Finite Group Databasebeta

← #376122

Subgroups — #376122

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#441C211yesmaximal, characteristic, complemented
2C2×#2001C211yesmaximal, characteristic
3C2×#1894C212yesmaximal
4C2×#1753C211yesmaximal, characteristic
5#13969C212yesmaximal
6#11257C211yesmaximal, characteristic, complemented
7#11095C211yesmaximal, characteristic
8#11024C211yesmaximal, characteristic, complemented
9#10926C212yesmaximal, complemented
10#10821C211yesmaximal, characteristic, complemented
11#10789C212yesmaximal, complemented
12C23×Q16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×C22⋊Q8C2212yes—
15C22×C4×Q8C2212yes—
16C2×#441—28no—
17C2×#441—24no—
18C22×#106C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×#97C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#380C2212yes—
27C2×C4×C22⋊C4C2211yescharacteristic
28#2001C418yescomplemented
29#1894—28no—
30#1753—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1441—22no—
39#1439—22no—
40#1437—22no—
41#1436—22no—
42C4⋊#95C2212yes—
43C4⋊#95C2212yes—
44C23×Q8C2312yes—
45C24×C4C2311yescharacteristic
46C22×Q16—24no—
47C22×Q16—24no—
48C22×Q16—22no—
49C22×Q16—22no—
50C22×Q16C2×C414yes—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—24no—
54C22×M16—22no—
55C22×M16—42no—
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—24no—
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—42no—
64C2×C22⋊Q8—42no—
65C2×C22⋊Q8C2×C418yes—
66C2×C4×Q8—24no—
67C2×C4×Q8—24no—
68C2×C4×Q8—24no—
69C2×C4×Q8—24no—
70C2×C4×Q8—42no—
71C2×C4×Q8—42no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D412yes—
80C22×C42—22no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#441—232no—
84C2×#106—22no—
85C2×#106—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103C2×C418yes—
94C2×#103D418yes—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8C2×C414yes—
98C2×#97—24no—
99C2×#97D414yes—
100C2×C4×C8—22no—
101C2×C4×C8—42no—
102#391—24no—
103#391—42no—
104#380—24no—
105#380—42no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D412yes—
116#340—24no—
117#340—22no—
118#340—24no—
119#340—42no—
120#340C2×C414yes—
121#336—22no—
122#336—22no—
123C25C2×D411yescharacteristic
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8C22×C414yes—
136C22×Q8C2×D414yes—
137C22×Q8#4114yes—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4C22×C412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4#4112yes—
155C23×C4C2411yescharacteristic
156C2×Q16—28no—
157C2×Q16—216no—
158C2×Q16—28no—
159C2×Q16—216no—
160C2×Q16—28no—
161C2×M16—28no—
162C2×M16—48no—
163C2×M16—48no—
164C2×M16—44no—
165C2×M16—44no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8C22×C412yes—
180C22×C8C22×C412yes—
181C22×C8C2×D412yes—
182C22×C8C2×D414yes—
183C22×C8C2×D412yes—
184C22×C8#4112yes—
185C22×C8#4112yes—
186C22⋊Q8—216no—
187C22⋊Q8—48no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C4×Q8—216no—
191C4×Q8—48no—
192C4×Q8—48no—
193C4×Q8—48no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4C22×C414yes—
217C2×C4⋊C4C2×D414yes—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—82no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—44no—
234C2×C42—44no—
235C2×C42—44no—
236C2×C42C2×D412yes—
237C2×C42#4112yes—
238#106—28no—
239#103—216no—
240#103—48no—
241#103—48no—
242C22⋊C8—28no—
243#97—28no—
244C4×C8—44no—
245#95—24no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—82no—
252#95—82no—
253C24—21no—
254C24—21no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24C4×D412yes—
262C24C22×D411yescharacteristic
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8C4×D418yes—
291C22×C4—24no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
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—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4C4×D412yes—
354C22×C4#12114yes—
355C22×C4#13714yes—
356C22×C4C23×C411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D412yes—
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4112yes—
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4H(5,𝔽2)12yes—
365C22×C4H(5,𝔽2)12yes—
366Q16—232no—
367M16—416no—
368C2×C8—24no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—22no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—22no—
377C2×C8—24no—
378C2×C8—22no—
379C2×C8—24no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8C4×D414yes—
387C2×C8#12014yes—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—416no—
393C4⋊C4—416no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C22⋊C4—48no—
403C22⋊C4—416no—
404C22⋊C4—88no—
405C42—22no—
406C42—22no—
407C42—24no—
408C42—416no—
409C42—41no—
410C42—41no—
411C42—42no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—41no—
418C23—42no—
419C23—41no—
420C23—41no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23C2×C4×D411yescharacteristic
434C23C2×#12111yescharacteristic
435C23#53211yescharacteristic
436C23D4212yes—
437C23#54612yes—
438C23C2×#13712yes—
439C23#57512yes—
440C23#57712yes—
441C23#57812yes—
442Q8—28no—
443Q8—28no—
444Q8—44no—
445Q8—44no—
446Q8—44no—
447Q8—44no—
448Q8—44no—
449Q8—44no—
450C2×C4—28no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4C2×C4×D411yescharacteristic
504C2×C4C2×C4×D412yes—
505C2×C4C4×#4111yescharacteristic
506C2×C4#51812yes—
507C2×C4C2×#12011yescharacteristic
508C2×C4#52511yescharacteristic
509C2×C4#52511yescharacteristic
510C2×C4#52512yes—
511C2×C4#52911yescharacteristic
512C2×C4#53511yescharacteristic
513C2×C4#53511yescharacteristic
514C2×C4#54012yes—
515C2×C4#54612yes—
516C2×C4#54612yes—
517C2×C4#54612yes—
518C2×C4#55112yes—
519C2×C4#55312yes—
520C2×C4#55312yes—
521C8—24no—
522C8—24no—
523C8—42no—
524C8—42no—
525C22—42no—
526C22—42no—
527C22—44no—
528C22—44no—
529C22—42no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22C4×#12111yescharacteristic
535C22#235812yes—
536C22#240811yescharacteristic
537C22#246412yes—
538C22#253511yescharacteristic
539C22C4×#13712yes—
540C22#295011yescharacteristic
541C22#295211yescharacteristic
542C22#305612yes—
543C22#306312yes—
544C22#317812yes—
545C22#318311yescharacteristic
546C22#318511yescharacteristic
547C22#327512yes—
548C22#327612yes—
549C22#327712yes—
550C22#327812yes—
551C22#331212yes—
552C22#331312yes—
553C22#331612yes—
554C22#331712yes—
555C4—42no—
556C4—48no—
557C4—44no—
558C4—44no—
559C4—84no—
560C4—84no—
561C4—162no—
562C4C4×#12011yescharacteristic
563C4#232811yescharacteristic
564C4#236412yes—
565C4#240411yescharacteristic
566C4#245712yes—
567C4#259811yescharacteristic
568C2—82no—
569C2#1417511yescharacteristic
570C2#2153011yescharacteristic
571C2#2153911yescharacteristic
572C2#2286712yes—
573C2#2286912yes—
574C2#2427612yes—
575C2#2698812yes—
576C2#2698912yes—
577C2#2815312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#441C2semidirect
6. #11257C2semidirect
8. #11024C2semidirect
9. #10926C2semidirect
10. #10821C2semidirect
11. #10789C2semidirect
28. #2001C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4412
C2×#20012
C2×#18942
C2×#17532
#139692
#112572
#110952
#110242
#109262
#108212
#107892