Finite Group Databasebeta

← #191989

Subgroups — #191989

Order 512.

962 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2994C211yesmaximal, characteristic
2#21953C211yesmaximal, characteristic, complemented
3#21388C211yesmaximal, characteristic
4C2×#1573C211yesmaximal, characteristic, complemented
5C2×#1494C211yesmaximal, characteristic
6#8850C211yesmaximal, characteristic, complemented
7#8849C211yesmaximal, characteristic, complemented
8#8848C211yesmaximal, characteristic, complemented
9#8847C211yesmaximal, characteristic, complemented
10#8846C211yesmaximal, characteristic, complemented
11#8845C211yesmaximal, characteristic, complemented
12#8844C211yesmaximal, characteristic, complemented
13#8843C211yesmaximal, characteristic, complemented
14C2×#1279C211yesmaximal, characteristic, complemented
15#7849C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2994—24no—
18#2994—24no—
19C2×#446C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#435—21no—
23C2×C8×D4—21no—
24C2×C8×D4C2211yescharacteristic
25C2×#433—21no—
26C2×#432C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#424—21no—
30C22×C4⋊C8C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1989C411yescharacteristic
34#1989C411yescharacteristic
35#1983C411yescharacteristic
36#1983C411yescharacteristic
37#1940—21no—
38#1940—21no—
39#1939C2211yescharacteristic
40#1939C2211yescharacteristic
41#1939C2211yescharacteristic
42#1939C2211yescharacteristic
43#1874C411yescharacteristic
44#1874C411yescharacteristic
45#1872C411yescharacteristic
46#1872C411yescharacteristic
47#1797—21no—
48#1797—21no—
49C4×#103C2211yescharacteristic
50C4×#103C2211yescharacteristic
51C4×#102C2211yescharacteristic
52C4×#102C2211yescharacteristic
53#1573—24no—
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58#1494—24no—
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×M16—21no—
72C23×C8—21no—
73C23×C8—21no—
74C23×C8—21no—
75#529—22no—
76#529—22no—
77#529D414yes—
78#529Q814yes—
79C2×#126—21no—
80C2×#125C2×C411yescharacteristic
81C2×#124C2×C411yescharacteristic
82C2×#123—21no—
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×C4×Q8C2311yescharacteristic
86C2×C4×D4—21no—
87C2×C4×D4C2311yescharacteristic
88C2×#117—21no—
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C4⋊C4C2×C411yescharacteristic
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93#446—22no—
94#446—22no—
95C8×Q8—22no—
96C8×Q8—22no—
97#436—22no—
98#436—22no—
99#435—28no—
100C8×D4—28no—
101C8×D4—22no—
102C8×D4—22no—
103#433—28no—
104#432—22no—
105#432—22no—
106#431—22no—
107#431—22no—
108#431—22no—
109#431—22no—
110#424—28no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8C2×C411yescharacteristic
120C2×C4⋊C8C2×C411yescharacteristic
121C2×C4⋊C8D411yescharacteristic
122C2×C4⋊C8D411yescharacteristic
123C2×C4⋊C8D414yes—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8C2×C411yescharacteristic
140C2×C22⋊C8C2×C411yescharacteristic
141C2×C22⋊C8D411yescharacteristic
142C2×C22⋊C8D411yescharacteristic
143C2×#97—21no—
144C2×#97—21no—
145C2×C4×C8—21no—
146C2×C4×C8—22no—
147C2×C4×C8—22no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4C2×C411yescharacteristic
158C4×C4⋊C4C2×C411yescharacteristic
159C4×C4⋊C4C2311yescharacteristic
160C4×C4⋊C4C2311yescharacteristic
161C2×#95—21no—
162#340C2×C411yescharacteristic
163#340C2×C411yescharacteristic
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#340C2×C411yescharacteristic
169#340C2×C411yescharacteristic
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#335—22no—
175#335—22no—
176#334—22no—
177#334—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182C22×Q8C22×C411yescharacteristic
183C22×D4—21no—
184C22×D4C22×C411yescharacteristic
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4#4111yescharacteristic
191C2×M16—24no—
192C2×M16—81no—
193C2×M16—81no—
194C22×C8—24no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—24no—
200C22×C8—24no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—42no—
204C22×C8—42no—
205C22×C8—81no—
206C22×C8—81no—
207C22×C8—81no—
208C22×C8—81no—
209C22×C8C22×C411yescharacteristic
210C22×C8C22×C411yescharacteristic
211C22×C8C22×C411yescharacteristic
212C22×C8C22×C411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C22×C8C2×D411yescharacteristic
217#126—24no—
218#126—24no—
219#125—22no—
220#125C4⋊C414yes—
221#124—22no—
222#124C4⋊C414yes—
223#123—24no—
224#123—24no—
225C22⋊Q8—22no—
226C22⋊Q8C4⋊C414yes—
227#121—22no—
228#121C4⋊C414yes—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8C2×D412yes—
232C4×Q8C2×Q812yes—
233C4×D4—24no—
234C4×D4—24no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4C2×D412yes—
238C4×D4C2×Q812yes—
239#117—24no—
