Finite Group Databasebeta

← #207291

Subgroups — #207291

Order 512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35253C211yesmaximal, characteristic
2#20628C211yesmaximal, characteristic
3#13801C211yesmaximal, characteristic
4#9913C212yesmaximal, complemented
5#9911C212yesmaximal, complemented
6#9910C212yesmaximal, complemented
7#9909C212yesmaximal, complemented
8#8023C211yesmaximal, characteristic
9#8019C211yesmaximal, characteristic
10#8016C211yesmaximal, characteristic
11#8012C211yesmaximal, characteristic
12C2×#544C411yescharacteristic, complemented
13C22×#117C2211yescharacteristic
14C2×#424—21no—
15C2×#407C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17#2857C2212yes—
18#2851C2212yes—
19#2629C412yescomplemented
20#2628C412yescomplemented
21#2442—21no—
22#2441—21no—
23#2440—21no—
24#2439—21no—
25#2438—22no—
26#2437—22no—
27#2317C411yescharacteristic, complemented
28#2317C411yescharacteristic, complemented
29#2316C411yescharacteristic, complemented
30#2280C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4—21no—
34C2×C4×C22⋊C4C2211yescharacteristic
35#1798—22no—
36#1797C2212yes—
37#1797C2212yes—
38#1729—21no—
39#1437—22no—
40C4⋊#95—21no—
41#1434C2212yes—
42C4×#95C2212yes—
43#1294C2211yescharacteristic
44#1294C2211yescharacteristic
45#1294C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53C24×C4C2311yescharacteristic
54C22×M16—21no—
55C23×C8—21no—
56#544—44no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×#125—21no—
60C2×#125C2×C412yes—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×#117—44no—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C412yes—
69C2×#117D411yescharacteristic
70C2×#117D412yes—
71C2×#117D411yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4D412yes—
81C22×C42—21no—
82C22×C42—22no—
83C22×C42C2312yes—
84#424—44no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87#407—22no—
88#407—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8C2311yescharacteristic
94C2×C22⋊C8C2311yescharacteristic
95C2×C22⋊C8C2312yes—
96#400—22no—
97#399—22no—
98#398—22no—
99#398—22no—
100#396—22no—
101#396—22no—
102#395—22no—
103#393—22no—
104#391—22no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—22no—
110#389C2×C412yes—
111#389C2×C412yes—
112#389C2×C412yes—
113#389C2×C412yes—
114#387—22no—
115#387—22no—
116#385—22no—
117#385—22no—
118#385—22no—
119#385—22no—
120#384—21no—
121#384C2×C412yes—
122#382—21no—
123#382C2×C412yes—
124#381—21no—
125#380—22no—
126#380—21no—
127#380—21no—
128#380—22no—
129#380—22no—
130#380C2×C411yescharacteristic
131#380C2×C412yes—
132#380C2×C411yescharacteristic
133#380Q814yescomplemented
134#380C2311yescharacteristic
135#380C2312yes—
136#380C2311yescharacteristic
137#379—21no—
138C4×C4⋊C4—44no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4Q814yescomplemented
143C2×#95—21no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
149C2×#95C2311yescharacteristic
150#336—22no—
151#336—22no—
152C25C2×D411yescharacteristic
153C22×Q8—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×Q812yes—
169C23×C4C2×Q812yes—
170C23×C4#4112yes—
171C23×C4#4112yes—
172C23×C4C2411yescharacteristic
173C2×M16—42no—
174C2×M16—42no—
175C2×M16—42no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8—42no—
180C22×C8—42no—
181C22×C8—42no—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184#125—44no—
185#125—44no—
186C22⋊Q8—48no—
187C22⋊Q8—82no—
188C22⋊Q8—82no—
189#117—24no—
190#117—24no—
191#117—44no—
192#117—44no—
193#117—42no—
194#117—42no—
195#117—42no—
196#117—42no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
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—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4C22⋊C411yescharacteristic
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—81no—
241C2×C22⋊C4—81no—
242C2×C22⋊C4—81no—
243C2×C22⋊C4—81no—
244C2×C22⋊C4C22⋊C412yes—
245C2×C22⋊C4C22⋊C412yes—
246C2×C22⋊C4C22⋊C412yes—
247C2×C22⋊C4C22⋊C412yes—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—24no—
251C2×C42—44no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—82no—
257C2×C42C22×C412yes—
