Finite Group Databasebeta

← #117744

Subgroups — #117744

Order 512.

910 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1834C211yesmaximal, characteristic
2C2×#1803C211yesmaximal, characteristic
3C22×Q8⋊C8C211yesmaximal, characteristic
4#13926C211yesmaximal, characteristic
5#10849C211yesmaximal, characteristic
6C2×#1296C211yesmaximal, characteristic
7C2×#1279C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9C22×C4⋊C8C2211yescharacteristic
10C22×C4⋊C8C2211yescharacteristic
11C22×#103—21no—
12C22×C4×C8C2211yescharacteristic
13C2×#391—21no—
14C2×#384C2211yescharacteristic
15C2×#384C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#1834—22no—
18#1834—22no—
19#1803—22no—
20#1803—22no—
21C2×#340—22no—
22C2×#340—21no—
23C2×#340—22no—
24C2×Q8⋊C8—22no—
25C2×Q8⋊C8—22no—
26C2×Q8⋊C8C414yescomplemented
27#1441—22no—
28C4⋊#95—22no—
29C4⋊#95—21no—
30#1296—22no—
31#1279—22no—
32C23×Q8D411yescharacteristic
33C23×C8—21no—
34C23×C8—21no—
35C23×C8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8Q812yes—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D412yes—
56C22×C4⋊C4D412yes—
57C22×C42—21no—
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2×C412yes—
71C2×C4⋊C8D412yes—
72C2×#103—24no—
73C2×#103—42no—
74C2×#103—42no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8C2×C412yes—
79C2×C4×C8D412yes—
80C2×C4×C8D412yes—
81#391—42no—
82#391—42no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384D412yes—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C2×#95—24no—
96C2×#95—21no—
97C2×#95—24no—
98#340—24no—
99#340—28no—
100#340—44no—
101Q8⋊C8—28no—
102C22×Q8—21no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8C4⋊C412yes—
114C22×Q8D812yes—
115C22×Q8Q1612yes—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—41no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—42no—
130C22×C8—42no—
131C22×C8—41no—
132C22×C8—41no—
133C22×C8—41no—
134C22×C8—41no—
135C22×C8—41no—
136C22×C8—42no—
137C22×C8—42no—
138C22×C8—41no—
139C22×C8—41no—
140C22×C8—41no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×Q8—41no—
156C4×Q8—41no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4C4⋊C412yes—
200C2×C4⋊C4SD1614yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—41no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42—41no—
210C2×C42—41no—
211C2×C42—41no—
212C2×C42—41no—
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×Q811yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42#4111yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42#4111yescharacteristic
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—41no—
230C4⋊C8—41no—
231C4⋊C8—41no—
232C4⋊C8—41no—
233#103—48no—
234C4×C8—22no—
235C4×C8—22no—
236C4×C8—22no—
237C4×C8C22⋊C414yes—
238#95—48no—
239#95—41no—
240#95—41no—
241#95—84no—
242C24#12011yescharacteristic
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8#10714yes—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—81no—
299C22×C4—82no—
300C22×C4—81no—
301C22×C4—82no—
302C22×C4C2×C4⋊C411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4#12011yescharacteristic
306C22×C4#12011yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4C22⋊Q811yescharacteristic
314C22×C4C22⋊Q811yescharacteristic
315C22×C4#12311yescharacteristic
316C22×C4#12311yescharacteristic
317C22×C4#12311yescharacteristic
318C22×C4#12411yescharacteristic
319C22×C4#12711yescharacteristic
320C22×C4C2×D811yescharacteristic
321C22×C4C2×SD1612yes—
322C22×C4C2×Q1611yescharacteristic
323C22×C4#13511yescharacteristic
324C22×C4#13511yescharacteristic
325C22×C4#13512yes—
326C22×C4AGL(1,ℤ/8)11yescharacteristic
327C22×C4AGL(1,ℤ/8)11yescharacteristic
328C22×C4#13711yescharacteristic
329C22×C4#13711yescharacteristic
330C22×C4#13712yes—
331C22×C4#13712yes—
332C2×C8—24no—
333C2×C8—44no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—44no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—44no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—82no—
343C2×C8—81no—
344C2×C8—81no—
345C2×C8—81no—
346C2×C8—81no—
347C2×C8—81no—
348C2×C8—81no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—44no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—82no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4—81no—
