Finite Group Databasebeta

← #360650

Subgroups — #360650

Order 512.

1340 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic
2C2×#1836C211yesmaximal, characteristic
3C2×#1805C211yesmaximal, characteristic
4C2×#1763C211yesmaximal, characteristic
5#13954C211yesmaximal, characteristic
6#11067C212yesmaximal
7#11067C212yesmaximal
8#10885C212yesmaximal
9#10884C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10835C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#97C2211yescharacteristic
18#2355—24no—
19#2355—24no—
20C2×#391C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#387—22no—
24C2×#384C2211yescharacteristic
25C2×#382—21no—
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1836—24no—
32#1805—24no—
33#1763C418yescomplemented
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1444—21no—
44#1444—21no—
45C4⋊#95—22no—
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C414yes—
54C2×C4×Q8D414yes—
55C2×C4×Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C42—21no—
69C22×C42—21no—
70C22×C42—22no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—22no—
79C2×#103—22no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103C2×C418yes—
83C2×#97—22no—
84C2×#97—22no—
85C2×#97C2×C414yes—
86#391—24no—
87#391—24no—
88#389—24no—
89#389—24no—
90#389—42no—
91#389—42no—
92#387—216no—
93#384—22no—
94#384D414yes—
95#382—28no—
96#382—22no—
97#382—22no—
98C4×C4⋊C4—28no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4C2×C414yes—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95D412yes—
114#340—22no—
115#340—22no—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340C2×C418yes—
121#336—42no—
122#336—42no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8C22×C412yes—
131C22×Q8C2×D412yes—
132C22×Q8C2×Q812yes—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8C22×C414yes—
161C4×Q8—28no—
162C4×Q8—28no—
163C4×Q8—28no—
164C4×Q8C4⋊C4116yescomplemented
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
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—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—81no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C414yes—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×Q812yes—
207C2×C4⋊C4#4114yes—
208C2×C4⋊C4#4114yes—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—28no—
219C2×C42—24no—
220C2×C42—81no—
221C2×C42—81no—
222C2×C42—82no—
223C2×C42—82no—
224C2×C42C22×C412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×Q812yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230C4⋊C8—44no—
231#103—216no—
232#103—44no—
233#103—44no—
234#97—28no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—24no—
239#95—24no—
240#95—44no—
241#95—44no—
242#95—42no—
243#95—42no—
244C24C22×D411yescharacteristic
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—24no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—24no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8C2×C4⋊C414yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4C4×D412yes—
319C22×C4C4×D412yes—
320C22×C4#12012yes—
321C22×C4#12012yes—
322C22×C4C22⋊Q812yes—
323C22×C4C22⋊Q812yes—
324C22×C4#13714yes—
325C22×C4C23×C411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×Q811yescharacteristic
329C22×C4C2×#4112yes—
330C22×C4C2×#4112yes—
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4H(5,𝔽2)12yes—
334C22×C42−1+412yes—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C4⋊C4—28no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—216no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—44no—
363C4⋊C4—48no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—84no—
373C4⋊C4—88no—
374C4⋊C4—88no—
375C4⋊C4C2×C4⋊C414yes—
376C4⋊C4#11718yes—
377C42—216no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—84no—
383C42—88no—
384C42C2×C4⋊C414yes—
385C42C4×D414yes—
386C42C22⋊Q814yes—
387C42#12314yes—
388C23C2×C4×D411yescharacteristic
389C23C2×#12011yescharacteristic
390C23C2×C22⋊Q811yescharacteristic
391C23#54612yes—
392C23#54812yes—
393C23C2×#13712yes—
394C23#57512yes—
395C23#57611yescharacteristic
396C23#57711yescharacteristic
397C23#57711yescharacteristic
398C23#57811yescharacteristic
399Q8—24no—
400Q8—24no—
401Q8—44no—
402Q8—44no—
403Q8—42no—
404Q8—42no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4#38514yes—
466C2×C4#38514yes—
467C2×C4C22×C4⋊C411yescharacteristic
468C2×C4C2×#11712yes—
469C2×C4C2×C4×D411yescharacteristic
470C2×C4#52012yes—
471C2×C4C2×C22⋊Q811yescharacteristic
472C2×C4C2×#12311yescharacteristic
473C2×C4#53211yescharacteristic
474C2×C4#53211yescharacteristic
475C2×C4#53312yes—
476C2×C4#53511yescharacteristic
477C2×C4#53611yescharacteristic
478C2×C4#54412yes—
479C2×C4#54712yes—
480C2×C4#55012yes—
481C2×C4#55212yes—
482C2×C4#55312yes—
483C2×C4#55512yes—
484C2×C4#55612yes—
485C8—44no—
486C8—81no—
487C8—81no—
488C22C2×#38511yescharacteristic
489C22#235212yes—
490C22#243111yescharacteristic
491C22#243211yescharacteristic
492C22#243912yes—
493C22C4×#13712yes—
494C22#294611yescharacteristic
495C22#294711yescharacteristic
496C22#300411yescharacteristic
497C22#300511yescharacteristic
498C22#301211yescharacteristic
499C22#301311yescharacteristic
500C22#307812yes—
501C22#308311yescharacteristic
502C22#308411yescharacteristic
503C22#331112yes—
504C22#331312yes—
505C22#331412yes—
506C22#331612yes—
507C22#337012yes—
508C22#337112yes—
509C22#337212yes—
510C22#337412yes—
511C4—24no—
512C4—24no—
513C4—44no—
514C4—44no—
515C4—44no—
516C4—48no—
517C4—84no—
518C4—82no—
519C4—161no—
520C4—161no—
521C4#228511yescharacteristic
522C4#231711yescharacteristic
523C4#235412yes—
524C4#244112yes—
525C4#249911yescharacteristic
526C4#250011yescharacteristic
527C2#1387911yescharacteristic
528C2#2087411yescharacteristic
529C2#2087811yescharacteristic
530C2#2277612yes—
531C2#2277712yes—
532C2#2557911yescharacteristic
533C2#2558011yescharacteristic
534C2#2565711yescharacteristic
535C2#2565811yescharacteristic
536C2#2603412yes—
537C2#2603512yes—
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
33. #1763C4semidirect
164. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#18362
C2×#18052
C2×#17632
#139542
#110672
#108852
#108842
#108492
#108352