Finite Group Databasebeta

← #374552

Subgroups — #374552

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic, complemented
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1750C211yesmaximal, characteristic
6#11208C212yesmaximal, complemented
7#11207C212yesmaximal, complemented
8#10843C211yesmaximal, characteristic, complemented
9#10842C211yesmaximal, characteristic, complemented
10#10731C212yesmaximal
11#10730C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×C22⋊C8C2212yes—
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2628—24no—
20#2628C418yescomplemented
21C2×#400—21no—
22C2×#399—21no—
23C2×#395—21no—
24C2×#393—21no—
25#2317—24no—
26#2317—24no—
27C2×#389—21no—
28C2×#389—21no—
29C2×#389C2212yes—
30C2×#385—21no—
31C2×#385—21no—
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×#380C2212yes—
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×C4—21no—
52C24×C4C2311yescharacteristic
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2×C414yescomplemented
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—24no—
68C2×#117—42no—
69C2×#117C2×C414yes—
70C2×#117D414yes—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—24no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4—21no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4D411yescharacteristic
93C22×C22⋊C4D411yescharacteristic
94C22×C42—21no—
95C22×C42C2311yescharacteristic
96C22×C42C2311yescharacteristic
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8Q818yescomplemented
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103C2×C4×C8—22no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106#400—28no—
107#399—28no—
108#395—28no—
109#393—28no—
110#389—28no—
111#389—28no—
112#389—24no—
113#389C2×C418yescomplemented
114#385—28no—
115#385—28no—
116#384—22no—
117#384—22no—
118#384—22no—
119#384D414yes—
120#382—22no—
121#382—22no—
122#382—22no—
123#382—22no—
124#382C2×C414yescomplemented
125#382D414yes—
126#380—24no—
127#380C2×C418yes—
128C2×#95—21no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136#340—24no—
137#340—24no—
138#340—24no—
139#340—24no—
140#340—42no—
141#340—42no—
142#340—42no—
143#340—42no—
144#336—24no—
145C25C2×D411yescharacteristic
146C22×Q8—24no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—82no—
150C22×Q8—82no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—24no—
164C23×C4—42no—
165C23×C4—42no—
166C23×C4—82no—
167C23×C4—82no—
168C23×C4C22×C412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D412yes—
175C23×C4C2×Q814yes—
176C23×C4#4114yes—
177C23×C4C2411yescharacteristic
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8C2×Q814yes—
191C22×C8C2×Q814yes—
192C4⋊Q8—28no—
193C4⋊Q8—28no—
194C4⋊Q8—44no—
195C4⋊Q8—44no—
196#117—28no—
197#117—28no—
198#117—28no—
199#117—28no—
200#117—48no—
201#117—48no—
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—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4—48no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4—82no—
243C2×C4⋊C4—82no—
244C2×C4⋊C4—82no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4—82no—
247C2×C4⋊C4C22⋊C414yes—
248C2×C4⋊C4C22⋊C414yes—
249C2×C4⋊C4C22×C414yes—
250C2×C4⋊C4C22×C414yes—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2×D412yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—82no—
264C2×C22⋊C4—82no—
265C2×C22⋊C4C22⋊C414yes—
266C2×C22⋊C4C22⋊C414yes—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—24no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42C22×C412yes—
275C2×C42C22×C412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42#4112yes—
281C2×C42#4112yes—
282C22⋊C8—216no—
283#97—44no—
284C4×C8—44no—
285#95—24no—
286#95—24no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—28no—
291#95—24no—
292#95—24no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—21no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24C2×C22⋊C412yes—
