Finite Group Databasebeta

← #111840

Subgroups — #111840

Order 512.

998 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3C2×#1503C211yesmaximal, characteristic
4C2×#1479C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic
6C2×#1303C211yesmaximal, characteristic
7#8043C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10#3078—22no—
11#3078—22no—
12#3078—22no—
13#3078C414yescomplemented
14C2×#479—22no—
15C2×#477—21no—
16C2×#475—21no—
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#428—21no—
20C2×#423C2211yescharacteristic
21C2×#419—21no—
22C2×#391—21no—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1990—21no—
26#1956—21no—
27#1872—22no—
28#1803—22no—
29#1503—22no—
30#1503—22no—
31C2×#328—22no—
32#1479—22no—
33#1479—22no—
34C2×Q8⋊C8—22no—
35#1441—22no—
36C4⋊#95—22no—
37C4⋊#95—21no—
38#1303—22no—
39#1296—22no—
40C23×Q8D411yescharacteristic
41C22×M16—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533—21no—
47#533—21no—
48#533C2×C412yes—
49#533D412yes—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D411yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—21no—
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×Q8C2×C412yes—
68C2×C4×Q8D412yes—
69C2×#117—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D412yes—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79#479—28no—
80#479—28no—
81#477—24no—
82#477—24no—
83#475—24no—
84#475—24no—
85#431—21no—
86#431—21no—
87#431—21no—
88#431—21no—
89#431—21no—
90#431—21no—
91#431—21no—
92#431D412yes—
93#428—24no—
94#428—24no—
95C2×#107—22no—
96C2×#106—22no—
97#423—21no—
98#423—21no—
99#423C2×C412yes—
100#423D412yes—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8D412yes—
104C2×C4⋊C8D412yes—
105#419—24no—
106#419—24no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—24no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114#391—42no—
115#391—42no—
116#384—21no—
117#384—21no—
118#384—21no—
119#384—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C2×#95—24no—
125C2×#95—24no—
126C2×#95D412yes—
127#340—24no—
128#328—28no—
129Q8⋊C8—28no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8C22⋊C412yes—
144C22×Q8SD1612yes—
145C22×Q8Q1612yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—41no—
156C2×M16—41no—
157C2×M16—41no—
158C2×M16—41no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—41no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8C22⋊C412yes—
169C4⋊Q8C22⋊C412yes—
170#125—22no—
171#125—22no—
172#125—41no—
173#125—41no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—41no—
179C22⋊Q8—41no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—22no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206#117—22no—
207#117—22no—
208#117—41no—
209#117—41no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4C22⋊C412yes—
244C2×C22⋊C4—41no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—41no—
248C2×C42—41no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42—41no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—41no—
257C2×C42—41no—
258C2×C42C22×C411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42#4111yescharacteristic
268C2×C42#4111yescharacteristic
269C2×C42#4111yescharacteristic
270#107—28no—
271#106—28no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—41no—
279C4⋊C8C22⋊C414yes—
280#103—28no—
281#103—28no—
282#103—84no—
283C22⋊C8—22no—
284C22⋊C8—41no—
285C22⋊C8—41no—
286C22⋊C8—41no—
287C4×C8—22no—
288C4×C8—41no—
289C4×C8—41no—
290C4×C8—41no—
291#95—22no—
292#95—22no—
293#95—48no—
294#95—84no—
295C24#12011yescharacteristic
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—81no—
330C2×Q8—81no—
331C2×Q8#10314yescomplemented
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C4×D411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4#12011yescharacteristic
367C22×C4#12011yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4#12111yescharacteristic
370C22×C4#12111yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12111yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12311yescharacteristic
378C22×C4#12711yescharacteristic
379C22×C4#12711yescharacteristic
