Finite Group Databasebeta

← #454437

Subgroups — #454437

Order 512.

1916 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×#2857C211yesmaximal, characteristic
2C2×#2857C211yesmaximal, characteristic
3C2×#2043C212yesmaximal, complemented
4C2×#2024C212yesmaximal, complemented
5#11207C212yesmaximal, complemented
6#11050C212yesmaximal, complemented
7#10970C214yesmaximal
8#10712C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8—21no—
12C22×C22⋊Q8C2212yes—
13C23×C42C2211yescharacteristic
14C22×#103C2214yes—
15C22×#103C2214yes—
16C22×C22⋊C8C2212yes—
17#2857—28no—
18#2857—28no—
19C2×#391—21no—
20C2×#391C2212yes—
21C2×#389—21no—
22C2×#389—21no—
23C2×#389—21no—
24C2×#389C2212yes—
25C2×#384C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#380—21no—
28C2×#380—21no—
29C2×#380—21no—
30C2×#380C2212yes—
31#2043—28no—
32#2024—28no—
33C2×#340C2214yes—
34C2×#340C2214yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C23×Q8—21no—
38C23×Q8C2312yes—
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8C2314yes—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—42no—
48C2×C4⋊Q8—42no—
49C2×C22⋊Q8—28no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—24no—
52C2×C22⋊Q8—28no—
53C2×C22⋊Q8—44no—
54C2×C22⋊Q8—44no—
55C2×C22⋊Q8—44no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C22⋊C4—22no—
69C22×C22⋊C4—22no—
70C22×C42—24no—
71C22×C42—24no—
72C22×C42—42no—
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#103—28no—
78C2×#103—28no—
79C2×#103—28no—
80C2×#103—44no—
81C2×#103—44no—
82C2×#103—44no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—42no—
85C2×C22⋊C8—42no—
86#391—28no—
87#391—24no—
88#391D418yes—
89#389—28no—
90#389—28no—
91#389—28no—
92#389—24no—
93#389D418yes—
94#384—24no—
95#384—24no—
96#380—28no—
97#380—28no—
98#380—28no—
99#380—24no—
100#380—24no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105#340—28no—
106#340—44no—
107#336—24no—
108#336—24no—
109C25C2×D411yescharacteristic
110C22×Q8—24no—
111C22×Q8—22no—
112C22×Q8—24no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8C2×D414yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4—41no—
149C23×C4—41no—
150C23×C4SD1614yes—
151C23×C4Q1614yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4#4114yes—
158C23×C4C2411yescharacteristic
159C22×C8—24no—
160C22×C8—24no—
161C22×C8—24no—
162C22×C8—24no—
163C22×C8—24no—
164C22×C8—24no—
165C22×C8—24no—
166C4⋊Q8—416no—
167C4⋊Q8—416no—
168C22⋊Q8—416no—
169C22⋊Q8—416no—
170C22⋊Q8—432no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C2×D414yes—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D414yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—416no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42C2×D414yes—
230C2×C42C2×D414yes—
231#103—416no—
232#103—416no—
233C22⋊C8—48no—
234#95—28no—
235#95—28no—
236#95—28no—
237#95—28no—
238C24—21no—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24#12412yes—
251C24C2×SD1612yes—
252C24C2×Q1612yes—
253C24C22×D411yescharacteristic
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—28no—
257C2×Q8—28no—
258C2×Q8—48no—
259C2×Q8—48no—
260C2×Q8—48no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8—44no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×Q8—84no—
274C2×Q8—84no—
275C2×Q8—84no—
276C2×Q8—84no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—28no—
304C22×C4—28no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
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×C4—42no—
351C22×C4#12012yes—
352C22×C4#12012yes—
353C22×C4#12012yes—
354C22×C4#12012yes—
355C22×C4#12412yes—
356C22×C4C2×SD1612yes—
357C22×C4C2×SD1612yes—
358C22×C4C2×Q1612yes—
359C22×C4C2×Q1612yes—
360C22×C4AGL(1,ℤ/8)12yes—
