Finite Group Databasebeta

← #363190

Subgroups — #363190

Order 512.

2766 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#1996C212yesmaximal, complemented
3C2×#1799C211yesmaximal, characteristic
4C2×C4×#99C211yesmaximal, characteristic
5#13904C212yesmaximal, complemented
6#10966C214yesmaximal
7#10726C214yesmaximal, complemented
8C2×#552—28no—
9C2×#552—24no—
10C2×#552—24no—
11C2×#552—24no—
12C2×#552—24no—
13C22×#126C412yescomplemented
14C22×#125C412yescomplemented
15C22×#125C411yescharacteristic, complemented
16C22×#123C2212yes—
17C22×C22⋊Q8C411yescharacteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C411yescharacteristic, complemented
20C22×C4×D4C2211yescharacteristic
21C22×#117C411yescharacteristic, complemented
22C22×#117C2212yes—
23C23×C4⋊C4C2211yescharacteristic
24C22×#99C2212yes—
25C22×#99C2212yes—
26C2×#381C2218yes—
27C2×#380C2212yes—
28C2×C4×C22⋊C4C2212yes—
29#1996—28no—
30#1799—24no—
31C4×#99—24no—
32C2×#342C2218yes—
33C2×#342C2212yes—
34C2×#342C2212yes—
35#1445—24no—
36C4⋊#95—24no—
37C23×Q8C2×C411yescharacteristic
38C23×D4C2311yescharacteristic
39C24×C4C2×C411yescharacteristic
40C24×C4C2312yes—
41C24×C4C2311yescharacteristic
42#552—232no—
43#552—232no—
44#552—232no—
45#552—232no—
46C2×#126—28no—
47C2×#126—28no—
48C2×#126—28no—
49C2×#125—28no—
50C2×#125—28no—
51C2×#125—28no—
52C2×#125—24no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—24no—
56C2×#123—24no—
57C2×#123—24no—
58C2×#123—28no—
59C2×#123—28no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8D418yes—
69C2×C4×Q8—24no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—24no—
73C2×C4×D4—22no—
74C2×C4×D4—24no—
75C2×C4×D4—22no—
76C2×C4×D4D414yes—
77C2×C4×D4D414yes—
78C2×#117—24no—
79C2×#117—28no—
80C2×#117—24no—
81C2×#117—22no—
82C2×#117—22no—
83C2×#117—28no—
84C2×#117—24no—
85C2×#117—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—22no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4C2×C412yes—
91C22×C4⋊C4C2×C412yes—
92C22×C4⋊C4C2×C412yes—
93C22×C4⋊C4C2×C412yes—
94C22×C4⋊C4C2×C411yescharacteristic
95C22×C4⋊C4C2×C411yescharacteristic
96C22×C4⋊C4C2×C412yes—
97C22×C4⋊C4C2×C411yescharacteristic
98C22×C4⋊C4C2×C411yescharacteristic
99C22×C4⋊C4C2×C412yes—
100C22×C4⋊C4D414yes—
101C22×C4⋊C4D414yes—
102C22×C22⋊C4C2×C412yes—
103C22×C22⋊C4C2×C412yes—
104C22×C22⋊C4C2×C411yescharacteristic
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2312yes—
109C22×C42—22no—
110C22×C42C2×C412yes—
111C22×C42C2×C412yes—
112C22×C42C2×C411yescharacteristic
113C2×#99—24no—
114C2×#99—28no—
115C2×#99—24no—
116C2×#99—44no—
117#381—216no—
118#381—48no—
119#380—24no—
120#380—24no—
121C4×C22⋊C4—24no—
122C4×C22⋊C4—24no—
123C2×#95—216no—
124C2×#95—22no—
125C2×#95—22no—
126#342—24no—
127#342—216no—
128#342—24no—
129C25C22×C411yescharacteristic
130C25C2411yescharacteristic
131C22×Q8—22no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—44no—
135C22×Q8C22⋊C414yes—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4C2×D412yes—
142C22×D4C2×D412yes—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—28no—
147C23×C4—24no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4C22⋊C414yes—
157C23×C4C22×C412yes—
158C23×C4C22×C412yes—
159C23×C4C22×C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C412yes—
162C23×C4C22×C412yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D418yes—
167C23×C4C2×D412yes—
168C23×C4C2×D414yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172#126—264no—
173#125—264no—
174#125—216no—
175#125—216no—
176#123—232no—
177#123—232no—
178C22⋊Q8—216no—
179C22⋊Q8—216no—
180C22⋊Q8—28no—
181C22⋊Q8—28no—
182C22⋊Q8—28no—
183C22⋊Q8—28no—
184C4×Q8—216no—
185C4×Q8—216no—
186C4×D4—28no—
187C4×D4—28no—
188C4×D4—28no—
189C4×D4—28no—
190#117—216no—
191#117—216no—
192#117—232no—
193#117—232no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C412yes—
229C2×C4⋊C4C22⋊C418yes—
230C2×C4⋊C4C22⋊C414yes—
231C2×C4⋊C4C22⋊C414yes—
232C2×C4⋊C4C22⋊C418yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4C22⋊C414yes—
256C2×C22⋊C4C2×D412yes—
257C2×C22⋊C4C2×D412yes—
258C2×C22⋊C4C2×D418yes—
259C2×C22⋊C4#4118yes—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—48no—
