Finite Group Databasebeta

← #370096

Subgroups — #370096

Order 512.

1654 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
1C22×#428C211yesmaximal, characteristic
2C2×#1947C212yesmaximal, complemented
3C2×#1854C211yesmaximal, characteristic
4C2×#1851C211yesmaximal, characteristic
5#13880C211yesmaximal, characteristic
6#13801C211yesmaximal, characteristic
7#11323C212yesmaximal, complemented
8#11231C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428—28no—
16C2×#428—24no—
17C22×#107C2211yescharacteristic
18C22×#107C2212yes—
19C22×#106C2212yes—
20C22×#106C2211yescharacteristic
21C22×C4⋊C8C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C2×#385—22no—
24C2×#384C2212yes—
25C2×#380C2211yescharacteristic
26C2×#380C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1947—28no—
30#1854—24no—
31#1851C418yescomplemented
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1444—21no—
40#1444—21no—
41#1437—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44#1434—21no—
45C4×#95—21no—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—24no—
51C22×M16—24no—
52C22×M16C2×C414yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C2×#117—22no—
57C2×#117—24no—
58C2×#117—42no—
59C2×#117D414yes—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4D412yes—
77C22×C42—21no—
78C22×C42—22no—
79C22×C42C2312yes—
80#428—232no—
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—22no—
84C2×#107—24no—
85C2×#107D418yes—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—24no—
89C2×#106—22no—
90C2×#106—42no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—24no—
93C2×C4⋊C8C2×C418yescomplemented
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8C2×C414yes—
97#385—48no—
98#384—24no—
99#384—42no—
100#380—22no—
101#380—22no—
102#380—22no—
103#380Q814yes—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—42no—
106C4×C22⋊C4C2×C414yes—
107C4×C22⋊C4Q814yes—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95D411yescharacteristic
116C2×#95D411yescharacteristic
117#340—24no—
118#340—24no—
119#340—22no—
120#340—22no—
121#340—24no—
122#340—42no—
123#336C2×C418yescomplemented
124C25C2×D411yescharacteristic
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—44no—
136C23×C4—42no—
137C23×C4—44no—
138C23×C4—81no—
139C23×C4—81no—
140C23×C4C22×C412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×Q812yes—
146C23×C4C2×Q812yes—
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4C2411yescharacteristic
150C2×M16—216no—
151C2×M16—28no—
152C2×M16—28no—
153C2×M16—28no—
154C2×M16—216no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8C22×C412yes—
168C22×C8C22×C412yes—
169C22×C8C22×C414yes—
170C22×C8C2×D414yes—
171C22×C8#4114yes—
172#117—28no—
173#117—44no—
174#117—48no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
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—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×Q814yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—81no—
222C2×C22⋊C4—81no—
223C2×C22⋊C4C4⋊C418yes—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42—82no—
233C2×C42C22×C414yes—
234C2×C42C2×Q814yes—
235#107—216no—
236#107—44no—
237#106—216no—
238#106—44no—
239C4⋊C8—216no—
240C22⋊C8—28no—
241#95—24no—
242#95—44no—
243#95—44no—
244#95—42no—
245#95—44no—
246#95—44no—
247#95—82no—
248#95C4⋊C414yes—
249#95C4⋊C414yes—
250C24—22no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—21no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24C2×C4⋊C412yes—
259C24#12112yes—
260C24C22⋊Q812yes—
261C24C22×D411yescharacteristic
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—48no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—88no—
324C22×C4—88no—
325C22×C4—84no—
326C22×C4—84no—
327C22×C4C2×C4⋊C414yes—
328C22×C4C2×C4⋊C412yes—
329C22×C4C2×C4⋊C414yes—
330C22×C4C4×D414yes—