375C4⋊C4—81no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—81no—
384C42—81no—
385C42—81no—
386C42—81no—
387C42C2×C22⋊C411yescharacteristic
388C42#11711yescharacteristic
389C42C4×D411yescharacteristic
390C42C4×D411yescharacteristic
391C42#12111yescharacteristic
392C42#12111yescharacteristic
393C42C22⋊Q811yescharacteristic
394C42C22⋊Q811yescharacteristic
395C42C22⋊Q811yescharacteristic
396C42C22⋊Q811yescharacteristic
397C42#12311yescharacteristic
398C42#12311yescharacteristic
399C42#12411yescharacteristic
400C42#12411yescharacteristic
401C42#12611yescharacteristic
402C42#12611yescharacteristic
403C23#38511yescharacteristic
404C23#39211yescharacteristic
405C23#39411yescharacteristic
406C23#44711yescharacteristic
407C23#44812yes—
408C23#44911yescharacteristic
409C23#45111yescharacteristic
410C23#45211yescharacteristic
411C23#45212yes—
412C23#45412yes—
413C23#45512yes—
414Q8—24no—
415Q8—24no—
416Q8—42no—
417Q8—42no—
418Q8—42no—
419Q8—42no—
420Q8—42no—
421Q8—42no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—162no—
458C2×C4—162no—
459C2×C4#38011yescharacteristic
460C2×C4#38511yescharacteristic
461C2×C4#38611yescharacteristic
462C2×C4#38811yescharacteristic
463C2×C4#38811yescharacteristic
464C2×C4#39011yescharacteristic
465C2×C4#39311yescharacteristic
466C2×C4#39411yescharacteristic
467C2×C4#39611yescharacteristic
468C2×C4#39611yescharacteristic
469C2×C4#39711yescharacteristic
470C2×C4#39911yescharacteristic
471C2×C4C2×#10711yescharacteristic
472C2×C4#42711yescharacteristic
473C2×C4#42811yescharacteristic
474C2×C4#42811yescharacteristic
475C2×C4C4×SD1612yes—
476C2×C4#44112yes—
477C2×C4#44711yescharacteristic
478C2×C4#44911yescharacteristic
479C2×C4#45111yescharacteristic
480C2×C4#45211yescharacteristic
481C2×C4#46112yes—
482C2×C4#46412yes—
483C2×C4#46512yes—
484C2×C4#46611yescharacteristic
485C2×C4#46611yescharacteristic
486C2×C4#46711yescharacteristic
487C2×C4#46711yescharacteristic
488C2×C4#46811yescharacteristic
489C2×C4#46811yescharacteristic
490C2×C4#46911yescharacteristic
491C2×C4#47011yescharacteristic
492C2×C4#47012yes—
493C2×C4#47512yes—
494C2×C4#47912yes—
495C2×C4#48011yescharacteristic
496C2×C4#48011yescharacteristic
497C2×C4#48211yescharacteristic
498C2×C4#48211yescharacteristic
499C2×C4#48312yes—
500C2×C4#48411yescharacteristic
501C2×C4#48411yescharacteristic
502C2×C4#48511yescharacteristic
503C2×C4#48511yescharacteristic
504C2×C4#48512yes—
505C2×C4#49211yescharacteristic
506C2×C4#49511yescharacteristic
507C2×C4#49711yescharacteristic
508C2×C4#49711yescharacteristic
509C8—44no—
510C8—82no—
511C8—82no—
512C22#143911yescharacteristic
513C22#162612yes—
514C22#162712yes—
515C22#163612yes—
516C22#163812yes—
517C22#164012yes—
518C22#164112yes—
519C22#164312yes—
520C22#164512yes—
521C22#185211yescharacteristic
522C22#185411yescharacteristic
523C22#186012yes—
524C22#186512yes—
525C22#199911yescharacteristic
526C22#200111yescharacteristic
527C22#200311yescharacteristic
528C22#200411yescharacteristic
529C22#201311yescharacteristic
530C22#201511yescharacteristic
531C22#203311yescharacteristic
532C22#203512yes—
533C22#203611yescharacteristic
534C22#204012yes—
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—44no—
540C4—82no—
541C4—84no—
542C4—162no—
543C4—162no—
544C4#180811yescharacteristic
545C4#180911yescharacteristic
546C4#185211yescharacteristic
547C4#185411yescharacteristic
548C4#192712yes—
549C4#196811yescharacteristic
550C4#197111yescharacteristic
551C4#203311yescharacteristic
552C4#203611yescharacteristic
553C4#206612yes—
554C4#207511yescharacteristic
555C4#207611yescharacteristic
556C4#209611yescharacteristic
557C4#209711yescharacteristic
558C2#889112yes—
559C2#889212yes—
560C2#1005111yescharacteristic
561C2#1006111yescharacteristic
562C2#1006311yescharacteristic
563C2#1007311yescharacteristic
564C2#1043312yes—
565C2#1043812yes—
566C2#1107911yescharacteristic
567C2#1109611yescharacteristic
568C2#1112211yescharacteristic
subgroupG/H
center, socle, prime radicalC24#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
26. C2×Q8⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18342
C2×#18032
C22×Q8⋊C82
#139262
#108492
C2×#12962
C2×#12792