Finite Group Databasebeta

← #374469

Subgroups — #374469

Order 512.

1508 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#2314C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic, complemented
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1750C211yesmaximal, characteristic
6#11200C212yesmaximal
7#11199C212yesmaximal
8#10885C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10842C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×C22⋊C8C2212yes—
15C22×#97C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2628—24no—
18#2628C418yescomplemented
19C2×#400—21no—
20C2×#399—21no—
21C2×#395—21no—
22C2×#393—21no—
23#2314—24no—
24#2314—24no—
25C2×#389C2212yes—
26C2×#388—21no—
27C2×#388—21no—
28C2×#387—21no—
29C2×#387—21no—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1993—24no—
39#1990—24no—
40#1750—24no—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#340C2212yes—
49C2×#336C2212yes—
50C23×Q8—21no—
51C24×C4C2311yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117C2×C414yes—
65C2×#117D414yes—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4D411yescharacteristic
82C22×C22⋊C4D411yescharacteristic
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90C2×#97—22no—
91C2×#97—22no—
92C2×#97—22no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8—22no—
96#400—28no—
97#399—28no—
98#395—28no—
99#393—28no—
100#389—24no—
101#389C2×C418yes—
102#388—28no—
103#388—28no—
104#387—28no—
105#387—28no—
106#384—22no—
107#384D414yes—
108#382—22no—
109#382—22no—
110#382C2×C414yes—
111#382D414yes—
112#380—24no—
113#380—22no—
114#380—22no—
115#380—22no—
116#380—22no—
117#380C2×C418yes—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—24no—
140#336—24no—
141C25C2×D411yescharacteristic
142C22×Q8—24no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—82no—
146C22×Q8—82no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C22×C412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8#4114yes—
182C22×C8#4114yes—
183C4⋊Q8—28no—
184C4⋊Q8—28no—
185C4⋊Q8—44no—
186C4⋊Q8—44no—
187#117—28no—
188#117—28no—
189#117—28no—
190#117—28no—
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—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—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C22⋊C414yes—
228C2×C4⋊C4C22⋊C414yes—
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C2×D412yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D412yes—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—82no—
243C2×C22⋊C4—82no—
244C2×C22⋊C4—82no—
245C2×C22⋊C4—82no—
246C2×C22⋊C4C22⋊C414yes—
247C2×C22⋊C4C22⋊C414yes—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—82no—
255C2×C42—82no—
256C2×C42C22×C412yes—
257C2×C42C22×C412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42#4112yes—
263C2×C42#4112yes—
264C22⋊C8—216no—
265#97—44no—
266C4×C8—44no—
267#95—24no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—28no—
272#95—24no—
273#95—24no—
274#95—28no—
275#95—24no—
276#95—24no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—21no—
282C24—21no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24C2×C22⋊C412yes—
288C24C22×D411yescharacteristic
289C2×Q8—48no—
290C2×Q8—88no—
291C2×Q8—88no—
292C2×Q8—84no—
293C2×Q8—84no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
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—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C412yes—
367C22×C4#12012yes—
368C22×C4#12012yes—
369C22×C4#12312yes—
370C22×C4#12312yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C42−1+412yes—
380C22×C42−1+412yes—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C4⋊C4—28no—
396C4⋊C4—28no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—48no—
414C4⋊C4—84no—
415C4⋊C4—84no—
416C4⋊C4—88no—
417C22⋊C4—28no—
418C22⋊C4—28no—
419C22⋊C4—88no—
420C22⋊C4—84no—
421C22⋊C4—84no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—84no—
427C42—84no—
428C42C4×D414yes—
429C42C4×D414yes—
430C42#12114yes—
431C42#12114yes—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—22no—
436C23—22no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23C22×C22⋊C411yescharacteristic
456C23C2×#12011yescharacteristic
457C23C2×#12311yescharacteristic
458C23#53211yescharacteristic
459C23#53211yescharacteristic
460C23#53611yescharacteristic
461C23#53611yescharacteristic
462C23#57612yes—
463C23#57712yes—
464C23#57712yes—
465C23#57812yes—
466Q8—816no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
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—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4#38614yes—
531C2×C4#38614yes—
532C2×C4C2×C4×D411yescharacteristic
533C2×C4C2×C4×D411yescharacteristic
534C2×C4C4×#4112yes—
535C2×C4#51912yes—
536C2×C4C2×#12111yescharacteristic
537C2×C4C2×#12111yescharacteristic
538C2×C4#52512yes—
539C2×C4#53612yes—
540C2×C4#54712yes—
541C2×C4#54712yes—
542C2×C4#54812yes—
543C2×C4#54812yes—
544C2×C4#55212yes—
545C2×C4#55212yes—
546C2×C4#55612yes—
547C2×C4#55612yes—
548C8—44no—
549C8—44no—
550C22—42no—
551C22—44no—
552C22—44no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22—82no—
560C22—82no—
561C22C2×#38611yescharacteristic
562C22#229711yescharacteristic
563C22#231011yescharacteristic
564C22#243211yescharacteristic
565C22#243211yescharacteristic
566C22#249911yescharacteristic
567C22#249911yescharacteristic
568C22#289712yes—
569C22#289812yes—
570C22#300811yescharacteristic
571C22#300911yescharacteristic
572C22#301011yescharacteristic
573C22#301111yescharacteristic
574C22#309612yes—
575C22#309712yes—
576C22#317312yes—
577C22#317512yes—
578C22#317812yes—
579C22#317912yes—
580C22#324112yes—
581C22#324212yes—
582C22#324312yes—
583C22#324412yes—
584C4—24no—
585C4—24no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4—82no—
593C4—84no—
594C4—84no—
595C4#236212yes—
596C4#236212yes—
597C4#241512yes—
598C4#241512yes—
599C2—82no—
600C2#1413811yescharacteristic
601C2#2091311yescharacteristic
602C2#2091411yescharacteristic
603C2#2141212yes—
604C2#2184312yes—
605C2#2568011yescharacteristic
606C2#2568111yescharacteristic
607C2#2568511yescharacteristic
608C2#2568611yescharacteristic
609C2#2788612yes—
610C2#2788712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1993C2semidirect
4. C2×#1990C2semidirect
8. #10885C2semidirect
9. #10884C2semidirect
10. #10843C2semidirect
11. #10842C2semidirect
18. #2628C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#23142
C2×#19932
C2×#19902
C2×#17502
#112002
#111992
#108852
#108842
#108432
#108422