Finite Group Databasebeta

← #196124

Subgroups — #196124

Order 512.

1220 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60248C211yesmaximal, characteristic
2C22×#441C211yesmaximal, characteristic
3#17449C211yesmaximal, characteristic
4#17449C211yesmaximal, characteristic
5#17365C211yesmaximal, characteristic
6#9092C214yesmaximal, complemented
7#9088C214yesmaximal, complemented
8#7919C211yesmaximal, characteristic
9#7861C211yesmaximal, characteristic
10C23×Q16C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13#2984—22no—
14#2984—22no—
15#2983—22no—
16#2983C2212yes—
17#2982C2212yes—
18Q8×M16—22no—
19Q8×M16—22no—
20C2×#441—22no—
21C2×#441—22no—
22C2×#441—22no—
23C2×#441—22no—
24C2×#441C412yes—
25C2×#441C412yes—
26C2×#441C2212yes—
27C2×#441C2212yes—
28#2920—22no—
29#2920C2211yescharacteristic
30#2920C2211yescharacteristic
31#2920C2211yescharacteristic
32#2920C2211yescharacteristic
33C22×#106C411yescharacteristic
34C22×#103C411yescharacteristic
35C22×#103C411yescharacteristic
36C22×#97C2211yescharacteristic
37#1491—22no—
38#1491—22no—
39#1487—22no—
40#1487—22no—
41#1479C2212yes—
42#1479C2214yes—
43#1479C2212yes—
44#1467—22no—
45#1463—22no—
46#1461C2214yes—
47#1281C2214yes—
48#1275C2214yes—
49C23×Q8C2×C411yescharacteristic
50C23×Q8C2×C411yescharacteristic
51C22×Q16—22no—
52C22×Q16—22no—
53C22×Q16—28no—
54C22×Q16C2×C412yes—
55C22×Q16C2×C412yes—
56C23×C8C2×C411yescharacteristic
57C23×C8C2×C411yescharacteristic
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×C4×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×Q8C2×C412yes—
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8C2×C411yescharacteristic
77C2×C4×Q8D412yes—
78C2×C4×Q8D412yes—
79C2×C4×Q8D412yes—
80C2×C4×Q8D412yes—
81C2×C4×Q8C2312yes—
82C2×C4×Q8C2311yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C42—21no—
89C22×C42—21no—
90C22×C42C2311yescharacteristic
91#446—28no—
92#446—28no—
93#446—24no—
94#446—24no—
95C8×Q8—24no—
96C8×Q8—24no—
97#441—24no—
98#441—24no—
99#441—24no—
100#441—24no—
101#441—24no—
102#441—24no—
103#441—24no—
104#441—24no—
105#432—24no—
106#432—21no—
107#432—21no—
108#432—21no—
109#432—24no—
110#432—21no—
111#432—21no—
112#432—21no—
113#432—21no—
114#432D412yes—
115#431—24no—
116#431—21no—
117#431—24no—
118#431—21no—
119#431—21no—
120#431D412yes—
121#431C2312yes—
122#431C2312yes—
123#431C2312yes—
124#431C2312yes—
125C2×#106—22no—
126C2×#106—22no—
127C2×#106C2×C412yes—
128C2×#106C2×C412yes—
129#423—24no—
130#423—21no—
131#423—21no—
132#423—21no—
133#423—21no—
134C2×#103—22no—
135C2×#103—22no—
136C2×#103—22no—
137C2×#103—22no—
138C2×#103C2×C412yes—
139C2×#103C2×C412yes—
140C2×#103C2×C412yes—
141C2×#103C2×C412yes—
142#407—21no—
143#407—21no—
144#407—21no—
145#407D412yes—
146C4×M16—24no—
147C4×M16—21no—
148C4×M16—21no—
149C4×M16—21no—
150C4×M16D412yes—
151C2×#97—21no—
152C2×#97—21no—
153C2×#97—21no—
154C2×#97—21no—
155C2×#97C2×C412yes—
156C2×#97C2312yes—
157Q8⋊C8—24no—
158Q8⋊C8—24no—
159Q8⋊C8—28no—
160#324—28no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8C22⋊C412yes—
174C22×Q8C22⋊C412yes—
175C22×Q8C22×C411yescharacteristic
176C22×Q8C22×C412yes—
177C22×Q8C22×C411yescharacteristic
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4#4111yescharacteristic
185C2×Q16—216no—
186C2×Q16—216no—
187C2×Q16—24no—
188C2×Q16—24no—
189C2×Q16—24no—
190C2×Q16—24no—
191C2×Q16—24no—
192C2×Q16M1618yescomplemented
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—41no—
196C2×M16—41no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—24no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—42no—
203C22×C8C22×C412yes—
204C22×C8C22×C412yes—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—22no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—22no—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—41no—
230C4×Q8—41no—
231C4×Q8—41no—
232C4×Q8—41no—
233C4×Q8—41no—
234C4×Q8—41no—
235C4×Q8—41no—
236C4×Q8—41no—
237C4×Q8C22⋊C412yes—
238C4×Q8C22⋊C412yes—
239C4×Q8C22⋊C412yes—
240C4×Q8C22⋊C412yes—
241C4×Q8C2×D412yes—
242C4×Q8C2×D412yes—
243C4×Q8C2×D412yes—
244C4×Q8C2×D412yes—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4C22⋊C412yes—
268C2×C4⋊C4C22⋊C412yes—
269C2×C4⋊C4C22×C412yes—
