Finite Group Databasebeta

← #194332

Subgroups — #194332

Order 512.

1139 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2C2×Q8×M16C211yesmaximal, characteristic
3#21306C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5#17486C211yesmaximal, characteristic
6#8990C218yesmaximal, complemented
7#8095C211yesmaximal, characteristic
8#7882C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#3104—21no—
12#3104—21no—
13#3104—22no—
14#3104C412yescomplemented
15#3104C412yescomplemented
16#3104C2214yes—
17C2×#484—21no—
18C2×#477C412yescomplemented
19C2×#462C412yescomplemented
20C2×#451—21no—
21Q8×M16—22no—
22Q8×M16—22no—
23Q8×M16—22no—
24Q8×M16C2214yes—
25C2×#446—22no—
26C2×#446—21no—
27C2×C8×Q8—21no—
28C2×C4×Q16—22no—
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×#423—21no—
32C2×#423C2211yescharacteristic
33C2×C4×M16—21no—
34C2×C4×M16C2211yescharacteristic
35#2285C2211yescharacteristic
36#1986—22no—
37#1986C2214yes—
38#1824—21no—
39#1803—22no—
40#1787—21no—
41C4×#103C2214yes—
42#1733—21no—
43#1733—21no—
44#1493C2214yes—
45#1487—22no—
46#1481—22no—
47#1303C2214yes—
48#1278C2214yes—
49C23×Q8D411yescharacteristic
50C22×Q16—22no—
51C22×M16—21no—
52C22×M16D411yescharacteristic
53C22×M16D411yescharacteristic
54#533—21no—
55#533—21no—
56#533—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533C2×C412yes—
61#533C2312yes—
62C2×C4⋊Q8C2×C411yescharacteristic
63C2×C4⋊Q8C2×C411yescharacteristic
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8C2×C412yes—
76C2×C4×Q8C2×C411yescharacteristic
77C2×C4×Q8C2×C411yescharacteristic
78C2×C4×Q8D411yescharacteristic
79C2×C4×Q8D411yescharacteristic
80C2×C4×Q8D412yes—
81C2×C4×Q8D412yes—
82C2×C4×Q8C2312yes—
83C2×#117—21no—
84C2×#117—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4D411yescharacteristic
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#484—22no—
90#484—22no—
91#484—24no—
92#477—22no—
93#477—22no—
94#477C2×C414yes—
95#477C2×C414yes—
96#462—22no—
97#462—22no—
98#462C2×C414yes—
99#462C2×C414yes—
100#451—22no—
101#451—22no—
102#451—24no—
103#446—28no—
104#446—24no—
105#446—24no—
106#446—28no—
107C8×Q8—24no—
108C8×Q8—24no—
109C4×Q16—24no—
110C4×Q16—24no—
111C4×Q16—24no—
112C4×Q16—24no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—21no—
118#431—21no—
119#431C2×C412yes—
120#431C2312yes—
121C2×#107—22no—
122#423—24no—
123#423—24no—
124#423—21no—
125#423—21no—
126#423D412yes—
127#423C2312yes—
128C2×C4⋊C8—22no—
129C2×C4⋊C8—22no—
130C2×C4⋊C8—21no—
131C2×C4⋊C8—21no—
132C2×C4⋊C8C2×C412yes—
133C2×C4⋊C8D412yes—
134C2×#103—22no—
135C2×#103—22no—
136C2×#103—24no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C4×M16—24no—
140C4×M16—24no—
141C4×M16—21no—
142C4×M16—21no—
143C4×M16D412yes—
144C4×M16C2312yes—
145C2×#97—22no—
146C2×#97—21no—
147C2×C4×C8—22no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8C2312yes—
151#384—21no—
152#382—21no—
153#380—21no—
154#380—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4C2312yes—
157#340—24no—
158#339—24no—
159#339—24no—
160Q8⋊C8—28no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—44no—
167C22×Q8C22⋊C411yescharacteristic
168C22×Q8C22⋊C412yes—
169C22×Q8C22⋊C411yescharacteristic
170C22×Q8C2×D412yes—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4#4111yescharacteristic
176C2×Q16—24no—
177C2×Q16—24no—
178C2×Q16—48no—
179C2×M16—21no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—21no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—42no—
187C2×M16—42no—
188C2×M16Q1614yescomplemented
189C2×M16C2×D412yes—
190C2×M16C2×D412yes—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—22no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8C22×C412yes—
203C4⋊Q8C22×C412yes—
204#125—22no—
205#125—22no—
206#125—41no—
207#125—41no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—41no—
213C22⋊Q8—41no—
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8—22no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—22no—
222C4×Q8—21no—
223C4×Q8—22no—
224C4×Q8—21no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—22no—
235C4×Q8—22no—
236C4×Q8—41no—
237C4×Q8—41no—
238C4×Q8—41no—
239C4×Q8—41no—
240C4×Q8C22⋊C412yes—
241C4×Q8C22⋊C412yes—
242C4×Q8C22⋊C412yes—
243C4×Q8C22⋊C412yes—
244C4×Q8C22×C412yes—
245C4×Q8C22×C412yes—
246C4×Q8C2×D412yes—
247C4×Q8C2×D412yes—
248#117—22no—
249#117—22no—
250#117—41no—
251#117—41no—
252#117—42no—
253#117—42no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—41no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4C22⋊C412yes—
275C2×C4⋊C4C22⋊C411yescharacteristic
276C2×C4⋊C4C22⋊C411yescharacteristic
277C2×C4⋊C4C2×D412yes—
278C2×C22⋊C4—41no—
279C2×C22⋊C4—41no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—41no—
