Finite Group Databasebeta

← #192635

Subgroups — #192635

Order 512.

1088 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34196C211yesmaximal, characteristic
2#21987C211yesmaximal, characteristic
3#20800C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×#1491C211yesmaximal, characteristic
6#8888C212yesmaximal, complemented
7#8887C212yesmaximal, complemented
8#8884C212yesmaximal, complemented
9#8883C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic
11#7895C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#428—22no—
15C22×C4⋊C8C2211yescharacteristic
16C22×#103—21no—
17C2×C4×M16C2211yescharacteristic
18#2355—21no—
19#2355—21no—
20#2355C2211yescharacteristic
21#2355C2211yescharacteristic
22Q8×C4⋊C4C2211yescharacteristic
23Q8×C4⋊C4C2211yescharacteristic
24#2344—22no—
25#2344—21no—
26Q8×C22⋊C4—21no—
27#2320—22no—
28#2320—21no—
29#2285—21no—
30#2285C2211yescharacteristic
31C2×#384C2211yescharacteristic
32#2283—21no—
33#2283C2211yescharacteristic
34#2276—21no—
35#1948C2212yes—
36#1947C2212yes—
37#1853—21no—
38#1852—21no—
39#1847—22no—
40#1847—21no—
41#1845C2212yes—
42#1844C2212yes—
43#1836—21no—
44#1803—21no—
45#1803—21no—
46#1803C2212yes—
47#1803C2212yes—
48C2×#340—21no—
49#1491C418yescomplemented
50C2×Q8⋊C8C2212yes—
51C2×Q8⋊C8C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C23×Q8D411yescharacteristic
57C22×M16—21no—
58C23×C8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C414yes—
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8Q814yes—
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×#117—22no—
72C2×#117D412yes—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#432—22no—
81#432C2×C414yes—
82#428—48no—
83C2×#107—24no—
84C2×#107—22no—
85C2×#106—24no—
86C2×#106—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—44no—
95C2×C22⋊C8—21no—
96C4×M16—22no—
97C4×M16C2×C414yes—
98C2×#97—21no—
99C2×#97—21no—
100C2×C4×C8C2312yes—
101#391—22no—
102#391—22no—
103#391—21no—
104#391—21no—
105#389—21no—
106#389—21no—
107#389—22no—
108#389—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#384—21no—
116#384—21no—
117#384—21no—
118#384—21no—
119#384—21no—
120#384C2312yes—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#383—22no—
124#383—21no—
125#382—22no—
126#382—21no—
127#382—21no—
128#382D412yes—
129#382D411yescharacteristic
130#382D411yescharacteristic
131#381—22no—
132#381—21no—
133#380—22no—
134#380—21no—
135#380—21no—
136#380D411yescharacteristic
137#380D411yescharacteristic
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—22no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C4×C4⋊C4—21no—
149C4×C4⋊C4C2311yescharacteristic
150C4×C4⋊C4C2311yescharacteristic
151C4×C22⋊C4—22no—
152C4×C22⋊C4—21no—
153#376—22no—
154#376—21no—
155#340—22no—
156#340—22no—
157#340—22no—
158#340—22no—
159Q8⋊C8C2×C418yescomplemented
160Q8⋊C8C2×C418yescomplemented
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8C4⋊C414yes—
168C22×Q8C2×D411yescharacteristic
169C22×Q8C2×D411yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—41no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C2×M16—24no—
177C2×M16—24no—
178C2×M16—24no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—42no—
183C22×C8—42no—
184C22×C8—42no—
185C4×Q8—24no—
186C4×Q8—24no—
187C4×Q8—24no—
188C4×Q8—24no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8C4⋊C414yes—
200C4×Q8C4⋊C414yes—
201C4×Q8C22×C412yes—
202C4×Q8C22×C412yes—
203C4×Q8C2×Q812yes—
204C4×Q8C2×Q812yes—
205#117—44no—
206#117—84no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4C4⋊C414yes—
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D412yes—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—21no—
245C2×C42—24no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—24no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42C22×C412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×Q812yes—
