Finite Group Databasebeta

← #219801

Subgroups — #219801

Order 512.

1406 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#550C211yesmaximal, characteristic
2#22795C211yesmaximal, characteristic, complemented
3C4×C8×D4C211yesmaximal, characteristic, complemented
4C2×#1476C212yesmaximal, complemented
5#9674C211yesmaximal, characteristic, complemented
6#9638C211yesmaximal, characteristic, complemented
7#9072C212yesmaximal, complemented
8#8940C212yesmaximal, complemented
9#8564C211yesmaximal, characteristic, complemented
10#8312C212yesmaximal, complemented
11C4×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×#550C411yescharacteristic
13C2×C4×#41C2212yes—
14C2×C8×D4—21no—
15C2×#431C2212yes—
16C22×C4×C8C2212yes—
17#2366—21no—
18#2366C411yescharacteristic
19#2363—22no—
20#2363C412yes—
21#2354—21no—
22#2354C411yescharacteristic
23Q8×C4⋊C4C411yescharacteristic
24#2341C412yes—
25C4×#127—21no—
26C4×#127C2211yescharacteristic
27C4×#121—22no—
28C4×#121C2212yes—
29C42×Q8C2211yescharacteristic
30C42×D4—21no—
31C42×D4C2211yescharacteristic
32#1954—21no—
33#1923—22no—
34#1842C2211yescharacteristic
35#1797C2212yes—
36C8×C4⋊C4—21no—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C8×C22⋊C4—22no—
41C4×C22⋊C8C2212yes—
42C42×C8C2211yescharacteristic
43#1476—28no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2211yescharacteristic
46C2×Q8⋊C8C2211yescharacteristic
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49C2×D4⋊C8C2212yes—
50C2×D4⋊C8C2212yes—
51#1279C2211yescharacteristic
52#1279C2212yes—
53#1279C2211yescharacteristic
54C22×#41C2×C412yes—
55C23×C8—22no—
56#550—24no—
57#550C818yescomplemented
58C2×#127—21no—
59C2×#127C2×C411yescharacteristic
60C2×#121—22no—
61C2×#121C2×C412yes—
62C4×#41—24no—
63C4×#41—24no—
64C4×#41—42no—
65C4×#41—42no—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8C2×C412yes—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×D4—22no—
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×C4×D4C2×C412yes—
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4D411yescharacteristic
80C2×C4×D4D411yescharacteristic
81C2×C4×D4C2312yes—
82C2×#117—22no—
83C2×#117C2×C412yes—
84C22×C42—22no—
85C22×C42C2312yes—
86C8×D4—28no—
87#431—24no—
88#431—42no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2312yes—
94C2×C4⋊C8C2311yescharacteristic
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C4×C8—21no—
99C2×C4×C8—24no—
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—42no—
105C2×C4×C8C2312yes—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#390—21no—
109#390—21no—
110#390—22no—
111#390—22no—
112#390—22no—
113#390—21no—
114#390—21no—
115#390C2×C412yes—
116#390C2×C412yes—
117#389—22no—
118#389C2×C412yes—
119#388—22no—
120#388—22no—
121#388—22no—
122#386—22no—
123#386—22no—
124#386—22no—
125#384—21no—
126#384C2×C411yescharacteristic
127#380—22no—
128#380C2×C412yes—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2×C411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—22no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4C2×C412yes—
144C4×C22⋊C4D412yes—
145C4×C22⋊C4D412yes—
146C43—21no—
147C43C2311yescharacteristic
148#336—22no—
149#336—21no—
150#336—22no—
151#336—21no—
152Q8⋊C8—24no—
153Q8⋊C8—22no—
154Q8⋊C8D414yescomplemented
155D4⋊C8—24no—
156D4⋊C8—24no—
157D4⋊C8—42no—
158D4⋊C8—42no—
159C2×#41—24no—
160C2×#41—24no—
161C2×#41—24no—
162C2×#41—42no—
163C2×#41—42no—
164C2×#41C2×C818yescomplemented
165C22×Q8C22×C411yescharacteristic
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4C22⋊C411yescharacteristic
171C22×D4C22×C412yes—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4C22×C412yes—
176C23×C4C2×D412yes—
177C23×C4C2×D412yes—
178C22×C8—21no—
179C22×C8—28no—
180C22×C8—22no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—44no—
186C22×C8—44no—
187#127—28no—
188#127—22no—
189#127C2×C814yescomplemented
190#121—24no—
191#121—216no—
192#121C2×C818yescomplemented
193C4×Q8—22no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—41no—
199C4×Q8—42no—
200C4×Q8C2×C814yescomplemented
201C4×Q8C2×D412yes—
202C4×Q8#4112yes—
203C4×D4—28no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—22no—
212C4×D4—42no—
213C4×D4—44no—
214C4×D4—44no—
215C4×D4—44no—
216C4×D4—44no—
217C4×D4—44no—
218C4×D4—42no—
219C4×D4—42no—
220C4×D4—81no—
221C4×D4—81no—
222C4×D4—82no—
223C4×D4—82no—
224C4×D4C2×C814yes—
225C4×D4M1614yes—
226#117—24no—
227#117—44no—
228#117—44no—
229#117—42no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22×C412yes—
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4C22⋊C412yes—
254C2×C22⋊C4C22⋊C412yes—
255C2×C22⋊C4C22⋊C412yes—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42C22×C412yes—
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C2×D412yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2411yescharacteristic
280C4⋊C8—24no—
281C4⋊C8—22no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—22no—
