Finite Group Databasebeta

← #455355

Subgroups — #455355

Order 512.

1314 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C2×#2056C212yesmaximal
3C2×#1878C211yesmaximal, characteristic
4#15191C211yesmaximal, characteristic
5#13931C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11322C211yesmaximal, characteristic
8#11232C212yesmaximal
9#11190C212yesmaximal
10#11124C212yesmaximal
11#11077C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#477—28no—
16C2×#477—24no—
17C22×#107C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27#2056—28no—
28#1878—24no—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1446—22no—
36#1445—22no—
37#1442—21no—
38#1442—21no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42#1434—21no—
43#1434—21no—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×Q8C2311yescharacteristic
48C22×Q16—22no—
49C22×Q16—22no—
50C22×Q16—24no—
51C22×Q16—22no—
52C22×Q16—42no—
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55C23×C8C2311yescharacteristic
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C4×Q8—22no—
61C2×C4×Q8—24no—
62C2×C4×Q8—42no—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C42—21no—
71C22×C42C2312yes—
72C22×C42C2311yescharacteristic
73#477—232no—
74C2×#107—24no—
75C2×#107—24no—
76C2×#107—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—42no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—22no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—22no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103D418yes—
92#391—24no—
93#391—24no—
94#391—22no—
95#391—22no—
96#391—42no—
97#391—42no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95D411yescharacteristic
111C2×#95D412yes—
112C2×#95D411yescharacteristic
113#340—24no—
114#340—22no—
115#340—42no—
116#340—42no—
117#336—22no—
118#336—42no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—44no—
137C22×Q8—44no—
138C22×Q8—42no—
139C22×Q8—81no—
140C22×Q8—81no—
141C22×Q8C2×D414yes—
142C22×Q8C2×D412yes—
143C22×Q8#4112yes—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C2×Q16—28no—
156C2×Q16—48no—
157C2×Q16—48no—
158C2×Q16—44no—
159C2×Q16—44no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8C2×D414yes—
176C22×C8#4112yes—
177C4⋊Q8—28no—
178C4⋊Q8—48no—
179C4⋊Q8—44no—
180C4×Q8—28no—
181C4×Q8—44no—
182C4×Q8—48no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D414yes—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—44no—
217C2×C42C2×D412yes—
218C2×C42#4112yes—
219#107—216no—
220C4⋊C8—28no—
221C4⋊C8—48no—
222#103—216no—
223#103—44no—
224#103—48no—
225#103—44no—
226#95—22no—
227#95—42no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—42no—
234#95—42no—
235#95—42no—
236#95—82no—
237#95—82no—
238#95—81no—
239#95—82no—
240#95Q1614yescomplemented
241C24C22×D411yescharacteristic
242C2×Q8—22no—
243C2×Q8—24no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—48no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×Q8—88no—
281C2×Q8—88no—
282C2×Q8#12014yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
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—22no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4#12112yes—
337C22×C4#12112yes—
338C22×C4C2×Q1614yes—
339C22×C4C2×Q1612yes—
340C22×C4#13712yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C22×D412yes—
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4H(5,𝔽2)12yes—
347C22×C4H(5,𝔽2)12yes—
348C22×C42−1+412yes—
349C22×C42−1+411yescharacteristic
350C22×C42−1+411yescharacteristic
351Q16—416no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—48no—
374C4⋊C4—48no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C42—22no—
386C42—22no—
387C42—24no—
388C42—48no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—42no—
393C23C2×#12111yescharacteristic
394C23#53712yes—
395C23D4212yes—
396C23#54711yescharacteristic
397C23#54811yescharacteristic
398C23C22×Q1611yescharacteristic
399C23C2×#13711yescharacteristic
400C23#57512yes—
401C23#57611yescharacteristic
402C23#57711yescharacteristic
403C23#57812yes—
404Q8—24no—
405Q8—24no—
406Q8—42no—
407Q8—42no—
408Q8—44no—
409Q8—44no—
410Q8—42no—
411Q8—42no—
412Q8—44no—
413Q8—816no—
414Q8—81no—
415Q8—81no—
416Q8—82no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4#46214yes—
475C2×C4C2×#12011yescharacteristic
476C2×C4#52511yescharacteristic
477C2×C4#53411yescharacteristic
478C2×C4#53511yescharacteristic
479C2×C4#53611yescharacteristic
480C2×C4#53611yescharacteristic
481C2×C4#53612yes—
482C2×C4#53612yes—
483C2×C4#53912yes—
484C2×C4#54212yes—
485C2×C4#54212yes—
486C2×C4#54612yes—
487C2×C4#54612yes—
488C2×C4#54612yes—
489C2×C4#55211yescharacteristic
490C2×C4#55611yescharacteristic
491C2×C4#56212yes—
492C2×C4#56212yes—
493C8—44no—
494C8—42no—
495C8—81no—
496C22#241011yescharacteristic
497C22#241511yescharacteristic
498C22#262011yescharacteristic
499C22#267212yes—
500C22#268112yes—
501C22C2×#46211yescharacteristic
502C22#304011yescharacteristic
503C22#304411yescharacteristic
504C22#304511yescharacteristic
505C22#324912yes—
506C22#325212yes—
507C22#325812yes—
508C22#326212yes—
509C22#326812yes—
510C22#327012yes—
511C22#327512yes—
512C22D4×Q1612yes—
513C22#333811yescharacteristic
514C22#334912yes—
515C22#335611yescharacteristic
516C22#336311yescharacteristic
517C22#337511yescharacteristic
518C22#337712yes—
519C4—24no—
520C4—42no—
521C4—44no—
522C4—44no—
523C4—82no—
524C4—84no—
525C4—82no—
526C4—82no—
527C4—161no—
528C4—162no—
529C4#240411yescharacteristic
530C4#261411yescharacteristic
531C4#263811yescharacteristic
532C4#268712yes—
533C4#270012yes—
534C4#281611yescharacteristic
535C2#1626311yescharacteristic
536C2#2436011yescharacteristic
537C2#2437111yescharacteristic
538C2#2468211yescharacteristic
539C2#2469911yescharacteristic
540C2#2983011yescharacteristic
541C2#2984011yescharacteristic
542C2#3067212yes—
543C2#3068812yes—
544C2#3115112yes—
545C2#3116912yes—
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
240. #95Q16semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×#20562
C2×#18782
#151912
#139312
#113842
#113222
#112322
#111902
#111242
#110772