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—22no—
275C2×C42—22no—
276C2×C42—22no—
277C2×C42—22no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—41no—
283C2×C42—41no—
284C2×C42—41no—
285C2×C42—41no—
286C2×C42C22×C411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42#4111yescharacteristic
295C2×C42#4111yescharacteristic
296C2×C42#4111yescharacteristic
297C2×C42C2411yescharacteristic
298#106—24no—
299#106—24no—
300C4⋊C8—28no—
301C4⋊C8—28no—
302C4⋊C8—22no—
303C4⋊C8—22no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—41no—
309C4⋊C8—41no—
310#103—24no—
311#103—24no—
312#103—24no—
313#103—24no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—22no—
318C22⋊C8—22no—
319C22⋊C8—41no—
320C22⋊C8—41no—
321C22⋊C8C2×D414yes—
322#97—28no—
323#97—22no—
324#97—22no—
325#97—22no—
326#97—22no—
327#97—22no—
328#97—22no—
329#97—41no—
330#97—41no—
331C4×C8—28no—
332C4×C8—22no—
333C4×C8—22no—
334C4×C8—22no—
335C4×C8C2×D414yes—
336C24C4×D411yescharacteristic
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—41no—
364C2×Q8—41no—
365C2×Q8C2×C22⋊C412yes—
366C2×Q8C2×C22⋊C412yes—
367C2×Q8C2×M1614yes—
368C2×Q8C2×M1614yes—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—82no—
390C22×C4—82no—
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C4×D411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4C4×D411yescharacteristic
396C22×C4C4×D411yescharacteristic
397C22×C4C4×D411yescharacteristic
398C22×C4#12011yescharacteristic
399C22×C4#12011yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12311yescharacteristic
405C22×C4#12311yescharacteristic
406C22×C4C23×C411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4111yescharacteristic
409Q16—216no—
410Q16—216no—
411M16—44no—
412M16—44no—
413M16—82no—
414M16—82no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8C2×M1614yes—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4C2×C22⋊C412yes—
449C4⋊C4C2×C22⋊C412yes—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—42no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42C2×C22⋊C411yescharacteristic
463C42C2×C22⋊C411yescharacteristic
464C42C4×D411yescharacteristic
465C42C4×D411yescharacteristic
466C42#12011yescharacteristic
467C42#12011yescharacteristic
468C42#12111yescharacteristic
469C42#12111yescharacteristic
470C42#12111yescharacteristic
471C42#12111yescharacteristic
472C42#12311yescharacteristic
473C42#12311yescharacteristic
474C42C22×D411yescharacteristic
475C42C22×D411yescharacteristic
476C42C22×D411yescharacteristic
477C42C2×#4111yescharacteristic
478C23—22no—
479C23—22no—
480C23—22no—
481C23#38611yescharacteristic
482C23#38611yescharacteristic
483C23C2×C4×D411yescharacteristic
484Q8—22no—
485Q8—24no—
486Q8—22no—
487Q8—24no—
488Q8—22no—
489Q8—24no—
490Q8—44no—
491Q8—44no—
492Q8—44no—
493Q8—44no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497Q8#40714yes—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4#38611yescharacteristic
539C2×C4#38611yescharacteristic
540C2×C4#38611yescharacteristic
541C2×C4#38611yescharacteristic
542C2×C4#38611yescharacteristic
543C2×C4#38611yescharacteristic
544C2×C4C22×C22⋊C411yescharacteristic
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×#12011yescharacteristic
547C2×C4C2×#12111yescharacteristic
548C2×C4C2×#12111yescharacteristic
549C2×C4C2×#12311yescharacteristic
550C2×C4C22×M1612yes—
551C2×C4#56812yes—
552C8—24no—
553C8—44no—
554C8—44no—
555C8—42no—
556C8—82no—
557C8—82no—
558C22—28no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22#190212yes—
566C22#190512yes—
567C22C2×#38611yescharacteristic
568C22D4×M1612yes—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—21no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—24no—
580C4—42no—
581C4—42no—
582C4—44no—
583C4—42no—
584C4—82no—
585C4C2×#40711yescharacteristic
586C4#287911yescharacteristic
587C4D4×M1612yes—
588C2—22no—
589C2—22no—
590C2—22no—
591C2#2089811yescharacteristic
592C2C2×#190211yescharacteristic
593C2C2×#190511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9092C2semidirect
7. #9088C2semidirect
192. C2×Q16M16semidirect

Maximal subgroups

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

subgroupindex
#602482
C22×#4412
#174492
#173652
#90922
#90882
#79192
#78612