Finite Group Databasebeta

← #470272

Subgroups — #470272

Order 512.

1368 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2C2×#2016C211yesmaximal, characteristic
3C2×#1928C211yesmaximal, characteristic
4#15201C211yesmaximal, characteristic
5#11380C211yesmaximal, characteristic
6#11190C212yesmaximal
7#11124C212yesmaximal
8#11024C212yesmaximal
9#11024C212yesmaximal
10#11010C211yesmaximal, characteristic
11#10807C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20C22×#103C2212yes—
21#2568—24no—
22#2568—24no—
23C2×#400—22no—
24C2×#391—21no—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#382—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31#2016—24no—
32#1928—24no—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1446—21no—
41#1442—22no—
42#1442—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8D411yescharacteristic
46C23×Q8C2312yes—
47C23×Q8C2311yescharacteristic
48C22×Q16—24no—
49C22×Q16—24no—
50C22×Q16—24no—
51C22×Q16—42no—
52C22×Q16—42no—
53C22×Q16—42no—
54C22×Q16—42no—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8D414yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—42no—
65C2×C4×Q8—42no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42—21no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77C2×#107—22no—
78C2×#107—42no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—22no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93#400—216no—
94#391—28no—
95#391—24no—
96#391—24no—
97#391—24no—
98#391—24no—
99#391—22no—
100#391—22no—
101#391—42no—
102#391—42no—
103#382—28no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4D414yes—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—21no—
119#340—24no—
120#340—22no—
121#340—42no—
122#336—22no—
123#336—22no—
124#336—42no—
125#336—42no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—21no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8Q1614yescomplemented
153C22×Q8C2×D414yes—
154C22×Q8#4112yes—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—22no—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2411yescharacteristic
166C2×Q16—48no—
167C2×Q16—48no—
168C2×Q16—48no—
169C2×Q16—48no—
170C2×Q16—48no—
171C2×Q16—48no—
172C2×Q16—48no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8#4114yes—
189C22×C8#4112yes—
190C4⋊Q8—48no—
191C4⋊Q8—44no—
192C4⋊Q8—44no—
193C4×Q8—48no—
194C4×Q8—44no—
195C4×Q8—44no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—44no—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237#107—44no—
238#103—28no—
239#103—44no—
240#103—48no—
241#103—48no—
242#103—48no—
243#95—28no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—24no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—42no—
252#95—42no—
253#95—42no—
254#95—81no—
255#95—82no—
256C24C22×D411yescharacteristic
257C2×Q8—28no—
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—22no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—48no—
269C2×Q8—44no—
270C2×Q8—48no—
271C2×Q8—44no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—41no—
281C2×Q8—42no—
282C2×Q8—41no—
283C2×Q8—42no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×Q8—42no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4#12012yes—
362C22×C4#12012yes—
363C22×C4C2×Q1614yes—
364C22×C4C2×Q1612yes—
365C22×C4#13712yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4112yes—
371C22×C4H(5,𝔽2)12yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C42−1+411yescharacteristic
376Q16—432no—
377C2×C8—22no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—42no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—42no—
393C2×C8—42no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—44no—
399C4⋊C4—48no—
400C4⋊C4—48no—
401C4⋊C4—48no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C42—24no—
413C42—22no—
414C42—22no—
415C42—48no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—42no—
420C23C2×#12011yescharacteristic
421C23#53412yes—
422C23#54811yescharacteristic
423C23#55012yes—
424C23#55011yescharacteristic
425C23C22×Q1611yescharacteristic
426C23C2×#13711yescharacteristic
427C23#57512yes—
428C23#57711yescharacteristic
429C23#57811yescharacteristic
430C23#57812yes—
431Q8—416no—
432Q8—42no—
433Q8—42no—
434Q8—44no—
435Q8—44no—
436Q8—44no—
437Q8—44no—
438Q8—44no—
439Q8—82no—
440Q8—81no—
441Q8—82no—
442Q8—82no—
443Q8—82no—
444Q8—81no—
445Q8—82no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4#45114yes—
504C2×C4#52511yescharacteristic
505C2×C4#53211yescharacteristic
506C2×C4#53412yes—
507C2×C4#53511yescharacteristic
508C2×C4#53511yescharacteristic
509C2×C4#53511yescharacteristic
510C2×C4#53711yescharacteristic
511C2×C4#53812yes—
512C2×C4#53812yes—
513C2×C4#53812yes—
514C2×C4#54212yes—
515C2×C4#54611yescharacteristic
516C2×C4#54612yes—
517C2×C4#54711yescharacteristic
518C2×C4#55112yes—
519C2×C4#55512yes—
520C2×C4H(3,𝔽4)12yes—
521C2×C4#56212yes—
522C8—42no—
523C8—44no—
524C8—81no—
525C22#240711yescharacteristic
526C22#243111yescharacteristic
527C22#261311yescharacteristic
528C22#273012yes—
529C22#273312yes—
530C22C2×#45111yescharacteristic
531C22#300511yescharacteristic
532C22#301111yescharacteristic
533C22#301311yescharacteristic
534C22#318912yes—
535C22#319612yes—
536C22#320112yes—
537C22#320212yes—
538C22#336311yescharacteristic
539C22#337411yescharacteristic
540C22#337712yes—
541C22#339512yes—
542C22#339511yescharacteristic
543C22#339812yes—
544C22#340011yescharacteristic
545C22#340112yes—
546C22#340212yes—
547C22#340212yes—
548C4—24no—
549C4—44no—
550C4—42no—
551C4—82no—
552C4—84no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—161no—
557C4—162no—
558C4#261011yescharacteristic
559C4#263111yescharacteristic
560C4#266211yescharacteristic
561C4#268412yes—
562C4#268612yes—
563C4#281911yescharacteristic
564C2#1668011yescharacteristic
565C2#2419111yescharacteristic
566C2#2420111yescharacteristic
567C2#2563411yescharacteristic
568C2#2563811yescharacteristic
569C2#2959711yescharacteristic
570C2#2960511yescharacteristic
571C2#3260212yes—
572C2#3261912yes—
573C2#3267012yes—
574C2#3267612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
152. C22×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#20162
C2×#19282
#152012
#113802
#111902
#111242
#110242
#110102
#108072