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42#4111yescharacteristic
299C2×C42#4111yescharacteristic
300C2×C42#4111yescharacteristic
301C2×C42C2411yescharacteristic
302#107—24no—
303#107—24no—
304C4⋊C8—28no—
305C4⋊C8—28no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—41no—
311C4⋊C8—41no—
312C4⋊C8C22×C414yes—
313C4⋊C8C2×D414yes—
314#103—24no—
315#103—24no—
316#103—24no—
317#103—24no—
318#103—48no—
319C22⋊C8—22no—
320C22⋊C8—22no—
321C22⋊C8—41no—
322C22⋊C8—41no—
323#97—28no—
324#97—22no—
325#97—22no—
326C4×C8—28no—
327C4×C8—22no—
328C4×C8—22no—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8—22no—
332C4×C8—22no—
333#95—22no—
334#95—41no—
335C24#12111yescharacteristic
336C2×Q8—24no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8—21no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—82no—
355C2×Q8C2×C22⋊C412yes—
356C2×Q8C2×C22⋊C412yes—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—81no—
371C22×C4—82no—
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4C4×D411yescharacteristic
378C22×C4#12011yescharacteristic
379C22×C4#12011yescharacteristic
380C22×C4#12111yescharacteristic
381C22×C4#12111yescharacteristic
382C22×C4#12111yescharacteristic
383C22×C4#12111yescharacteristic
384C22×C4#12111yescharacteristic
385C22×C4#12311yescharacteristic
386C22×C4#12311yescharacteristic
387C22×C4C2×Q1612yes—
388C22×C4#13512yes—
389C22×C4#13712yes—
390C22×C4#13712yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C2×#4111yescharacteristic
394Q16—48no—
395Q16—48no—
396M16—22no—
397M16—24no—
398M16—22no—
399M16—22no—
400M16—42no—
401M16—42no—
402M16—42no—
403M16—44no—
404M16C2×Q1614yescomplemented
405C2×C8—22no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—22no—
409C2×C8—24no—
410C2×C8—44no—
411C2×C8—42no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—42no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4C2×C22⋊C412yes—
443C4⋊C4C2×C22⋊C412yes—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—82no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—42no—
459C42C2×C22⋊C411yescharacteristic
460C42C2×C22⋊C411yescharacteristic
461C42C4×D411yescharacteristic
462C42C4×D411yescharacteristic
463C42C4×D411yescharacteristic
464C42C4×D411yescharacteristic
465C42#12011yescharacteristic
466C42#12011yescharacteristic
467C42#12111yescharacteristic
468C42#12111yescharacteristic
469C42#12311yescharacteristic
470C42#12311yescharacteristic
471C42C23×C411yescharacteristic
472C42C22×D411yescharacteristic
473C42C22×D411yescharacteristic
474C42C2×#4111yescharacteristic
475C23—41no—
476C23—41no—
477C23#38611yescharacteristic
478C23#38611yescharacteristic
479C23#46212yes—
480C23#46412yes—
481C23C2×#12111yescharacteristic
482Q8—24no—
483Q8—22no—
484Q8—22no—
485Q8—22no—
486Q8—44no—
487Q8—44no—
488Q8—42no—
489Q8—44no—
490Q8—44no—
491Q8—82no—
492Q8—82no—
493Q8#42114yes—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#38611yescharacteristic
526C2×C4#38611yescharacteristic
527C2×C4#38611yescharacteristic
528C2×C4#38611yescharacteristic
529C2×C4#38611yescharacteristic
530C2×C4#38611yescharacteristic
531C2×C4C4×Q1612yes—
532C2×C4C4×Q1612yes—
533C2×C4#44112yes—
534C2×C4#44112yes—
535C2×C4#45112yes—
536C2×C4#45112yes—
537C2×C4#45212yes—
538C2×C4#45212yes—
539C2×C4#46212yes—
540C2×C4#46412yes—
541C2×C4C22×C22⋊C411yescharacteristic
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C2×#12011yescharacteristic
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×#12311yescharacteristic
547C2×C4C22×Q1611yescharacteristic
548C2×C4C2×#13511yescharacteristic
549C2×C4C2×#13711yescharacteristic
550C2×C4C2×#13711yescharacteristic
551C8—24no—
552C8—44no—
553C8—42no—
554C8—44no—
555C8—82no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—42no—
563C22—42no—
564C22#187812yes—
565C22#187812yes—
566C22C2×#38611yescharacteristic
567C22C2×#46211yescharacteristic
568C22C2×#46411yescharacteristic
569C4—21no—
570C4—24no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—21no—
577C4—42no—
578C4—42no—
579C4—42no—
580C4—82no—
581C4—82no—
582C4—162no—
583C4#187812yes—
584C4#187812yes—
585C4C2×#42112yes—
586C4C2×C4×Q1611yescharacteristic
587C4C2×#44111yescharacteristic
588C4C2×#45111yescharacteristic
589C4C2×#45211yescharacteristic
590C2—21no—
591C2—22no—
592C2—22no—
593C2—21no—
594C2#2090211yescharacteristic
595C2C2×#187811yescharacteristic
596C2#2093911yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8990C2semidirect
14. #3104C4semidirect
15. #3104C4semidirect
18. C2×#477C4semidirect
19. C2×#462C4semidirect
188. C2×M16Q16semidirect
404. M16C2×Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
C2×Q8×M162
#213062
#207342
#174862
#89902
#80952
#78822