286C4⋊C8—42no—
287C4⋊C8—81no—
288C4⋊C8C2×D412yes—
289C22⋊C8—28no—
290C22⋊C8—44no—
291C22⋊C8—82no—
292C4×C8—24no—
293C4×C8—22no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—22no—
298C4×C8—44no—
299C4×C8—44no—
300C4×C8—42no—
301C4×C8—42no—
302C4×C8C2×D412yes—
303#95—22no—
304#95—22no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95C22⋊C412yes—
309#95C22⋊C412yes—
310#95C22⋊C412yes—
311C24—22no—
312C24C2×C22⋊C412yes—
313#41—216no—
314#41—48no—
315#41—48no—
316#41—48no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8C4×D412yes—
321C2×Q8C22×C812yes—
322C2×Q8#13112yes—
323C2×D4—28no—
324C2×D4—28no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—81no—
339C2×D4—81no—
340C2×D4—82no—
341C2×D4—82no—
342C2×D4C22⋊C814yes—
343C2×D4C22×C814yes—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—28no—
360C22×C4—24no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—42no—
364C22×C4—44no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—82no—
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C22×C814yes—
383C22×C4C2×M1614yes—
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—24no—
392C2×C8—22no—
393C2×C8—416no—
394C2×C8—41no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—42no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—42no—
404C2×C8—41no—
405C2×C8—42no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—44no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—81no—
420C4⋊C4—82no—
421C4⋊C4C22⋊C814yes—
422C4⋊C4#10414yes—
423C4⋊C4C4×D412yes—
424C4⋊C4C22×C812yes—
425C22⋊C4—28no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—82no—
435C22⋊C4—82no—
436C22⋊C4—82no—
437C22⋊C4C22⋊C818yes—
438C42—22no—
439C42—22no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—24no—
451C42—22no—
452C42—41no—
453C42—44no—
454C42—44no—
455C42—42no—
456C42—41no—
457C42—41no—
458C42—42no—
459C42—81no—
460C42C22×C812yes—
461C42C2×M1612yes—
462C42#13512yes—
463C42#13512yes—
464C42C22×D411yescharacteristic
465C42C2×#4111yescharacteristic
466C42H(5,𝔽2)11yescharacteristic
467C42H(5,𝔽2)11yescharacteristic
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23C2×C22⋊C814yes—
482C23C22×C22⋊C411yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—22no—
486Q8C8×D414yescomplemented
487D4—24no—
488D4—24no—
489D4—28no—
490D4—416no—
491D4—42no—
492D4—42no—
493D4—48no—
494D4—84no—
495D4—88no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—416no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—88no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4C2×C22⋊C812yes—
544C2×C4C2×C22⋊C814yes—
545C2×C4C2×C22⋊C812yes—
546C2×C4#41612yes—
547C2×C4#41612yes—
548C2×C4C2×#10414yes—
549C2×C4C2×#10412yes—
550C2×C4C2×C4×D411yescharacteristic
551C2×C4#51811yescharacteristic
552C2×C4D4211yescharacteristic
553C2×C4D4211yescharacteristic
554C2×C4#54611yescharacteristic
555C2×C4#54611yescharacteristic
556C2×C4C23×C811yescharacteristic
557C2×C4C22×M1611yescharacteristic
558C2×C4C2×#13111yescharacteristic
559C2×C4#56811yescharacteristic
560C2×C4C2×#13511yescharacteristic
561C2×C4C2×#13511yescharacteristic
562C2×C4#57611yescharacteristic
563C2×C4#57711yescharacteristic
564C8—24no—
565C8—42no—
566C8—44no—
567C8—42no—
568C8—82no—
569C8—81no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22D4×C22⋊C411yescharacteristic
585C22C22×C22⋊C811yescharacteristic
586C22#288011yescharacteristic
587C22C2×#41611yescharacteristic
588C22#289711yescharacteristic
589C22C22×#10411yescharacteristic
590C22#290111yescharacteristic
591C4—22no—
592C4—22no—
593C4—24no—
594C4—24no—
595C4—21no—
596C4—21no—
597C4—24no—
598C4—42no—
599C4—41no—
600C4—42no—
601C4—42no—
602C4—81no—
603C4—82no—
604C4#147614yescomplemented
605C4C2×C8×D411yescharacteristic
606C4D4×M1611yescharacteristic
607C4#297211yescharacteristic
608C4#297411yescharacteristic
609C4#328411yescharacteristic
610C4#328711yescharacteristic
611C4#330411yescharacteristic
612C4#330511yescharacteristic
613C2—44no—
614C2—84no—
615C2C2×#147611yescharacteristic
616C2#1743811yescharacteristic
617C2#1751011yescharacteristic
618C2#1752011yescharacteristic
619C2D4×C22⋊C811yescharacteristic
620C2#2241811yescharacteristic
621C2D4×#10411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#550C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22795C2semidirect
3. C4×C8×D4C2semidirect
4. C2×#1476C2semidirect
5. #9674C2semidirect
6. #9638C2semidirect
7. #9072C2semidirect
8. #8940C2semidirect
9. #8564C2semidirect
10. #8312C2semidirect
11. C4×Q8⋊C8C2semidirect
57. #550C8semidirect
154. Q8⋊C8D4semidirect
164. C2×#41C2×C8semidirect
189. #127C2×C8semidirect
192. #121C2×C8semidirect
200. C4×Q8C2×C8semidirect
486. Q8C8×D4semidirect
604. C4#1476semidirect

Maximal subgroups

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

subgroupindex
C4×#5502
#227952
C4×C8×D42
C2×#14762
#96742
#96382
#90722
#89402
#85642
#83122
C4×Q8⋊C82