Finite Group Databasebeta

← #380988

Subgroups — #380988

Order 512.

1564 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×#1728C211yesmaximal, characteristic
3C4×#380C211yesmaximal, characteristic
4#11386C212yesmaximal, complemented
5#11370C212yesmaximal, complemented
6#11226C214yesmaximal, complemented
7#10732C214yesmaximal
8C23×M16C2212yes—
9C2×#533C2211yescharacteristic
10C23×C42C2211yescharacteristic
11#2857C411yescharacteristic, complemented
12#2857C411yescharacteristic, complemented
13#2857C411yescharacteristic, complemented
14#2857C411yescharacteristic, complemented
15#2855—21no—
16#2855—21no—
17#2851C412yescomplemented
18#2851C412yescomplemented
19#2848—21no—
20#2848—21no—
21C4×C22⋊Q8—21no—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25#2296C2212yes—
26#2285C2211yescharacteristic
27#2280C2212yes—
28C2×#380—21no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C22⋊C4C2212yes—
31C2×#339C2218yes—
32C2×#339C2218yes—
33#1728—21no—
34#1728—21no—
35#1728C2212yes—
36#1728C2212yes—
37C2×#336—21no—
38C2×#336—21no—
39C4⋊#95—21no—
40#1434—21no—
41C22⋊#95—21no—
42C4×#95C2212yes—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—24no—
48C22×M16—44no—
49C22×M16—44no—
50C22×M16Q818yescomplemented
51C22×M16C2314yes—
52C23×C8—22no—
53#533—44no—
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×C4×Q8—21no—
64C2×C4×Q8—21no—
65C2×#117C2312yes—
66C2×#117C2312yes—
67C22×C4⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C42—44no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42D411yescharacteristic
80C22×C42C2314yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#391—21no—
84#391—21no—
85#391—21no—
86#391—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#384C2×C411yescharacteristic
106#384C2×C411yescharacteristic
107#382C2×C412yes—
108#381—22no—
109#380—28no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380C2×C412yes—
117#380C2×C412yes—
118#380C2×C412yes—
119#380C2×C412yes—
120#379—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—44no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—28no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133#339—48no—
134#339—48no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336—22no—
139C25C2×D411yescharacteristic
140C22×Q8—21no—
141C22×Q8—21no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×Q814yes—
170C23×C4#4114yes—
171C23×C4C2411yescharacteristic
172C2×M16—28no—
173C2×M16—24no—
174C2×M16—28no—
175C2×M16—28no—
176C2×M16—416no—
177C2×M16—416no—
178C2×M16—44no—
179C2×M16—44no—
180C2×M16C2×Q818yes—
181C22×C8—28no—
182C22×C8—24no—
183C22×C8—44no—
184C22×C8—44no—
185C4⋊Q8—42no—
186C4⋊Q8—42no—
187#125—44no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194C4×Q8—44no—
195C4×Q8—44no—
196#117—28no—
197#117—24no—
198#117—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C22×C412yes—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—28no—
232C2×C42—24no—
233C2×C42—28no—
234C2×C42—24no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—42no—
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22⋊C411yescharacteristic
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C22×C411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261#95—28no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—44no—
273#95—44no—
274#95—44no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—41no—
280C24—41no—
281C24C2×C22⋊C411yescharacteristic
282C24C2×C22⋊C411yescharacteristic
283C24#12412yes—
284C24C4⋊Q812yes—
285C24C22×D411yescharacteristic
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—28no—
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—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—28no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—416no—
349C22×C4—416no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4#10414yes—
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C4×Q814yes—
368C22×C4C4×Q814yes—
369C22×C4C22⋊Q814yes—
370C22×C4C22⋊Q814yes—
371C22×C4#12412yes—
372C22×C4C4⋊Q812yes—
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×Q812yes—
376C22×C4C2×#4112yes—
377C22×C4H(5,𝔽2)14yes—
378C22×C4H(5,𝔽2)12yes—
379C22×C42−1+412yes—
380M16—216no—
381M16—416no—
382M16—48no—
383C2×C8—28no—
384C2×C8—416no—
385C2×C8—44no—
386C2×C8—44no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—416no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—44no—
404C22⋊C4—416no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—44no—
428C42—42no—
429C42—42no—
430C42—48no—
431C42—48no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—24no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23#39112yes—
457C23#39112yes—
458C23C2×#10412yes—
459C23#42112yes—
460C23C22×C22⋊C411yescharacteristic
461C23C2×#12411yescharacteristic
462C23C2×C4⋊Q811yescharacteristic
463C23#53412yes—
464C23#53712yes—
465Q8—84no—
466Q8—84no—
467C2×C4—28no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—48no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—84no—
527C2×C4—84no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4#39112yes—
531C2×C4#39112yes—
532C2×C4C2×#10412yes—
533C2×C4C2×#10412yes—
534C2×C4#42112yes—
535C2×C4#42112yes—
536C2×C4C2×C4×Q812yes—
537C2×C4#51812yes—
538C2×C4#52012yes—
539C2×C4C2×C22⋊Q812yes—
540C2×C4#53412yes—
541C2×C4#53511yescharacteristic
542C2×C4#53611yescharacteristic
543C2×C4#53814yes—
544C2×C4#53914yes—
545C2×C4#54314yes—
546C2×C4#54414yes—
547C8—48no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—48no—
566C22—48no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22#198614yes—
572C22C2×#39111yescharacteristic
573C22#230811yescharacteristic
574C22#231111yescharacteristic
575C22#263912yes—
576C22#265912yes—
577C22C22×#10411yescharacteristic
578C22C2×#42111yescharacteristic
579C22C2×#42111yescharacteristic
580C22C2×#42111yescharacteristic
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—22no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—88no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4#232914yes—
597C4#262612yes—
598C4#262712yes—
599C2—22no—
600C2—22no—
601C2—24no—
602C2—42no—
603C2#1428111yescharacteristic
604C2C2×#198611yescharacteristic
605C2#2130611yescharacteristic
606C2#2177311yescharacteristic
607C2#2177411yescharacteristic
608C2#2185611yescharacteristic
609C2#2185711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11386C2semidirect
5. #11370C2semidirect
6. #11226C2semidirect
11. #2857C4semidirect
12. #2857C4semidirect
13. #2857C4semidirect
14. #2857C4semidirect
17. #2851C4semidirect
18. #2851C4semidirect
50. C22×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#17282
C4×#3802
#113862
#113702
#112262
#107322