Finite Group Databasebeta

← #196748

Subgroups — #196748

Order 512.

1297 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2989C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic
3#20721C211yesmaximal, characteristic
4C2×#1556C211yesmaximal, characteristic, complemented
5C2×#1549C211yesmaximal, characteristic, complemented
6#9138C212yesmaximal, complemented
7#9137C212yesmaximal, complemented
8#9130C212yesmaximal, complemented
9#9129C212yesmaximal, complemented
10#8046C211yesmaximal, characteristic, complemented
11#8042C211yesmaximal, characteristic, complemented
12C2×#536—21no—
13C22×C4⋊Q8C2211yescharacteristic
14C2×C4×#41C2211yescharacteristic
15#2989—24no—
16#2989—24no—
17C2×#446—21no—
18C2×#436—21no—
19C2×#435—21no—
20C2×#435—21no—
21C2×#432C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C22×C4⋊C8C2211yescharacteristic
29#2857C2212yes—
30#2857C2212yes—
31#2621—21no—
32#2621—21no—
33#2547—21no—
34#2547C412yes—
35#2546—21no—
36#2546C412yes—
37#2486—21no—
38#2486—21no—
39#2486C412yes—
40#2486C412yes—
41#2310—21no—
42#2285C2211yescharacteristic
43#1818—21no—
44#1797C2212yes—
45#1797C2212yes—
46#1780—21no—
47#1750—21no—
48#1750—21no—
49#1556—24no—
50#1549—24no—
51C2×#332C2212yes—
52C2×#332C2212yes—
53C2×#332C2212yes—
54C2×#332C2212yes—
55#1296C2212yes—
56#1296C2212yes—
57#1296C2212yes—
58#1296C2212yes—
59C22×#41D411yescharacteristic
60C23×Q8—21no—
61C22×M16—21no—
62C22×M16—21no—
63C23×C8—21no—
64#536—48no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8C2×C412yes—
73C2×C4⋊Q8C2312yes—
74C2×#124—22no—
75C2×#123—22no—
76C2×#123—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—22no—
79C4×#41—22no—
80C4×#41—22no—
81C4×#41—22no—
82C4×#41—22no—
83C2×C4×Q8C2×C411yescharacteristic
84C2×C4×Q8D411yescharacteristic
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4D411yescharacteristic
89C2×C4×D4D411yescharacteristic
90C2×C4×D4D411yescharacteristic
91C2×#117—21no—
92C2×#117D411yescharacteristic
93C2×#117D411yescharacteristic
94C2×#117D411yescharacteristic
95C22×C4⋊C4—21no—
96C22×C4⋊C4C2×C412yes—
97C22×C42C2311yescharacteristic
98C22×C42C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#446—28no—
101#436—28no—
102#435—28no—
103#435—28no—
104#432—22no—
105#432—22no—
106#432—22no—
107#432—22no—
108#432—22no—
109#432D414yes—
110#431—22no—
111#431—22no—
112#431—22no—
113#431—22no—
114#423—22no—
115#423—22no—
116#423—22no—
117#423D414yes—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—22no—
122C2×C4⋊C8—22no—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×C22⋊C8—21no—
130C2×C22⋊C8—21no—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—22no—
133C2×C22⋊C8—22no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×#97—21no—
137C2×#97—21no—
138C2×#97—21no—
139C2×C4×C8—21no—
140C2×C4×C8—21no—
141#398—22no—
142#398—22no—
143#398—22no—
144#398—22no—
145#397—22no—
146#397—22no—
147#397—22no—
148#397—22no—
149#391—22no—
150#391—22no—
151#391—22no—
152#391—22no—
153#389—22no—
154#389—22no—
155#389C2×C414yes—
156#389C2×C414yes—
157#384C2312yes—
158#382—21no—
159#381—22no—
160#381—22no—
161#380—22no—
162#380—22no—
163#380—21no—
164#380C2×C414yes—
165#380C2×C414yes—
166#380C2×C412yes—
167C4×C4⋊C4—21no—
168C4×C4⋊C4C2×C412yes—
169#336—22no—
170#336—22no—
171#332—24no—
172#332—24no—
173#332—24no—
174#332—24no—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—24no—
178C2×#41—42no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8C22⋊C411yescharacteristic
186C22×D4C22⋊C411yescharacteristic
187C22×D4C22⋊C411yescharacteristic
188C22×D4C22⋊C411yescharacteristic
189C23×C4—22no—
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C2×M16—24no—
200C2×M16—24no—
201C2×M16—82no—
202C2×M16—82no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—24no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—82no—
