescharacteristic
509C2×C4#55611yescharacteristic
510C8—24no—
511C8—42no—
512C8—44no—
513C22C2×#38811yescharacteristic
514C22#236111yescharacteristic
515C22#236211yescharacteristic
516C22#248911yescharacteristic
517C22#249111yescharacteristic
518C22#253611yescharacteristic
519C22#253711yescharacteristic
520C22C4×#13512yes—
521C22#294712yes—
522C22#305011yescharacteristic
523C22#305011yescharacteristic
524C22#305711yescharacteristic
525C22#305911yescharacteristic
526C22#313112yes—
527C22#313712yes—
528C22#328212yes—
529C22#328512yes—
530C22#333612yes—
531C22#333712yes—
532C22#336012yes—
533C22#336412yes—
534C22#336412yes—
535C22#336512yes—
536C4—24no—
537C4—24no—
538C4—44no—
539C4—44no—
540C4—44no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—84no—
545C4—162no—
546C4C4×#12111yescharacteristic
547C4#232311yescharacteristic
548C4#234411yescharacteristic
549C4#234611yescharacteristic
550C4#248511yescharacteristic
551C4#248611yescharacteristic
552C4#249511yescharacteristic
553C4#249611yescharacteristic
554C2#1399411yescharacteristic
555C2#2108311yescharacteristic
556C2#2109011yescharacteristic
557C2#2300012yes—
558C2#2306412yes—
559C2#2773212yes—
560C2#2777212yes—
561C2#2818911yescharacteristic
562C2#2819011yescharacteristic
563C2#2822411yescharacteristic
564C2#2822611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #1805C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19332
C2×#18052
C2×C8×C4⋊C42
C2×C4×#1032
#141502
#139422
#112922
#111902
#108502
#108422
#108202