Finite Group Databasebeta

← #106971

Subgroups — #106971

Order 512.

916 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×M16C211yesmaximal, characteristic
2C2×#1803C211yesmaximal, characteristic
3C2×#1487C211yesmaximal, characteristic
4#10857C211yesmaximal, characteristic
5#8043C211yesmaximal, characteristic
6C2×#1292C211yesmaximal, characteristic
7C2×#1283C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9Q8×M16—22no—
10Q8×M16—22no—
11Q8×M16—22no—
12Q8×M16C414yescomplemented
13C2×#446—22no—
14C2×#446—21no—
15C2×C8×Q8—21no—
16C2×#431C2211yescharacteristic
17C2×#423—21no—
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C412yes—
22C2×C4×M16—21no—
23C2×C4×M16C2211yescharacteristic
24C2×#384C2211yescharacteristic
25#1803—21no—
26#1803—21no—
27#1803—21no—
28#1803—21no—
29C2×#343—21no—
30C2×#343—21no—
31C2×#340C412yes—
32#1487—22no—
33#1487C414yescomplemented
34C2×Q8⋊C8—21no—
35C2×Q8⋊C8—21no—
36#1296—22no—
37#1292—22no—
38#1283—22no—
39C23×Q8C2×C411yescharacteristic
40C22×M16—21no—
41C22×M16—21no—
42C22×M16D411yescharacteristic
43C22×M16D411yescharacteristic
44C23×C8C2×C412yes—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8D411yescharacteristic
57C2×C4×Q8D412yes—
58C2×C4×Q8Q812yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C412yes—
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66#446—28no—
67#446—24no—
68#446—24no—
69#446—28no—
70C8×Q8—24no—
71C8×Q8—24no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431C2×C412yes—
76#423—24no—
77#423—24no—
78#423—21no—
79#423—21no—
80#423—21no—
81#423—21no—
82#423—21no—
83#423C2×C412yes—
84#423C2×C412yes—
85#423D412yes—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8D412yes—
99C2×#103—22no—
100C2×#103—22no—
101C2×#103—44no—
102C2×C22⋊C8—21no—
103C4×M16—24no—
104C4×M16—24no—
105C4×M16—21no—
106C4×M16—21no—
107C4×M16C2×C412yes—
108C4×M16D412yes—
109C2×#97—22no—
110C2×#97—21no—
111C2×C4×C8—22no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114#384—21no—
115#384—21no—
116#384—21no—
117#384—21no—
118#343—42no—
119#343—42no—
120#340—22no—
121#340—22no—
122Q8⋊C8—24no—
123Q8⋊C8—24no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—41no—
130C22×Q8—41no—
131C22×Q8C4212yes—
132C22×Q8C22⋊C411yescharacteristic
133C22×Q8C22⋊C411yescharacteristic
134C22×Q8C4⋊C412yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4#4111yescharacteristic
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—21no—
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—22no—
152C2×M16—41no—
153C2×M16—41no—
154C2×M16—41no—
155C2×M16—41no—
156C2×M16—41no—
157C2×M16—41no—
158C2×M16—41no—
159C2×M16—41no—
160C2×M16—41no—
161C2×M16C22⋊C412yes—
162C2×M16C22⋊C412yes—
163C2×M16SD1612yes—
164C2×M16Q1612yes—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—41no—
172C22×C8—41no—
173C22×C8—42no—
174C22×C8—42no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—21no—
180C4×Q8—22no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—22no—
191C4×Q8C22⋊C412yes—
192C4×Q8C22⋊C412yes—
193C4×Q8C22⋊C412yes—
194C4×Q8C22⋊C412yes—
195C4×Q8C4⋊C412yes—
196C4×Q8C4⋊C412yes—
197C4×Q8C4⋊C412yes—
198C4×Q8C4⋊C412yes—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4C4212yes—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C4⋊C412yes—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×Q811yescharacteristic
244C2×C42#4111yescharacteristic
245C2×C42#4111yescharacteristic
246C2×C42#4111yescharacteristic
247C4⋊C8—28no—
248C4⋊C8—28no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—41no—
256C4⋊C8—41no—
257C4⋊C8—41no—
258C4⋊C8—41no—
259C4⋊C8—41no—
260C4⋊C8—41no—
261C4⋊C8—41no—
262C4⋊C8C22⋊C414yes—
263#103—44no—
264#103—44no—
265C22⋊C8—22no—
266C22⋊C8—41no—
267#97—28no—
268#97—22no—
269#97—22no—
270C4×C8—28no—
271C4×C8—22no—
272C4×C8—22no—
273C4×C8—22no—
274C4×C8—41no—
275C24C4×D411yescharacteristic
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8#9512yes—
297C2×Q8#9512yes—
298C2×Q8#10012yes—
299C2×Q8#10112yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4C2×C4211yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C4⋊C411yescharacteristic
327C22×C4#11711yescharacteristic
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4#12011yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4C22⋊Q811yescharacteristic
338C22×C4C22⋊Q811yescharacteristic
339C22×C4#12311yescharacteristic
340C22×C4#12311yescharacteristic
341C22×C4C2×SD1611yescharacteristic
342C22×C4C2×Q1611yescharacteristic
343C22×C4#13511yescharacteristic
344C22×C4#13511yescharacteristic
345C22×C4AGL(1,ℤ/8)11yescharacteristic