240#117—24no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—41no—
250C2×C4⋊C4—41no—
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C4⋊C4C22×C411yescharacteristic
254C2×C4⋊C4C22×C411yescharacteristic
255C2×C4⋊C4C22×C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—81no—
268C2×C42—81no—
269C2×C42C22×C411yescharacteristic
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D411yescharacteristic
273C2×C42C2×Q812yes—
274C2×C42#4112yes—
275C2×C42#4111yescharacteristic
276C2×C42C2411yescharacteristic
277C4⋊C8—24no—
278C4⋊C8—24no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—24no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—81no—
291C4⋊C8—81no—
292C4⋊C8—81no—
293C4⋊C8—81no—
294C4⋊C8C2×D412yes—
295C4⋊C8C2×D412yes—
296#103—22no—
297#103—22no—
298#103—41no—
299#103—41no—
300#102—22no—
301#102—22no—
302#102—41no—
303#102—41no—
304C22⋊C8—24no—
305C22⋊C8—24no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8—42no—
311#97—24no—
312#97—42no—
313C4×C8—24no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—42no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331C24—21no—
332C24C4×D411yescharacteristic
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8C2×C4⋊C412yes—
339C2×D4—24no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4C2×C4⋊C412yes—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4C2×C4⋊C412yes—
378C22×C4C4×D412yes—
379C22×C4C4×D411yescharacteristic
380C22×C4C4×D411yescharacteristic
381C22×C4C4×D411yescharacteristic
382C22×C4#12012yes—
383C22×C4C22⋊Q812yes—
384C22×C4C22⋊Q812yes—
385C22×C4#12312yes—
386C22×C4C23×C411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4111yescharacteristic
389M16—84no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—48no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—84no—
434C2×C8—84no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4C2×C4⋊C412yes—
451C4⋊C4C2×C4⋊C412yes—
452C4⋊C4#13112yes—
453C4⋊C4#13112yes—
454C22⋊C4—24no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—24no—
460C42—22no—
461C42—22no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—81no—
467C42—81no—
468C42C2×C4⋊C412yes—
469C42C4×D412yes—
470C42#12012yes—
471C42C22⋊Q812yes—
472C42C22⋊Q812yes—
473C42#12312yes—
474C42C22×D411yescharacteristic
475C42C22×D411yescharacteristic
476C42C22×Q811yescharacteristic
477C42C2×#4111yescharacteristic
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23#38512yes—
490C23#44312yes—
491C23#44412yes—
492C23C2×C4×D411yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497D4—22no—
498D4—22no—
499D4—48no—
500D4—41no—
501D4—41no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4#38512yes—
548C2×C4#38512yes—
549C2×C4#38512yes—
550C2×C4C22×C4⋊C411yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C2×#12011yescharacteristic
553C2×C4C2×C22⋊Q811yescharacteristic
554C2×C4C2×C22⋊Q811yescharacteristic
555C2×C4C2×#12311yescharacteristic
556C2×C4C2×#13111yescharacteristic
557C2×C4C2×#13111yescharacteristic
558C2×C4#56811yescharacteristic
559C2×C4#56811yescharacteristic
560C2×C4#57611yescharacteristic
561C2×C4#57711yescharacteristic
562C2×C4#57711yescharacteristic
563C2×C4#57811yescharacteristic
564C8—42no—
565C8—42no—
566C8—41no—
567C8—41no—
568C8—41no—
569C8—41no—
570C8—41no—
571C8—41no—
572C8—42no—
573C8—42no—
574C8—41no—
575C8—41no—
576C8—81no—
577C8—81no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22#186614yes—
592C22C2×#38511yescharacteristic
593C22#293311yescharacteristic
594C22#293311yescharacteristic
595C22#294611yescharacteristic
596C22#294711yescharacteristic
597C22C2×#44311yescharacteristic
598C22C2×#44411yescharacteristic
599C4—22no—
600C4—22no—
601C4—21no—
602C4—21no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—81no—
610C4—81no—
611C4#290611yescharacteristic
612C4#290611yescharacteristic
613C4#293111yescharacteristic
614C4#293211yescharacteristic
615C4#301211yescharacteristic
616C4#301311yescharacteristic
617C4#308311yescharacteristic
618C4#308411yescharacteristic
619C2—24no—
620C2—82no—
621C2#1801411yescharacteristic
622C2#1801711yescharacteristic
623C2#1802011yescharacteristic
624C2#1802311yescharacteristic
625C2#2083911yescharacteristic
626C2#2087811yescharacteristic
627C2C2×#186611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21953C2semidirect
4. C2×#1573C2semidirect
6. #8850C2semidirect
7. #8849C2semidirect
8. #8848C2semidirect
9. #8847C2semidirect
10. #8846C2semidirect
11. #8845C2semidirect
12. #8844C2semidirect
13. #8843C2semidirect
14. C2×#1279C2semidirect
15. #7849C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29942
#219532
#213882
C2×#15732
C2×#14942
#88502
#88492
#88482
#88472
#88462
#88452
#88442
#88432
C2×#12792
#78492