Finite Group Databasebeta

← #360697

Subgroups — #360697

Order 512.

1312 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic
2C2×#1847C211yesmaximal, characteristic
3C2×#1805C211yesmaximal, characteristic
4C2×#1763C211yesmaximal, characteristic
5#13955C211yesmaximal, characteristic
6#11124C212yesmaximal
7#11122C212yesmaximal
8#10908C212yesmaximal
9#10907C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10835C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2355—24no—
21#2355—24no—
22C2×#391C2212yes—
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#1847—24no—
32#1805—24no—
33#1763C418yescomplemented
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1442—21no—
42#1442—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45#1434—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⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42—22no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#107—24no—
72C2×#107—24no—
73C2×#107—42no—
74C2×#106—24no—
75C2×#106—24no—
76C2×#106—42no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103C2×C418yes—
87C2×#97—22no—
88C2×#97—22no—
89C2×#97C2×C414yes—
90#391—24no—
91#391—24no—
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×#95—22no—
114C2×#95—22no—
115C2×#95D412yes—
116#340—22no—
117#340—22no—
118#340—42no—
119#340—42no—
120#340C2×C418yes—
121#336—24no—
122#336—24no—
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—21no—
134C23×C4—21no—
135C23×C4—22no—
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—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8C22×C414yes—
159C22×C8#4114yes—
160C22×C8#4114yes—
161C4×Q8—28no—
162C4×Q8—28no—
163C4×Q8—28no—
164C4×Q8C4⋊C4116yes—
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—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—81no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4C22×C414yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×Q812yes—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—28no—
213C2×C42—24no—
214C2×C42—81no—
215C2×C42—81no—
216C2×C42—82no—
217C2×C42—82no—
218C2×C42C22×C412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×Q812yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224#107—48no—
225#106—48no—
226C4⋊C8—44no—
227#103—216no—
228#103—44no—
229#103—44no—
230#97—28no—
231#95—28no—
232#95—28no—
233#95—28no—
234#95—24no—
235#95—24no—
236#95—44no—
237#95—42no—
238#95—42no—
239#95—44no—
240#95—82no—
241#95—82no—
242C24C22×D411yescharacteristic
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8C2×C4⋊C414yes—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
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—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4C4×D412yes—
317C22×C4C4×D412yes—
318C22×C4#12012yes—
319C22×C4#12012yes—
320C22×C4C22⋊Q812yes—
321C22×C4C22⋊Q812yes—
322C22×C4#13714yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×Q811yescharacteristic
327C22×C4C2×#4112yes—
328C22×C4C2×#4112yes—
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)12yes—
332C22×C42−1+412yes—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8#11718yes—
350C4⋊C4—28no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—216no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—44no—
360C4⋊C4—48no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—84no—
372C4⋊C4C2×C4⋊C414yes—
373C42—216no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—84no—
379C42—88no—
380C42C2×C4⋊C414yes—
381C42C4×D414yes—
382C42C22⋊Q814yes—
383C42#12314yes—
384C23C2×C4×D411yescharacteristic
385C23C2×#12011yescharacteristic
386C23C2×C22⋊Q811yescharacteristic
387C23#54712yes—
388C23#55012yes—
389C23C2×#13712yes—
390C23#57512yes—
391C23#57611yescharacteristic
392C23#57711yescharacteristic
393C23#57711yescharacteristic
394C23#57811yescharacteristic
395Q8—24no—
396Q8—24no—
397Q8—44no—
398Q8—44no—
399Q8—42no—
400Q8—42no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
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#38514yes—
462C2×C4#38514yes—
463C2×C4C22×C4⋊C411yescharacteristic
464C2×C4C2×#11712yes—
465C2×C4C2×C4×D411yescharacteristic
466C2×C4#52012yes—
467C2×C4C2×C22⋊Q811yescharacteristic
468C2×C4C2×#12311yescharacteristic
469C2×C4#53211yescharacteristic
470C2×C4#53211yescharacteristic
471C2×C4#53312yes—
472C2×C4#53511yescharacteristic
473C2×C4#53611yescharacteristic
474C2×C4#54412yes—
475C2×C4#54612yes—
476C2×C4#54812yes—
477C2×C4#55212yes—
478C2×C4#55312yes—
479C2×C4#55512yes—
480C2×C4#55612yes—
481C8—28no—
482C8—81no—
483C8—81no—
484C22C2×#38511yescharacteristic
485C22#235412yes—
486C22#243111yescharacteristic
487C22#243211yescharacteristic
488C22#244112yes—
489C22C4×#13712yes—
490C22#294611yescharacteristic
491C22#294711yescharacteristic
492C22#300411yescharacteristic
493C22#300511yescharacteristic
494C22#301211yescharacteristic
495C22#301311yescharacteristic
496C22#307812yes—
497C22#308311yescharacteristic
498C22#308411yescharacteristic
499C22#334412yes—
500C22#334612yes—
501C22#334912yes—
502C22#335012yes—
503C22#339812yes—
504C22#340012yes—
505C22#340112yes—
506C22#340312yes—
507C4—24no—
508C4—24no—
509C4—44no—
510C4—44no—
511C4—44no—
512C4—84no—
513C4—84no—
514C4—82no—
515C4—161no—
516C4—161no—
517C4#228511yescharacteristic
518C4#231711yescharacteristic
519C4#235212yes—
520C4#243912yes—
521C4#249911yescharacteristic
522C4#250011yescharacteristic
523C2#1387911yescharacteristic
524C2#2087411yescharacteristic
525C2#2087811yescharacteristic
526C2#2281812yes—
527C2#2281912yes—
528C2#2563911yescharacteristic
529C2#2564011yescharacteristic
530C2#2571111yescharacteristic
531C2#2571211yescharacteristic
532C2#2610012yes—
533C2#2610112yes—
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

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#18472
C2×#18052
C2×#17632
#139552
#111242
#111222
#109082
#109072
#108492
#108352