Finite Group Databasebeta

← #402293

Subgroups — #402293

Order 512.

1290 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2072C212yesmaximal
2C2×#2056C212yesmaximal
3C2×#1878C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5#15199C211yesmaximal, characteristic
6#13817C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic
8#11322C211yesmaximal, characteristic
9#11077C211yesmaximal, characteristic
10#10932C212yesmaximal
11#10849C212yesmaximal
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2212yes—
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#391C2212yes—
23C2×#384C2212yes—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26#2072—28no—
27#2056—28no—
28#1878—24no—
29#1803—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1444—21no—
38#1444—22no—
39#1444—21no—
40#1436—22no—
41C4⋊#95C2211yescharacteristic
42#1434—21no—
43C4×#95—21no—
44C4×#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×Q8C2311yescharacteristic
47C22×Q16—22no—
48C22×Q16—22no—
49C22×Q16—24no—
50C22×Q16—22no—
51C22×Q16—42no—
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C4×Q8—22no—
56C2×C4×Q8—24no—
57C2×C4×Q8—42no—
58C2×C4×Q8D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42—22no—
70C22×C42C2311yescharacteristic
71C22×C42C2312yes—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—24no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—42no—
76C2×C4⋊C8—42no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—22no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—22no—
85C2×#103—24no—
86C2×#103—42no—
87#391—24no—
88#391—24no—
89#391—22no—
90#391—22no—
91#391—24no—
92#391D418yes—
93#384—22no—
94#384—24no—
95#384—22no—
96#384—24no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—42no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95D411yescharacteristic
109C2×#95D412yes—
110C2×#95D411yescharacteristic
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—22no—
116#336—24no—
117#336—22no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8C2×D412yes—
136C22×Q8C2×D414yes—
137C22×Q8#4112yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D412yes—
148C23×C4C2411yescharacteristic
149C2×Q16—28no—
150C2×Q16—48no—
151C2×Q16—48no—
152C2×Q16—44no—
153C2×Q16—44no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8#4112yes—
171C4×Q8—28no—
172C4×Q8—44no—
173C4×Q8—48no—
174C2×C4⋊C4—21no—
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—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—81no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4#4114yes—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—44no—
215C2×C42—44no—
216C2×C42—44no—
217C2×C42—82no—
218C2×C42C2×D412yes—
219C2×C42C2×D414yes—
220C2×C42#4112yes—
221C2×C42#4114yes—
222C4⋊C8—48no—
223C4⋊C8—44no—
224#103—48no—
225#103—48no—
226#103—44no—
227#103—44no—
228#95—28no—
229#95—28no—
230#95—22no—
231#95—24no—
232#95—44no—
233#95—44no—
234#95—42no—
235#95—44no—
236#95—42no—
237#95—42no—
238#95—81no—
239#95Q1614yescomplemented
240C24C22×D411yescharacteristic
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—22no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—42no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—42no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8#12714yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4#12114yes—
328C22×C4#12112yes—
329C22×C4#12114yes—
330C22×C4#12112yes—
331C22×C4C2×Q1614yes—
332C22×C4C2×Q1612yes—
333C22×C4#13712yes—
334C22×C4#13714yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D412yes—
337C22×C4C2×#4112yes—
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4112yes—
341C22×C4C2×#4111yescharacteristic
342C22×C4H(5,𝔽2)12yes—
343C22×C42−1+411yescharacteristic
344C22×C42−1+411yescharacteristic
345Q16—416no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—42no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—41no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—48no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—84no—
378C4⋊C4—88no—
379C4⋊C4—88no—
380C4⋊C4—84no—
381C4⋊C4#12314yes—
382C42—22no—
383C42—24no—
384C42—22no—
385C42—24no—
386C42—24no—
387C42—24no—
388C42—24no—
389C42—48no—
390C42—88no—
391C23C2×#12111yescharacteristic
392C23C2×#12112yes—
393C23#54612yes—
394C23#54711yescharacteristic
395C23#54811yescharacteristic
396C23C22×Q1611yescharacteristic
397C23C2×#13712yes—
398C23C2×#13711yescharacteristic
399C23#57611yescharacteristic
400C23#57711yescharacteristic
401C23#57712yes—
402Q8—24no—
403Q8—24no—
404Q8—42no—
405Q8—42no—
406Q8—44no—
407Q8—44no—
408Q8—42no—
409Q8—42no—
410Q8—44no—
411Q8—81no—
412Q8—81no—
413Q8—82no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4#46218yes—
471C2×C4#46214yes—
472C2×C4C2×#12311yescharacteristic
473C2×C4#52511yescharacteristic
474C2×C4#52912yes—
475C2×C4#52912yes—
476C2×C4C2×#12711yescharacteristic
477C2×C4#53211yescharacteristic
478C2×C4#53212yes—
479C2×C4#53411yescharacteristic
480C2×C4#53611yescharacteristic
481C2×C4#53912yes—
482C2×C4#54212yes—
483C2×C4#54612yes—
484C2×C4#54612yes—
485C2×C4#54712yes—
486C2×C4#54712yes—
487C2×C4#55112yes—
488C2×C4#55211yescharacteristic
489C2×C4#55611yescharacteristic
490C8—42no—
491C8—81no—
492C8—82no—
493C22#241211yescharacteristic
494C22#241511yescharacteristic
495C22#242512yes—
496C22#242712yes—
497C22#285611yescharacteristic
498C22C2×#46211yescharacteristic
499C22C2×#46212yes—
500C22#303911yescharacteristic
501C22#303912yes—
502C22#304311yescharacteristic
503C22#304312yes—
504C22#304511yescharacteristic
505C22#304512yes—
506C22#329912yes—
507C22#331412yes—
508C22#331612yes—
509C22#333112yes—
510C22#333811yescharacteristic
511C22#334412yes—
512C22#335611yescharacteristic
513C22#336311yescharacteristic
514C22#336912yes—
515C22#337511yescharacteristic
516C4—28no—
517C4—24no—
518C4—44no—
519C4—42no—
520C4—44no—
521C4—44no—
522C4—82no—
523C4—84no—
524C4—162no—
525C4—161no—
526C4#239511yescharacteristic
527C4#250111yescharacteristic
528C4#252112yes—
529C4#253911yescharacteristic
530C4#255312yes—
531C4#266311yescharacteristic
532C2#1483911yescharacteristic
533C2#2453211yescharacteristic
534C2#2454811yescharacteristic
535C2#2468211yescharacteristic
536C2#2469811yescharacteristic
537C2#2520612yes—
538C2#2521912yes—
539C2#2532412yes—
540C2#2533612yes—
541C2#3316311yescharacteristic
542C2#3316711yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
239. #95Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#20722
C2×#20562
C2×#18782
C2×#18032
#151992
#138172
#113572
#113222
#110772
#109322
#108492