302C24#12412yes—
303C24C4⋊Q812yes—
304C24C22×D411yescharacteristic
305C2×Q8—48no—
306C2×Q8—88no—
307C2×Q8—88no—
308C2×Q8—84no—
309C2×Q8—84no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—48no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4—88no—
378C22×C4—88no—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C412yes—
382C22×C4C2×C22⋊C412yes—
383C22×C4C2×C22⋊C412yes—
384C22×C4C4×Q818yes—
385C22×C4#12012yes—
386C22×C4#12012yes—
387C22×C4C22⋊Q818yes—
388C22×C4#12312yes—
389C22×C4#12312yes—
390C22×C4#12412yes—
391C22×C4C4⋊Q812yes—
392C22×C4C23×C411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×Q812yes—
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4112yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C42−1+412yes—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C4⋊C4—28no—
417C4⋊C4—28no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—48no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—48no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—88no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C22⋊C4—28no—
442C22⋊C4—28no—
443C22⋊C4—48no—
444C22⋊C4—88no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—48no—
450C42C4×D414yes—
451C42C4×D414yes—
452C42#12114yes—
453C42#12114yes—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23#39114yes—
477C23C22×C22⋊C411yescharacteristic
478C23C2×#12011yescharacteristic
479C23C2×#12311yescharacteristic
480C23C2×#12411yescharacteristic
481C23C2×C4⋊Q811yescharacteristic
482C23#53511yescharacteristic
483C23#53611yescharacteristic
484C23#57612yes—
485C23#57712yes—
486C23#57712yes—
487C23#57812yes—
488Q8—816no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—816no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4#38614yes—
553C2×C4#38614yes—
554C2×C4#39114yes—
555C2×C4C2×C4×D411yescharacteristic
556C2×C4C2×C4×D411yescharacteristic
557C2×C4C2×C4×Q812yes—
558C2×C4#52012yes—
559C2×C4C2×#12111yescharacteristic
560C2×C4C2×#12111yescharacteristic
561C2×C4C2×C22⋊Q812yes—
562C2×C4#53712yes—
563C2×C4#54612yes—
564C2×C4#54612yes—
565C2×C4D4×Q812yes—
566C2×C4D4×Q812yes—
567C2×C4#55212yes—
568C2×C4#55212yes—
569C2×C4#55412yes—
570C2×C4#55412yes—
571C8—44no—
572C8—44no—
573C22—24no—
574C22—28no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22C2×#38611yescharacteristic
585C22C2×#39111yescharacteristic
586C22#231111yescharacteristic
587C22#243411yescharacteristic
588C22#243511yescharacteristic
589C22#250211yescharacteristic
590C22#250311yescharacteristic
591C22#289712yes—
592C22#289812yes—
593C22#300811yescharacteristic
594C22#300911yescharacteristic
595C22#301011yescharacteristic
596C22#301111yescharacteristic
597C22#309612yes—
598C22#309712yes—
599C22#313612yes—
600C22#313712yes—
601C22#316312yes—
602C22#316412yes—
603C22#322412yes—
604C22#322512yes—
605C22#324312yes—
606C22#324412yes—
607C4—24no—
608C4—24no—
609C4—24no—
610C4—42no—
611C4—42no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4—84no—
617C4—84no—
618C4#236012yes—
619C4#236012yes—
620C4#241712yes—
621C4#241712yes—
622C2—44no—
623C2#1413911yescharacteristic
624C2#2091311yescharacteristic
625C2#2091411yescharacteristic
626C2#2132612yes—
627C2#2187112yes—
628C2#2582011yescharacteristic
629C2#2582111yescharacteristic
630C2#2586511yescharacteristic
631C2#2586611yescharacteristic
632C2#2795912yes—
633C2#2797112yes—
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
6. #11208C2semidirect
7. #11207C2semidirect
8. #10843C2semidirect
9. #10842C2semidirect
20. #2628C4semidirect
60. C2×C4⋊Q8C2×C4semidirect
99. C2×C22⋊C8Q8semidirect
113. #389C2×C4semidirect
124. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#23172
C2×#19932
C2×#19902
C2×#17502
#112082
#112072
#108432
#108422
#107312
#107302