346C22×C4#13711yescharacteristic
347C22×C4#13711yescharacteristic
348C22×C4#13711yescharacteristic
349M16—22no—
350M16—22no—
351M16—22no—
352M16—22no—
353M16—22no—
354M16—42no—
355M16—42no—
356M16—41no—
357M16—42no—
358M16—41no—
359M16—42no—
360M16—42no—
361M16—42no—
362M16—81no—
363M16—81no—
364M16#10314yescomplemented
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—24no—
370C2×C8—41no—
371C2×C8—42no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—81no—
380C2×C8—81no—
381C2×C8—81no—
382C2×C8—81no—
383C2×C8—82no—
384C2×C8—82no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—82no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—81no—
403C4⋊C4—81no—
404C4⋊C4—81no—
405C4⋊C4—81no—
406C4⋊C4#9512yes—
407C4⋊C4#9512yes—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42C2×C22⋊C411yescharacteristic
417C42C2×C22⋊C411yescharacteristic
418C42C2×C22⋊C411yescharacteristic
419C42C2×C4⋊C411yescharacteristic
420C42C2×C4⋊C411yescharacteristic
421C42#11711yescharacteristic
422C42C4×D411yescharacteristic
423C42C4×D411yescharacteristic
424C42#12011yescharacteristic
425C42#12011yescharacteristic
426C42#12111yescharacteristic
427C42#12111yescharacteristic
428C42C22⋊Q811yescharacteristic
429C42C22⋊Q811yescharacteristic
430C42#12311yescharacteristic
431C42#12311yescharacteristic
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23C4×C22⋊C411yescharacteristic
437C23#38511yescharacteristic
438C23#38611yescharacteristic
439C23C4×SD1611yescharacteristic
440C23C4×Q1611yescharacteristic
441C23#44011yescharacteristic
442C23#44111yescharacteristic
443Q8—22no—
444Q8—22no—
445Q8—22no—
446Q8—22no—
447Q8—22no—
448Q8—44no—
449Q8—44no—
450Q8—41no—
451Q8—41no—
452Q8#34314yescomplemented
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—162no—
491C2×C4—162no—
492C2×C4C2×#9511yescharacteristic
493C2×C4C4×C22⋊C411yescharacteristic
494C2×C4#37911yescharacteristic
495C2×C4#38011yescharacteristic
496C2×C4#38011yescharacteristic
497C2×C4#38111yescharacteristic
498C2×C4#38511yescharacteristic
499C2×C4#38511yescharacteristic
500C2×C4#38511yescharacteristic
501C2×C4#38611yescharacteristic
502C2×C4#38611yescharacteristic
503C2×C4#38611yescharacteristic
504C2×C4C2×#10011yescharacteristic
505C2×C4C2×#10111yescharacteristic
506C2×C4#41311yescharacteristic
507C2×C4#41311yescharacteristic
508C2×C4C2×#10311yescharacteristic
509C2×C4#41611yescharacteristic
510C2×C4#41711yescharacteristic
511C2×C4#41911yescharacteristic
512C2×C4C4×SD1611yescharacteristic
513C2×C4C4×Q1611yescharacteristic
514C2×C4#44011yescharacteristic
515C2×C4#44111yescharacteristic
516C2×C4#44811yescharacteristic
517C2×C4#44811yescharacteristic
518C2×C4#44911yescharacteristic
519C2×C4#44911yescharacteristic
520C2×C4#45111yescharacteristic
521C2×C4#45111yescharacteristic
522C2×C4#45211yescharacteristic
523C2×C4#45211yescharacteristic
524C2×C4#46011yescharacteristic
525C2×C4#46211yescharacteristic
526C2×C4#46411yescharacteristic
527C2×C4#46411yescharacteristic
528C2×C4#47511yescharacteristic
529C2×C4#47711yescharacteristic
530C2×C4#47911yescharacteristic
531C2×C4#47911yescharacteristic
532C8—24no—
533C8—42no—
534C8—44no—
535C8—82no—
536C8—82no—
537C8—82no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22C22⋊#9511yescharacteristic
551C22C4×#10311yescharacteristic
552C22#176311yescharacteristic
553C22#186311yescharacteristic
554C22#186511yescharacteristic
555C22#187811yescharacteristic
556C22#188011yescharacteristic
557C22#190011yescharacteristic
558C22#190111yescharacteristic
559C22#190211yescharacteristic
560C22#190311yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—24no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—162no—
572C4C2×#34311yescharacteristic
573C4#174211yescharacteristic
574C4#178711yescharacteristic
575C4#178811yescharacteristic
576C4#180311yescharacteristic
577C4#180511yescharacteristic
578C4#186311yescharacteristic
579C4#186511yescharacteristic
580C4#187811yescharacteristic
581C4#188011yescharacteristic
582C4#190011yescharacteristic
583C4#190111yescharacteristic
584C2—21no—
585C2—21no—
586C2—21no—
587C2—21no—
588C2#924811yescharacteristic
589C2#925011yescharacteristic
590C2#925211yescharacteristic
591C2#925411yescharacteristic
592C2#1073911yescharacteristic
593C2#1078411yescharacteristic
594C2#1078911yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. Q8×M16C4semidirect
33. #1487C4semidirect
364. M16#103semidirect
452. Q8#343semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×M162
C2×#18032
C2×#14872
#108572
#80432
C2×#12922
C2×#12832