Finite Group Databasebeta

← #400040

Subgroups — #400040

Order 512.

1540 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×#2365C211yesmaximal, characteristic
2C2×#2016C211yesmaximal, characteristic
3C2×#1928C211yesmaximal, characteristic
4C2×#1878C212yesmaximal
5#15192C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11124C212yesmaximal
8#11024C212yesmaximal
9#11010C211yesmaximal, characteristic
10#10807C211yesmaximal, characteristic
11#10789C212yesmaximal
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×#107C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2211yescharacteristic
21#2365—24no—
22#2365—24no—
23C2×#391C2212yes—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#389—21no—
27C2×#387—22no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2016—24no—
32#1928—24no—
33#1878—28no—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1445—21no—
42#1443—22no—
43#1442—21no—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×Q8C2312yes—
48C22×Q16—24no—
49C22×Q16—24no—
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—42no—
54C22×Q16—42no—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C2×C4⋊Q8—24no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8D414yes—
62C2×C4×Q8—24no—
63C2×C4×Q8—24no—
64C2×C4×Q8—24no—
65C2×C4×Q8—24no—
66C2×C4×Q8—42no—
67C2×C4×Q8—42no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C42—22no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×#107—22no—
82C2×#107—42no—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—22no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103—42no—
95C2×#103—42no—
96C2×#103D414yes—
97#391—24no—
98#391—24no—
99#391—24no—
100#391—22no—
101#391—22no—
102#391—42no—
103#389—28no—
104#387—216no—
105C4×C4⋊C4—28no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4D414yes—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95D412yes—
122#340—24no—
123#340—24no—
124#340—22no—
125#336—24no—
126#336—22no—
127#336—22no—
128#336—42no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—22no—
134C22×Q8—21no—
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—21no—
148C22×Q8—22no—
149C22×Q8C2×D412yes—
150C22×Q8C2×D414yes—
151C22×Q8#4114yes—
152C22×Q8#4114yes—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
164C2×Q16—216no—
165C2×Q16—48no—
166C2×Q16—48no—
167C2×Q16—48no—
168C2×Q16—48no—
169C2×Q16—48no—
170C2×Q16—48no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—21no—
186C22×C8C2×D412yes—
187C22×C8#4114yes—
188C4⋊Q8—28no—
189C4⋊Q8—28no—
190C4⋊Q8—216no—
191C4×Q8—48no—
192C4×Q8—48no—
193C4×Q8—48no—
194C4×Q8—48no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
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—22no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4Q1614yescomplemented
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4#4112yes—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—28no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242#107—44no—
243#103—28no—
244#103—44no—
245#103—48no—
246#103—48no—
247#103—48no—
248#95—28no—
249#95—28no—
250#95—24no—
251#95—24no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95—81no—
259#95—82no—
260C24C22×D411yescharacteristic
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—42no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—42no—
305C2×Q8—42no—
306C2×Q8#12318yes—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—41no—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4C2×Q1614yes—
363C22×C4C2×Q1612yes—
364C22×C4#13712yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C22×C4C2×#4112yes—
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)11yescharacteristic
375Q16—432no—
376C2×C8—22no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—42no—
393C4⋊C4—216no—
394C4⋊C4—216no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—28no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—44no—
402C4⋊C4—416no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4#12114yes—
412C42—24no—
413C42—22no—
414C42—22no—
415C42—48no—
416C42—416no—
417C42—42no—
418C42—42no—
419C42—42no—
420C42—42no—
421C23C2×#12011yescharacteristic
422C23#52512yes—
423C23D4211yescharacteristic
424C23#54611yescharacteristic
425C23#55012yes—
426C23C22×Q1611yescharacteristic
427C23C2×#13711yescharacteristic
428C23#57512yes—
429C23#57711yescharacteristic
430C23#57811yescharacteristic
431C23#57812yes—
432Q8—28no—
433Q8—28no—
434Q8—44no—
435Q8—44no—
436Q8—44no—
437Q8—44no—
438Q8—44no—
439Q8—44no—
440Q8—42no—
441Q8—44no—
442Q8—42no—
443Q8—81no—
444Q8—81no—
445Q8—82no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—82no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4#45114yes—
504C2×C4C2×#12111yescharacteristic
505C2×C4C2×#12312yes—
506C2×C4#52511yescharacteristic
507C2×C4#52511yescharacteristic
508C2×C4#52512yes—
509C2×C4#52912yes—
510C2×C4#53211yescharacteristic
511C2×C4#53511yescharacteristic
512C2×C4#53511yescharacteristic
513C2×C4#53812yes—
514C2×C4#53812yes—
515C2×C4D4211yescharacteristic
516C2×C4#54612yes—
517C2×C4#54612yes—
518C2×C4#54611yescharacteristic
519C2×C4#55112yes—
520C2×C4#55312yes—
521C2×C4#55312yes—
522C8—44no—
523C8—42no—
524C8—81no—
525C22#240811yescharacteristic
526C22#243111yescharacteristic
527C22#245612yes—
528C22#246712yes—
529C22#284911yescharacteristic
530C22C2×#45111yescharacteristic
531C22#300511yescharacteristic
532C22#301111yescharacteristic
533C22#301311yescharacteristic
534C22#310412yes—
535C22#311712yes—
536C22#312112yes—
537C22#312712yes—
538C22#326912yes—
539C22#327811yescharacteristic
540C22D4×Q1611yescharacteristic
541C22#329911yescharacteristic
542C22#331512yes—
543C22#332811yescharacteristic
544C22#339512yes—
545C22#339812yes—
546C22#340112yes—
547C22#340212yes—
548C4—24no—
549C4—44no—
550C4—44no—
551C4—48no—
552C4—42no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—161no—
557C4—162no—
558C4#239211yescharacteristic
559C4#241812yes—
560C4#244412yes—
561C4#253511yescharacteristic
562C4#260311yescharacteristic
563C4#263111yescharacteristic
564C2#1479511yescharacteristic
565C2#2424311yescharacteristic
566C2#2425311yescharacteristic
567C2#2563411yescharacteristic
568C2#2563811yescharacteristic
569C2#2659012yes—
570C2#2660612yes—
571C2#2706812yes—
572C2#2708212yes—
573C2#3301011yescharacteristic
574C2#3301911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
222. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#20162
C2×#19282
C2×#18782
#151922
#113462
#111242
#110242
#110102
#108072
#107892