361C22×C4#13712yes—
362C22×C4#13712yes—
363C22×C4#13712yes—
364C22×C4C22×D412yes—
365C22×C4C22×D412yes—
366C22×C4C2×#4112yes—
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C4H(5,𝔽2)12yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C42−1+411yescharacteristic
371C22×C42−1+412yes—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—44no—
376C2×C8—44no—
377C2×C8—44no—
378C2×C8—44no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—28no—
382C4⋊C4—28no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—416no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—416no—
396C22⋊C4—416no—
397C22⋊C4—416no—
398C42—28no—
399C42—28no—
400C42—44no—
401C42—44no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23#48714yes—
437C23C2×#12011yescharacteristic
438C23C2×#12011yescharacteristic
439C23C2×#12411yescharacteristic
440C23#53411yescharacteristic
441C23#53511yescharacteristic
442C23#53611yescharacteristic
443C23#53711yescharacteristic
444C23C22×SD1611yescharacteristic
445C23C22×Q1611yescharacteristic
446C23C2×AGL(1,ℤ/8)11yescharacteristic
447C23C2×#13711yescharacteristic
448C23C2×#13711yescharacteristic
449C23C2×#13711yescharacteristic
450C23#57511yescharacteristic
451C23#57511yescharacteristic
452Q8—48no—
453Q8—48no—
454Q8—816no—
455Q8—84no—
456Q8—84no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—416no—
485C2×C4—416no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4#44814yes—
521C2×C4#45014yes—
522C2×C4#45114yes—
523C2×C4#45114yes—
524C2×C4#53512yes—
525C2×C4#53612yes—
526C2×C4#53712yes—
527C2×C4#53712yes—
528C2×C4#53812yes—
529C2×C4#53912yes—
530C2×C4D4214yes—
531C2×C4#54614yes—
532C2×C4#54814yes—
533C2×C4#56214yes—
534C8—84no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22#240311yescharacteristic
551C22#243511yescharacteristic
552C22#243611yescharacteristic
553C22#261511yescharacteristic
554C22#261611yescharacteristic
555C22#263911yescharacteristic
556C22#281611yescharacteristic
557C22C2×#45011yescharacteristic
558C22C2×#44811yescharacteristic
559C22C2×#45111yescharacteristic
560C22C2×#45111yescharacteristic
561C22#300611yescharacteristic
562C22#300711yescharacteristic
563C22#300711yescharacteristic
564C22#300711yescharacteristic
565C22C2×#48711yescharacteristic
566C22#313511yescharacteristic
567C22#313911yescharacteristic
568C22#314111yescharacteristic
569C22#318711yescharacteristic
570C22#318911yescharacteristic
571C22#319311yescharacteristic
572C22#319511yescharacteristic
573C22#320611yescharacteristic
574C22#320911yescharacteristic
575C22#321411yescharacteristic
576C22#321611yescharacteristic
577C22#322911yescharacteristic
578C22#323011yescharacteristic
579C22#323711yescharacteristic
580C22#323711yescharacteristic
581C22#324611yescharacteristic
582C22#324911yescharacteristic
583C22#325411yescharacteristic
584C22#325511yescharacteristic
585C4—24no—
586C4—24no—
587C4—24no—
588C4—24no—
589C4—42no—
590C4—84no—
591C4—84no—
592C4—84no—
593C4—84no—
594C4#267614yes—
595C4#268114yes—
596C2—44no—
597C2#1625311yescharacteristic
598C2#2391911yescharacteristic
599C2#2392111yescharacteristic
600C2#2585611yescharacteristic
601C2#2586311yescharacteristic
602C2#2590211yescharacteristic
603C2#2591011yescharacteristic
604C2#2966311yescharacteristic
605C2#2966411yescharacteristic
606C2#2968811yescharacteristic
607C2#2969111yescharacteristic
608C2#3018711yescharacteristic
609C2#3019911yescharacteristic
610C2#3277111yescharacteristic
611C2#3277411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
3. C2×#2043C2semidirect
4. C2×#2024C2semidirect
5. #11207C2semidirect
6. #11050C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#20432
C2×#20242
#112072
#110502
#109702
#107122