263C2×C42#4112yes—
264C2×C42#4112yes—
265C2×C42C2411yescharacteristic
266#107—48no—
267#107—82no—
268#106—48no—
269#106—82no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—42no—
274C4⋊C8C22×C414yes—
275#103—44no—
276#103—44no—
277C22⋊C8—24no—
278#97—24no—
279#97—24no—
280C4×C8—22no—
281C4×C8C22×C414yes—
282#95—22no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—21no—
291#95—22no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—22no—
298#95—21no—
299#95—21no—
300C24#12011yescharacteristic
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—24no—
306C2×Q8—28no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8C2×C4⋊C412yes—
317C2×Q8C2×C4⋊C412yes—
318C2×Q8AGL(1,ℤ/8)12yes—
319C2×Q8#13712yes—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4C2×C4⋊C412yes—
349C22×C4C4×D412yes—
350C22×C4C4×D412yes—
351C22×C4#12011yescharacteristic
352C22×C4#12011yescharacteristic
353C22×C4#12011yescharacteristic
354C22×C4C22⋊Q812yes—
355C22×C4C22⋊Q812yes—
356C22×C4#12312yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360M16—216no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—84no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4C2×C4⋊C412yes—
405C4⋊C4C2×C4⋊C412yes—
406C22⋊C4—44no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—24no—
414C42—24no—
415C42—44no—
416C42—44no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—41no—
421C42—82no—
422C42—82no—
423C42—82no—
424C42C2×C4⋊C412yes—
425C42C4×D412yes—
426C42C4×D412yes—
427C42C22⋊Q812yes—
428C42C22⋊Q812yes—
429C42#12312yes—
430C42C23×C411yescharacteristic
431C42C22×Q811yescharacteristic
432C42C2×#4111yescharacteristic
433C42C2×#4111yescharacteristic
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23#38512yes—
439C23#45312yes—
440C23#45612yes—
441C23C2×#12011yescharacteristic
442Q8—22no—
443Q8—22no—
444Q8—22no—
445Q8—22no—
446Q8—22no—
447Q8—48no—
448Q8—41no—
449Q8—41no—
450Q8#42814yescomplemented
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—48no—
474C2×C4—48no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—164no—
503C2×C4#38512yes—
504C2×C4#38512yes—
505C2×C4#38512yes—
506C2×C4C22×C4⋊C411yescharacteristic
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×C4×D411yescharacteristic
509C2×C4C2×C22⋊Q811yescharacteristic
510C2×C4C2×C22⋊Q811yescharacteristic
511C2×C4C2×#12311yescharacteristic
512C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
513C2×C4C2×#13711yescharacteristic
514C2×C4#57511yescharacteristic
515C2×C4#57511yescharacteristic
516C2×C4#57712yes—
517C2×C4#57812yes—
518C8—28no—
519C8—44no—
520C8—82no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22#185914yes—
528C22C2×#38511yescharacteristic
529C22#300611yescharacteristic
530C22#300711yescharacteristic
531C22#301011yescharacteristic
532C22#301111yescharacteristic
533C22C2×#45311yescharacteristic
534C22C2×#45611yescharacteristic
535C4—22no—
536C4—24no—
537C4—21no—
538C4—21no—
539C4—22no—
540C4—22no—
541C4—42no—
542C4—42no—
543C4—42no—
544C4—81no—
545C4—82no—
546C4—81no—
547C4—82no—
548C4—81no—
549C4—81no—
550C4—162no—
551C4C2×#42811yescharacteristic
552C4#291111yescharacteristic
553C4#294712yes—
554C4#308212yes—
555C4#309411yescharacteristic
556C4#309511yescharacteristic
557C2—24no—
558C2#1856411yescharacteristic
559C2#1856511yescharacteristic
560C2#1858811yescharacteristic
561C2#1858911yescharacteristic
562C2#2084611yescharacteristic
563C2C2×#185911yescharacteristic
564C2#2087711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8888C2semidirect
7. #8887C2semidirect
8. #8884C2semidirect
9. #8883C2semidirect
49. #1491C4semidirect
159. Q8⋊C8C2×C4semidirect
160. Q8⋊C8C2×C4semidirect
450. Q8#428semidirect

Maximal subgroups

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

subgroupindex
#341962
#219872
#208002
C2×#18032
C2×#14912
#88882
#88872
#88842
#88832
#80442
#78952