Finite Group Databasebeta

← #380280

Subgroups — #380280

Order 512.

1450 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1854C211yesmaximal, characteristic
3C2×#1853C211yesmaximal, characteristic
4C2×#1818C211yesmaximal, characteristic
5#14343C211yesmaximal, characteristic
6#11325C212yesmaximal, complemented
7#11323C212yesmaximal, complemented
8#11268C212yesmaximal, complemented
9#11267C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10730C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×#106C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2317—24no—
21#2317—24no—
22C2×#389—21no—
23C2×#389—21no—
24C2×#385—22no—
25C2×#384C2212yes—
26C2×#382C2212yes—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29#1854—24no—
30#1853—24no—
31#1818C418yescomplemented
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1445—21no—
42#1445—21no—
43#1440—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—42no—
57C2×#117—42no—
58C2×#117C2×C414yes—
59C2×#117D414yes—
60C2×#117Q814yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D414yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D412yes—
80C22×C42—21no—
81C22×C42C2312yes—
82C2×#107—22no—
83C2×#107—42no—
84C2×#106—24no—
85C2×#106—22no—
86C2×#106—42no—
87C2×#106—42no—
88C2×#106—42no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8C2×C414yes—
94C2×C22⋊C8C2×C414yes—
95#389—28no—
96#389—28no—
97#385—216no—
98#384—24no—
99#384—24no—
100#382—24no—
101#382—24no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114#340—24no—
115#340—22no—
116#340—22no—
117#340—42no—
118#340—42no—
119#340—42no—
120#340—42no—
121#336C2×C418yescomplemented
122#336C2×C418yescomplemented
123C25C2×D411yescharacteristic
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—44no—
136C23×C4—81no—
137C23×C4—82no—
138C23×C4—81no—
139C23×C4C22×C412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×Q812yes—
146C23×C4#4112yes—
147C23×C4#4112yes—
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8C22×C414yes—
164C22×C8C22×C412yes—
165C22×C8C22×C412yes—
166#117—28no—
167#117—28no—
168#117—28no—
169#117—28no—
170#117—44no—
171#117—44no—
172#117—48no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
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—22no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C4⋊C414yes—
214C2×C4⋊C4C4⋊C414yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—81no—
225C2×C22⋊C4—81no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4C4⋊C418yes—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—24no—
233C2×C42—44no—
234C2×C42C22×C414yes—
235C2×C42C2×Q814yes—
236#107—44no—
237#106—48no—
238#106—44no—
239C22⋊C8—28no—
240C22⋊C8—28no—
241#95—28no—
242#95—28no—
243#95—42no—
244#95—42no—
245#95—82no—
246#95—82no—
247#95—82no—
248#95—82no—
249C24—22no—
250C24—22no—
251C24—21no—
252C24—21no—
253C24—22no—
254C24—21no—
255C24—22no—
256C24—21no—
257C24C2×C4⋊C412yes—
258C24#12012yes—
259C24#12312yes—
260C24C22×D411yescharacteristic
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—48no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—84no—
325C22×C4—84no—
326C22×C4—88no—
327C22×C4—88no—
328C22×C4C2×C4⋊C414yes—
329C22×C4C2×C4⋊C414yes—
330C22×C4C2×C4⋊C412yes—
331C22×C4C4×D414yes—
332C22×C4C4×D414yes—
333C22×C4#12012yes—
334C22×C4C22⋊Q814yes—
335C22×C4C22⋊Q814yes—
336C22×C4#12312yes—
337C22×C4C23×C411yescharacteristic
338C22×C4C22×D411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×Q811yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4H(5,𝔽2)12yes—
344C22×C4H(5,𝔽2)12yes—
345C22×C42−1+412yes—
346C22×C42−1+412yes—
347C2×C8—24no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—42no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C4⋊C4—28no—
365C4⋊C4—28no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—88no—
388C4⋊C4—88no—
389C22⋊C4—216no—
390C22⋊C4—48no—
391C22⋊C4—84no—
392C22⋊C4—84no—
393C42—24no—
394C42—24no—
395C42—24no—
396C42—24no—
397C42—48no—
398C23—24no—
399C23—22no—
400C23—22no—
401C23—24no—
402C23—22no—
403C23—22no—
404C23—24no—
405C23—24no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23#38514yes—
421C23C22×C4⋊C411yescharacteristic
422C23C2×#12011yescharacteristic
423C23C2×#12311yescharacteristic
424C23#53512yes—
425C23#53612yes—
426C23#57611yescharacteristic
427C23#57711yescharacteristic
428C23#57711yescharacteristic
429C23#57712yes—
430C23#57712yes—
431C23#57811yescharacteristic
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—28no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—816no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4#38514yes—
492C2×C4C2×C4×D411yescharacteristic
493C2×C4C2×C4×D411yescharacteristic
494C2×C4#52012yes—
495C2×C4#52012yes—
496C2×C4C2×C22⋊Q811yescharacteristic
497C2×C4C2×C22⋊Q811yescharacteristic
498C2×C4#53412yes—
499C2×C4#53411yescharacteristic
500C2×C4#53411yescharacteristic
501C2×C4#53611yescharacteristic
502C2×C4#53612yes—
503C2×C4#53611yescharacteristic
504C2×C4#53912yes—
505C2×C4#53912yes—
506C2×C4#53912yes—
507C2×C4#53912yes—
508C2×C4#54412yes—
509C2×C4#54412yes—
510C8—44no—
511C8—42no—
512C8—42no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—42no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—44no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22C2×#38511yescharacteristic
525C22#231712yes—
526C22#261311yescharacteristic
527C22#261411yescharacteristic
528C22#262312yes—
529C22#291212yes—
530C22#291211yescharacteristic
531C22#291311yescharacteristic
532C22#300911yescharacteristic
533C22#301011yescharacteristic
534C22#301211yescharacteristic
535C22#301311yescharacteristic
536C22#309611yescharacteristic
537C22#309711yescharacteristic
538C22#309912yes—
539C22#322112yes—
540C22#322212yes—
541C22#322412yes—
542C22#322512yes—
543C22#324112yes—
544C22#324212yes—
545C22#324412yes—
546C22#324412yes—
547C4—24no—
548C4—44no—
549C4—44no—
550C4—82no—
551C4—82no—
552C4—84no—
553C4—161no—
554C4—161no—
555C4—162no—
556C4#232212yes—
557C4#232211yescharacteristic
558C4#232211yescharacteristic
559C4#262712yes—
560C4#262711yescharacteristic
561C4#262711yescharacteristic
562C2—44no—
563C2#1426311yescharacteristic
564C2#2084811yescharacteristic
565C2#2084911yescharacteristic
566C2#2199012yes—
567C2#2199212yes—
568C2#2963111yescharacteristic
569C2#2963211yescharacteristic
570C2#2964711yescharacteristic
571C2#2964811yescharacteristic
572C2#2991612yes—
573C2#2991912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11325C2semidirect
7. #11323C2semidirect
8. #11268C2semidirect
9. #11267C2semidirect
31. #1818C4semidirect
121. #336C2×C4semidirect
122. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#18542
C2×#18532
C2×#18182
#143432
#113252
#113232
#112682
#112672
#107312
#107302