211C4⋊Q8—24no—
212C4⋊Q8—24no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—48no—
218C4⋊Q8—42no—
219C4⋊Q8—42no—
220C4⋊Q8—42no—
221C4⋊Q8—42no—
222#124—48no—
223#123—48no—
224#123—48no—
225C22⋊Q8—48no—
226C22⋊Q8—48no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—22no—
243#117—22no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—22no—
248#117—22no—
249#117—44no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22×C414yes—
268C2×C4⋊C4C22×C414yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42#4112yes—
299C2×C42#4112yes—
300C2×C42C2411yescharacteristic
301C4⋊C8—24no—
302C4⋊C8—24no—
303C4⋊C8—24no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8C2×D412yes—
313C4⋊C8C2×D412yes—
314C4⋊C8C2×D412yes—
315C4⋊C8C2×D412yes—
316C22⋊C8—24no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319C22⋊C8—44no—
320C22⋊C8—42no—
321C22⋊C8—42no—
322C22⋊C8—44no—
323#97—24no—
324#97—24no—
325#97—42no—
326C4×C8—24no—
327C4×C8—42no—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334C24C2×C22⋊C411yescharacteristic
335C24C2×C22⋊C411yescharacteristic
336C24C2×C22⋊C411yescharacteristic
337#41—48no—
338#41—48no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—41no—
342C2×Q8—44no—
343C2×Q8—44no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8—88no—
350C2×Q8—88no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C4×D412yes—
401C22×C4C4×D412yes—
402C22×C4#12012yes—
403C22×C4#12112yes—
404C22×C4#12112yes—
405C22×C4#12312yes—
406C22×C4AGL(1,ℤ/8)14yes—
407C22×C4C23×C411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410M16—84no—
411M16—84no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—42no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—84no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—44no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—44no—
467C42C4×D412yes—
468C42C4×D412yes—
469C42#12012yes—
470C42#12112yes—
471C42#12112yes—
472C42#12312yes—
473C42C22×D411yescharacteristic
474C42C22×D411yescharacteristic
475C42C2×#4111yescharacteristic
476C42C2×#4111yescharacteristic
477C23—22no—
478C23—22no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23—21no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—22no—
489C23—21no—
490C23#38612yes—
491C23#41812yes—
492C23C22×C22⋊C411yescharacteristic
493Q8—44no—
494Q8—88no—
495Q8—84no—
496Q8—84no—
497D4—44no—
498D4—44no—
499D4—44no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—88no—
546C2×C4#38612yes—
547C2×C4#38612yes—
548C2×C4#38612yes—
549C2×C4#41812yes—
550C2×C4C2×C4×D411yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C2×#12011yescharacteristic
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×#12111yescharacteristic
555C2×C4C2×#12311yescharacteristic
556C2×C4C2×AGL(1,ℤ/8)12yes—
557C2×C4#57512yes—
558C2×C4#57712yes—
559C2×C4#57812yes—
560C8—42no—
561C8—42no—
562C8—42no—
563C8—42no—
564C8—82no—
565C8—82no—
566C22—24no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#38611yescharacteristic
582C22#288912yes—
583C22C2×#41811yescharacteristic
584C22#289611yescharacteristic
585C22#289812yes—
586C4—22no—
587C4—22no—
588C4—22no—
589C4—21no—
590C4—21no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—84no—
598C4—84no—
599C4#300911yescharacteristic
600C4#301111yescharacteristic
601C4#305512yes—
602C4#305512yes—
603C4#309712yes—
604C2—42no—
605C2—42no—
606C2—81no—
607C2#1775912yes—
608C2#1776712yes—
609C2#2091311yescharacteristic
610C2#2097311yescharacteristic
611C2#2099111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1556C2semidirect
5. C2×#1549C2semidirect
6. #9138C2semidirect
7. #9137C2semidirect
8. #9130C2semidirect
9. #9129C2semidirect
10. #8046C2semidirect
11. #8042C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29892
#353162
#207212
C2×#15562
C2×#15492
#91382
#91372
#91302
#91292
#80462
#80422