Finite Group Databasebeta

← #381868

Subgroups — #381868

Order 512.

1500 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2001C211yesmaximal, characteristic
2C2×#1971C211yesmaximal, characteristic, complemented
3C2×#1749C211yesmaximal, characteristic
4#14702C212yesmaximal
5#11257C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11095C211yesmaximal, characteristic
8#11024C211yesmaximal, characteristic, complemented
9#10970C212yesmaximal
10#10928C212yesmaximal, complemented
11#10821C211yesmaximal, characteristic, complemented
12C23×Q16C2211yescharacteristic
13C22×C22⋊Q8C2212yes—
14C23×C42C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C2×#391C2212yes—
24C2×#391C2212yes—
25C2×#380C2212yes—
26#2001C418yescomplemented
27#1971—24no—
28#1749—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1436—22no—
37#1436—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C22⋊#95—22no—
41C22⋊#95—22no—
42C23×Q8C2312yes—
43C24×C4D412yes—
44C24×C4C2311yescharacteristic
45C22×Q16—22no—
46C22×Q16—22no—
47C22×Q16—42no—
48C22×Q16—42no—
49C22×Q16C2×C414yescomplemented
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C2×C22⋊Q8—24no—
55C2×C22⋊Q8—42no—
56C2×C22⋊Q8—42no—
57C2×C22⋊Q8—42no—
58C2×C22⋊Q8—42no—
59C2×C22⋊Q8C2×C418yes—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C42—22no—
67C22×C42—22no—
68C22×C42—42no—
69C22×C42—42no—
70C22×C42D412yes—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×#106—22no—
74C2×#106—42no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103C2×C418yescomplemented
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8C2×C414yes—
90C2×#97—22no—
91C2×#97—42no—
92C2×#97—42no—
93C2×#97D414yes—
94#391—24no—
95#391—24no—
96#391—42no—
97#391—42no—
98#380—42no—
99#380—42no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109#340—22no—
110#340—42no—
111#340—42no—
112#340—42no—
113#340C2×C414yescomplemented
114#336—22no—
115#336—22no—
116C25C2×D411yescharacteristic
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8C22×C414yes—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4C22×C412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4#4112yes—
155C23×C4C2411yescharacteristic
156C2×Q16—28no—
157C2×Q16—48no—
158C2×Q16—44no—
159C2×Q16—44no—
160C2×Q16—48no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C22×C412yes—
176C22×C8C22×C412yes—
177C22×C8C2×D412yes—
178C22×C8C2×D412yes—
179C22×C8#4112yes—
180C22×C8#4112yes—
181C22⋊Q8—48no—
182C22⋊Q8—48no—
183C22⋊Q8—48no—
184C22⋊Q8—48no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
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—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4C22×C414yes—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—24no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—44no—
220C2×C42—44no—
221C2×C42—48no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42C2×D412yes—
225C2×C42#4112yes—
226#106—44no—
227#103—48no—
228#103—48no—
229#103—48no—
230C22⋊C8—28no—
231C22⋊C8—28no—
232#97—44no—
233#97—44no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243C24—21no—
244C24—21no—
245C24—22no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24C4×D412yes—
251C24#13714yes—
252C24C22×D411yescharacteristic
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—48no—
308C22×C4—48no—
309C22×C4—48no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—48no—
313C22×C4—48no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
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—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×C4C4×D412yes—
351C22×C4#12114yes—
352C22×C4#13714yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4H(5,𝔽2)12yes—
360C22×C4H(5,𝔽2)12yes—
361C22×C4H(5,𝔽2)12yes—
362C22×C4H(5,𝔽2)12yes—
363Q16—416no—
364C2×C8—24no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—24no—
368C2×C8—22no—
369C2×C8—24no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—24no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8C4×D414yes—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—416no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C22⋊C4—416no—
398C22⋊C4—88no—
399C42—22no—
400C42—22no—
401C42—24no—
402C42—48no—
403C42—41no—
404C42—41no—
405C42—42no—
406C23—24no—
407C23—24no—
408C23—24no—
409C23—24no—
410C23—22no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—22no—
416C23—41no—
417C23—41no—
418C23—42no—
419C23—41no—
420C23—41no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23#44114yes—
427C23C2×C4×D411yescharacteristic
428C23C2×#12111yescharacteristic
429C23#53211yescharacteristic
430C23#53412yes—
431C23#53512yes—
432C23C2×#13712yes—
433C23C2×#13712yes—
434C23#57512yes—
435C23#57512yes—
436Q8—44no—
437Q8—44no—
438Q8—44no—
439Q8—44no—
440Q8—82no—
441Q8—82no—
442Q8—82no—
443Q8—82no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—432no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
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×C4C2×C4×D411yescharacteristic
499C2×C4C4×#4111yescharacteristic
500C2×C4#51812yes—
501C2×C4#51812yes—
502C2×C4#52511yescharacteristic
503C2×C4#52911yescharacteristic
504C2×C4#53411yescharacteristic
505C2×C4#53511yescharacteristic
506C2×C4#53511yescharacteristic
507C2×C4#53511yescharacteristic
508C2×C4#53812yes—
509C2×C4#53812yes—
510C2×C4#53812yes—
511C2×C4#53812yes—
512C2×C4#53812yes—
513C2×C4#54012yes—
514C2×C4#54012yes—
515C2×C4#54012yes—
516C8—24no—
517C8—42no—
518C8—42no—
519C8—42no—
520C22—24no—
521C22—28no—
522C22—44no—
523C22—42no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—42no—
529C22C4×#12111yescharacteristic
530C22#232112yes—
531C22#261711yescharacteristic
532C22#261911yescharacteristic
533C22#266112yes—
534C22C2×#44111yescharacteristic
535C22#294311yescharacteristic
536C22C4×#13712yes—
537C22#305612yes—
538C22#305612yes—
539C22#318112yes—
540C22#318311yescharacteristic
541C22#318511yescharacteristic
542C22#319512yes—
543C22#319612yes—
544C22#319612yes—
545C22#319712yes—
546C22#321512yes—
547C22#321612yes—
548C22#321812yes—
549C22#321912yes—
550C4—44no—
551C4—44no—
552C4—42no—
553C4—84no—
554C4—84no—
555C4—162no—
556C4—162no—
557C4#232111yescharacteristic
558C4#232811yescharacteristic
559C4#233312yes—
560C4#263111yescharacteristic
561C4#263511yescharacteristic
562C4#264812yes—
563C2—44no—
564C2#1431111yescharacteristic
565C2#2152711yescharacteristic
566C2#2153911yescharacteristic
567C2#2204812yes—
568C2#2205012yes—
569C2#2974312yes—
570C2#2981512yes—
571C2#3032312yes—
572C2#3032612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1971C2semidirect
5. #11257C2semidirect
6. #11190C2semidirect
8. #11024C2semidirect
10. #10928C2semidirect
11. #10821C2semidirect
26. #2001C4semidirect
49. C22×Q16C2×C4semidirect
83. C2×#103C2×C4semidirect
113. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20012
C2×#19712
C2×#17492
#147022
#112572
#111902
#110952
#110242
#109702
#109282
#108212