380C22×C4C2×SD1611yescharacteristic
381C22×C4C2×Q1611yescharacteristic
382C22×C4#13511yescharacteristic
383C22×C4#13511yescharacteristic
384C22×C4AGL(1,ℤ/8)11yescharacteristic
385C22×C4#13711yescharacteristic
386C22×C4#13711yescharacteristic
387C22×C4#13711yescharacteristic
388M16—42no—
389M16—42no—
390M16—42no—
391M16—42no—
392C2×C8—24no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—81no—
400C2×C8—81no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—41no—
415C4⋊C4—42no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—82no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—81no—
440C42—81no—
441C42—81no—
442C42—81no—
443C42C2×C22⋊C411yescharacteristic
444C42C2×C22⋊C411yescharacteristic
445C42C4×D411yescharacteristic
446C42C4×D411yescharacteristic
447C42#12011yescharacteristic
448C42#12011yescharacteristic
449C42#12011yescharacteristic
450C42#12111yescharacteristic
451C42#12111yescharacteristic
452C42#12111yescharacteristic
453C42#12111yescharacteristic
454C42#12311yescharacteristic
455C42#12311yescharacteristic
456C42#12311yescharacteristic
457C42#12411yescharacteristic
458C42#12411yescharacteristic
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23#38611yescharacteristic
464C23#39211yescharacteristic
465C23#39211yescharacteristic
466C23#45011yescharacteristic
467C23#45111yescharacteristic
468C23#45211yescharacteristic
469C23#45211yescharacteristic
470Q8—24no—
471Q8—24no—
472Q8—42no—
473Q8—42no—
474Q8—42no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8—44no—
479Q8—82no—
480Q8—82no—
481Q8—82no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4#37911yescharacteristic
527C2×C4#38611yescharacteristic
528C2×C4#38611yescharacteristic
529C2×C4#38611yescharacteristic
530C2×C4#39011yescharacteristic
531C2×C4#39011yescharacteristic
532C2×C4#39211yescharacteristic
533C2×C4#39211yescharacteristic
534C2×C4#39411yescharacteristic
535C2×C4#39411yescharacteristic
536C2×C4#39411yescharacteristic
537C2×C4#39411yescharacteristic
538C2×C4C2×#10311yescharacteristic
539C2×C4#41611yescharacteristic
540C2×C4#41711yescharacteristic
541C2×C4#41911yescharacteristic
542C2×C4#44811yescharacteristic
543C2×C4#44811yescharacteristic
544C2×C4#44911yescharacteristic
545C2×C4#44911yescharacteristic
546C2×C4#45111yescharacteristic
547C2×C4#45111yescharacteristic
548C2×C4#45211yescharacteristic
549C2×C4#45211yescharacteristic
550C2×C4#45512yes—
551C2×C4#45612yes—
552C2×C4#46511yescharacteristic
553C2×C4#46511yescharacteristic
554C2×C4#46711yescharacteristic
555C2×C4#46711yescharacteristic
556C2×C4#46811yescharacteristic
557C2×C4#46811yescharacteristic
558C2×C4#47011yescharacteristic
559C2×C4#47011yescharacteristic
560C2×C4#48311yescharacteristic
561C2×C4#48411yescharacteristic
562C2×C4#48511yescharacteristic
563C2×C4#48511yescharacteristic
564C8—44no—
565C8—82no—
566C8—82no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22#143811yescharacteristic
580C22#187211yescharacteristic
581C22#187311yescharacteristic
582C22#200011yescharacteristic
583C22#200011yescharacteristic
584C22#200111yescharacteristic
585C22#200111yescharacteristic
586C22#200411yescharacteristic
587C22#200511yescharacteristic
588C22#200811yescharacteristic
589C22#200911yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—44no—
597C4—84no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4—162no—
602C4#178711yescharacteristic
603C4#178811yescharacteristic
604C4#188212yes—
605C4#189211yescharacteristic
606C4#189411yescharacteristic
607C4#200712yes—
608C4#203511yescharacteristic
609C4#203511yescharacteristic
610C4#203611yescharacteristic
611C4#203611yescharacteristic
612C2—21no—
613C2—21no—
614C2—21no—
615C2—21no—
616C2#1025011yescharacteristic
617C2#1025111yescharacteristic
618C2#1025211yescharacteristic
619C2#1025311yescharacteristic
620C2#1097011yescharacteristic
621C2#1101711yescharacteristic
622C2#1103611yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3078C4semidirect
331. C2×Q8#103semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
#218602
C2×#15032
C2×#14792
#139262
C2×#13032
#80432