258C2×C42C22×C412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C4⋊C8—44no—
262C4⋊C8—44no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—21no—
267C22⋊C8—22no—
268C22⋊C8—22no—
269C22⋊C8—42no—
270C22⋊C8—42no—
271#95—22no—
272#95—21no—
273#95—21no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—22no—
286#95—44no—
287#95—44no—
288#95—82no—
289#95C2×Q812yes—
290#95C2×Q812yes—
291#95C2×Q814yes—
292C24—21no—
293C24—21no—
294C24—21no—
295C24—21no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—41no—
300C24C2×C22⋊C411yescharacteristic
301C24C2×C22⋊C411yescharacteristic
302C24C22⋊Q812yes—
303C24C22⋊Q812yes—
304C24C22×D411yescharacteristic
305C2×Q8—44no—
306C2×Q8—82no—
307C2×Q8—82no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—81no—
362C22×C4—81no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C412yes—
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C4×Q812yes—
372C22×C4C4×Q812yes—
373C22×C4C4×Q812yes—
374C22×C4C4×Q812yes—
375C22×C4C22⋊Q812yes—
376C22×C4C22⋊Q812yes—
377C22×C4#12412yes—
378C22×C4#12412yes—
379C22×C4C4⋊Q812yes—
380C22×C4C4⋊Q812yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×Q811yescharacteristic
384C22×C4C22×Q811yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387M16—84no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—42no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8—84no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4#9914yes—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—48no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—42no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—81no—
436C22⋊C4—81no—
437C22⋊C4—84no—
438C22⋊C4—82no—
439C42—24no—
440C42—24no—
441C42—42no—
442C42—42no—
443C42—48no—
444C42—42no—
445C42—42no—
446C42—88no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—81no—
477C23—81no—
478C23—81no—
479C23#39112yes—
480C23#39112yes—
481C23C22×C22⋊C411yescharacteristic
482C23C2×C22⋊Q811yescharacteristic
483C23C2×C22⋊Q811yescharacteristic
484C23#57611yescharacteristic
485C23#57711yescharacteristic
486C23#57711yescharacteristic
487C23#57811yescharacteristic
488Q8—164no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—44no—
522C2×C4—44no—
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—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—82no—
559C2×C4—81no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—81no—
563C2×C4—162no—
564C2×C4—162no—
565C2×C4#39112yes—
566C2×C4#39112yes—
567C2×C4C2×#9914yes—
568C2×C4C2×#9912yes—
569C2×C4#41012yes—
570C2×C4C2×C4×Q811yescharacteristic
571C2×C4C2×C4×Q811yescharacteristic
572C2×C4C2×#12411yescharacteristic
573C2×C4C2×C4⋊Q811yescharacteristic
574C8—81no—
575C8—81no—
576C8—82no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—21no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22—82no—
607C22C2×#39111yescharacteristic
608C22C22×#9911yescharacteristic
609C22C2×#41011yescharacteristic
610C22#289711yescharacteristic
611C22#289811yescharacteristic
612C22#290111yescharacteristic
613C22#290111yescharacteristic
614C22#308111yescharacteristic
615C22#308211yescharacteristic
616C22#308311yescharacteristic
617C22#308411yescharacteristic
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—22no—
623C4—24no—
624C4—44no—
625C4—42no—
626C4—42no—
627C4—42no—
628C4—84no—
629C4—82no—
630C4—82no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—82no—
636C4#199614yes—
637C2—21no—
638C2—21no—
639C2—21no—
640C2—21no—
641C2—42no—
642C2—42no—
643C2#1760311yescharacteristic
644C2#1760511yescharacteristic
645C2#1760611yescharacteristic
646C2#1760711yescharacteristic
647C2#2131511yescharacteristic
648C2#2133011yescharacteristic
649C2C2×#199611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9913C2semidirect
5. #9911C2semidirect
6. #9910C2semidirect
7. #9909C2semidirect
12. C2×#544C4semidirect
19. #2629C4semidirect
20. #2628C4semidirect
27. #2317C4semidirect
28. #2317C4semidirect
29. #2316C4semidirect
133. #380Q8semidirect
142. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#352532
#206282
#138012
#99132
#99112
#99102
#99092
#80232
#80192
#80162
#80122