270C2×C42—44no—
271C2×C42C22⋊C414yes—
272#99—216no—
273#99—48no—
274#95—432no—
275#95—44no—
276#95—44no—
277C24—21no—
278C24—21no—
279C24—22no—
280C24—22no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24C2×C22⋊C412yes—
286C24C23×C411yescharacteristic
287C24C22×D412yes—
288C24C22×D411yescharacteristic
289C24C22×D411yescharacteristic
290C24C22×D412yes—
291C24C2×#4114yes—
292C24H(5,𝔽2)12yes—
293C242−1+412yes—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—48no—
297C2×Q8—44no—
298C2×Q8—48no—
299C2×Q8—44no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—44no—
311C22×C4—28no—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—216no—
336C22×C4—216no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—28no—
340C22×C4—44no—
341C22×C4—48no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—44no—
367C22×C4—48no—
368C22×C4C2×C22⋊C412yes—
369C22×C4C2×C22⋊C414yes—
370C22×C4C2×C22⋊C414yes—
371C22×C4C2×C22⋊C412yes—
372C22×C4C2×C22⋊C412yes—
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C412yes—
375C22×C4C2×C22⋊C412yes—
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C414yes—
378C22×C4C2×C22⋊C414yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4#12014yes—
383C22×C4#12014yes—
384C22×C4#12014yes—
385C22×C4#12014yes—
386C4⋊C4—216no—
387C4⋊C4—216no—
388C4⋊C4—216no—
389C4⋊C4—216no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—216no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—28no—
398C4⋊C4—216no—
399C4⋊C4—88no—
400C22⋊C4—216no—
401C22⋊C4—216no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—28no—
405C22⋊C4—28no—
406C22⋊C4—28no—
407C22⋊C4—28no—
408C22⋊C4—28no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—48no—
414C42—216no—
415C42—216no—
416C42—24no—
417C42—24no—
418C42—88no—
419C23—24no—
420C23—22no—
421C23—24no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—41no—
427C23—41no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—42no—
432C23—48no—
433C23—44no—
434C23—44no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—84no—
444C23—84no—
445C23C22×C22⋊C411yescharacteristic
446C23C22×C22⋊C411yescharacteristic
447C23C22×C22⋊C411yescharacteristic
448C23C4×#4112yes—
449C23#52012yes—
450C23C2×#12012yes—
451C23C2×#12012yes—
452C23#52512yes—
453C23#53212yes—
454C23#53511yescharacteristic
455C23#53611yescharacteristic
456C23#53712yes—
457C23#54614yes—
458C23#54714yes—
459C23#54814yes—
460C23#55014yes—
461Q8—48no—
462Q8—48no—
463D4—48no—
464D4—44no—
465D4—44no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—416no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—48no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—416no—
513C2×C4—48no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—88no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—816no—
520C2×C4—84no—
521C2×C4#37912yes—
522C2×C4#37914yes—
523C2×C4#37912yes—
524C2×C4#41014yes—
525C22—21no—
526C22—21no—
527C22—22no—
528C22—22no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—22no—
533C22—28no—
534C22—28no—
535C22—24no—
536C22—24no—
537C22—42no—
538C22—42no—
539C22—41no—
540C22—41no—
541C22—42no—
542C22—44no—
543C22—44no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—82no—
548C22—82no—
549C22—84no—
550C22—84no—
551C22C2×#37911yescharacteristic
552C22#229711yescharacteristic
553C22#231111yescharacteristic
554C22#233912yes—
555C22#234112yes—
556C22#240714yes—
557C22#243112yes—
558C22#243212yes—
559C22C2×#41012yes—
560C22#288412yes—
561C4—22no—
562C4—24no—
563C4—42no—
564C4—44no—
565C4—42no—
566C4—44no—
567C4—44no—
568C4—44no—
569C4—42no—
570C4—88no—
571C4—164no—
572C4#178214yes—
573C2—24no—
574C2—22no—
575C2—21no—
576C2—22no—
577C2—21no—
578C2—22no—
579C2—82no—
580C2#1393711yescharacteristic
581C2C2×#178211yescharacteristic
582C2#2058711yescharacteristic
583C2#2247814yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1996C2semidirect
5. #13904C2semidirect
7. #10726C2semidirect
13. C22×#126C4semidirect
14. C22×#125C4semidirect
15. C22×#125C4semidirect
17. C22×C22⋊Q8C4semidirect
19. C22×C4×Q8C4semidirect
21. C22×#117C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#19962
C2×#17992
C2×C4×#992
#139042
#109662
#107262