Finite Group Databasebeta

← #192377

Subgroups — #192377

Order 512.

1314 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic
2#21942C211yesmaximal, characteristic
3#21765C211yesmaximal, characteristic
4C2×#1508C211yesmaximal, characteristic
5#13965C211yesmaximal, characteristic
6#8871C214yesmaximal, complemented
7#8868C214yesmaximal, complemented
8#8019C211yesmaximal, characteristic
9#7909C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13#2329—22no—
14#2329—22no—
15#2329—22no—
16#2329C2214yes—
17C2×#391—21no—
18C2×#389—21no—
19C2×#387—21no—
20C2×#387—21no—
21C2×#385—21no—
22C2×#385—21no—
23C2×#385—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×C4×C22⋊C4—21no—
27C2×C4×C22⋊C4C2211yescharacteristic
28C22×#95C2211yescharacteristic
29#1894—22no—
30#1872C2212yes—
31#1872C2212yes—
32#1853—21no—
33#1853C2212yes—
34#1852—21no—
35#1852C2212yes—
36#1823—21no—
37#1809—22no—
38#1787—21no—
39#1508C418yescomplemented
40C2×#328C2214yes—
41#1444C2212yes—
42#1444C2212yes—
43C4⋊#95—22no—
44#1294C2214yes—
45#1280C2214yes—
46C23×Q8D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—21no—
51C22×M16—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8C2×C414yes—
57C2×C22⋊Q8D412yes—
58C2×C22⋊Q8Q814yes—
59C2×C22⋊Q8C2312yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42—21no—
77C2×#107—22no—
78C2×#106—22no—
79C2×#103—22no—
80C2×#103—22no—
81#407—22no—
82#407—22no—
83#407—22no—
84#407C2×C414yes—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8C2312yes—
88C2×C22⋊C8C2312yes—
89#391—24no—
90#391—24no—
91#389—24no—
92#389—24no—
93#387—24no—
94#387—24no—
95#387—24no—
96#387—24no—
97#385—24no—
98#385—24no—
99#385—44no—
100#385—44no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—21no—
105#380—21no—
106#380—21no—
107#380D412yes—
108#380C2312yes—
109C4×C22⋊C4—24no—
110C4×C22⋊C4—24no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4C2×C414yes—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95D412yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128#340—22no—
129#340—22no—
130#340—22no—
131#340—22no—
132#328C2×C4116yescomplemented
133C25C2×D411yescharacteristic
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—81no—
137C22×Q8—81no—
138C22×Q8C4⋊C414yescomplemented
139C22×Q8C2×D412yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—81no—
156C23×C4—81no—
157C23×C4—81no—
158C23×C4—81no—
159C23×C4C22×C412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×Q812yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4C2411yescharacteristic
171C2×M16—24no—
172C2×M16—42no—
173C2×M16—81no—
174C2×M16—81no—
175C2×M16—81no—
176C2×M16—81no—
177C22×C8—22no—
178C22×C8—22no—
179C22⋊Q8—24no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8—22no—
185C22⋊Q8—22no—
186C22⋊Q8C4⋊C418yescomplemented
187C22⋊Q8C22×C414yes—
188C22⋊Q8C2×Q814yes—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—81no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4—81no—
210C2×C4⋊C4—81no—
211C2×C4⋊C4—82no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C4⋊C414yescomplemented
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—81no—
237C2×C42—81no—
238C2×C42—82no—
239C2×C42—82no—
240#107—82no—
241#106—82no—
242#103—82no—
243#103—82no—
244C22⋊C8—24no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—42no—
249C22⋊C8C22×C414yes—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—44no—
262#95—44no—
263#95—41no—
264#95—41no—
265#95C22×C414yes—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—41no—
271C24—41no—
272C24C4×D412yes—
273C24#12011yescharacteristic
274C24#12011yescharacteristic
275C24C22⋊Q812yes—
276C24C22×D411yescharacteristic
277C2×Q8—22no—
278C2×Q8—24no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—81no—
282C2×Q8—81no—
283C2×Q8—84no—
284C2×Q8—84no—
285C2×Q8C2×C4⋊C414yes—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—88no—
353C22×C4—81no—
354C22×C4—81no—
355C22×C4—81no—
356C22×C4—81no—
357C22×C4—88no—
358C22×C4C2×C4⋊C412yes—
359C22×C4C2×C4⋊C412yes—
360C22×C4C4×D412yes—
361C22×C4C4×D412yes—
362C22×C4C4×D412yes—
363C22×C4#12011yescharacteristic
364C22×C4#12011yescharacteristic
365C22×C4C22⋊Q812yes—
366C22×C4C22⋊Q812yes—
367C22×C4C22⋊Q812yes—
368C22×C4#12312yes—
369C22×C4#12312yes—
370C22×C4#13714yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×Q811yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377M16—84no—
378M16—84no—
379C2×C8—24no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—28no—
389C4⋊C4—24no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—81no—
397C4⋊C4—81no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4—88no—
401C4⋊C4—88no—
402C4⋊C4C2×C4⋊C414yes—
403C22⋊C4—216no—
404C22⋊C4—24no—
405C22⋊C4—24no—
406C22⋊C4—28no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C42—84no—
412C42—84no—
413C42—84no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23#38512yes—
439C23#38512yes—
440C23#45312yes—
441C23#45612yes—
442C23C2×C4×D411yescharacteristic
443C23C2×#12011yescharacteristic
444C23C2×C22⋊Q811yescharacteristic
445C23C2×#13712yes—
446C23#57512yes—
447Q8—44no—
448Q8—82no—
449Q8—84no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—48no—
482C2×C4—48no—
483C2×C4—48no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—82no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4#38512yes—
520C2×C4#38512yes—
521C2×C4C22×C4⋊C411yescharacteristic
522C2×C4C2×C4×D411yescharacteristic
523C2×C4C2×C22⋊Q811yescharacteristic
524C2×C4C2×#12311yescharacteristic
525C8—44no—
526C8—82no—
527C22—21no—
528C22—21no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—21no—
539C22—22no—
540C22—21no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—48no—
545C22—48no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22#185914yes—
551C22C2×#38511yescharacteristic
552C22C4×#13712yes—
553C22#300411yescharacteristic
554C22#300511yescharacteristic
555C22C2×#45311yescharacteristic
556C22C2×#45611yescharacteristic
557C22#302611yescharacteristic
558C22#302611yescharacteristic
559C22#307812yes—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—22no—
565C4—22no—
566C4—48no—
567C4—44no—
568C4—44no—
569C4—44no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—161no—
574C4—161no—
575C2—22no—
576C2—22no—
577C2—24no—
578C2—42no—
579C2#1831811yescharacteristic
580C2#1832111yescharacteristic
581C2#1832711yescharacteristic
582C2#1832811yescharacteristic
583C2C2×#185911yescharacteristic
584C2#2087411yescharacteristic
585C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8871C2semidirect
7. #8868C2semidirect
39. #1508C4semidirect
132. #328C2×C4semidirect
138. C22×Q8C4⋊C4semidirect
186. C22⋊Q8C4⋊C4semidirect
215. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23292
#219422
#217652
C2×#15082
#139652
#88712
#88682
#80192
#79092