331C22×C4C4×Q814yes—
332C22×C4#12112yes—
333C22×C4C22⋊Q812yes—
334C22×C4#12514yes—
335C22×C4C4⋊Q814yes—
336C22×C4AGL(1,ℤ/8)14yes—
337C22×C4#13714yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D412yes—
341C22×C4C22×Q811yescharacteristic
342C22×C4C22×Q811yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4112yes—
345C22×C4C2×#4111yescharacteristic
346C22×C4H(5,𝔽2)12yes—
347C22×C42−1+412yes—
348M16—232no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—24no—
354C2×C8—24no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8C4×D418yes—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—24no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—48no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—88no—
387C4⋊C4—88no—
388C4⋊C4#12014yes—
389C22⋊C4—216no—
390C22⋊C4—48no—
391C22⋊C4—84no—
392C22⋊C4—84no—
393C42—24no—
394C42—24no—
395C42—24no—
396C42—24no—
397C42—48no—
398C42—88no—
399C23—24no—
400C23—22no—
401C23—22no—
402C23—24no—
403C23—24no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—24no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23#38914yes—
422C23C22×C4⋊C411yescharacteristic
423C23C2×#12111yescharacteristic
424C23C2×C22⋊Q811yescharacteristic
425C23D4212yes—
426C23#54712yes—
427C23C2×AGL(1,ℤ/8)12yes—
428C23C2×#13712yes—
429C23#57611yescharacteristic
430C23#57711yescharacteristic
431C23#57711yescharacteristic
432C23#57811yescharacteristic
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—28no—
446C2×C4—24no—
447C2×C4—28no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—816no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4#38914yes—
492C2×C4#42818yescomplemented
493C2×C4C2×C4×D411yescharacteristic
494C2×C4C2×C4×D412yes—
495C2×C4C2×C4×Q811yescharacteristic
496C2×C4#52012yes—
497C2×C4C2×#12011yescharacteristic
498C2×C4#52511yescharacteristic
499C2×C4#52512yes—
500C2×C4C2×#12511yescharacteristic
501C2×C4C2×C4⋊Q811yescharacteristic
502C2×C4#53411yescharacteristic
503C2×C4#53611yescharacteristic
504C2×C4#53912yes—
505C2×C4#54612yes—
506C2×C4D4×Q812yes—
507C2×C4D4×Q812yes—
508C2×C4#55212yes—
509C2×C4#55412yes—
510C2×C4#55412yes—
511C8—28no—
512C8—42no—
513C8—42no—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—42no—
518C22—42no—
519C22—42no—
520C22—42no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22C2×#38911yescharacteristic
526C22D4×C4⋊C412yes—
527C22#241011yescharacteristic
528C22#241311yescharacteristic
529C22#244912yes—
530C22C2×#42812yes—
531C22#291211yescharacteristic
532C22#291311yescharacteristic
533C22#303811yescharacteristic
534C22#303911yescharacteristic
535C22#304511yescharacteristic
536C22#304611yescharacteristic
537C22#307912yes—
538C22#308111yescharacteristic
539C22#308211yescharacteristic
540C22#326412yes—
541C22#327012yes—
542C22#327212yes—
543C22#327812yes—
544C22#334312yes—
545C22#334612yes—
546C22#335412yes—
547C22#335612yes—
548C4—28no—
549C4—24no—
550C4—44no—
551C4—44no—
552C4—84no—
553C4—82no—
554C4—84no—
555C4—161no—
556C4—161no—
557C4C4×#12011yescharacteristic
558C4#232211yescharacteristic
559C4#236012yes—
560C4#243411yescharacteristic
561C4#247312yes—
562C4#258311yescharacteristic
563C2—44no—
564C2#1404311yescharacteristic
565C2#2117111yescharacteristic
566C2#2117511yescharacteristic
567C2#2266112yes—
568C2#2266312yes—
569C2#2437211yescharacteristic
570C2#2437411yescharacteristic
571C2#2459411yescharacteristic
572C2#2459611yescharacteristic
573C2#2635512yes—
574C2#2635712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1947C2semidirect
7. #11323C2semidirect
8. #11231C2semidirect
9. #10907C2semidirect
31. #1851C4semidirect
93. C2×C4⋊C8C2×C4semidirect
123. #336C2×C4semidirect
492. C2×C4#428semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#19472
C2×#18542
C2×#18512
#138802
#138012
#113232
#112312
#109072
#107302
#107222