Finite Group Databasebeta

← #415255

Subgroups — #415255

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2660C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic
3C2×#1992C211yesmaximal, characteristic
4C2×#1991C211yesmaximal, characteristic
5C2×#1990C211yesmaximal, characteristic
6#14277C211yesmaximal, characteristic
7#11268C212yesmaximal
8#11267C212yesmaximal
9#11208C212yesmaximal
10#11207C212yesmaximal
11#10694C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2660—24no—
17#2660—24no—
18C2×#398—21no—
19C2×#398—21no—
20C2×#395—21no—
21C2×#395—21no—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1993—24no—
31#1992—24no—
32#1991—24no—
33#1990—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1444—21no—
45#1444—21no—
46#1442—21no—
47#1442—21no—
48C4⋊#95C2212yes—
49C23×Q8—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8D414yes—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125—22no—
63C2×#125D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#97—22no—
84C2×#97—22no—
85C2×#97—22no—
86C2×C4×C8—22no—
87C2×C4×C8—22no—
88C2×C4×C8—22no—
89#398—28no—
90#398—28no—
91#395—28no—
92#395—28no—
93#389—28no—
94#384—22no—
95#384—22no—
96#384—22no—
97#384—22no—
98#384—22no—
99#384—22no—
100#382—22no—
101#382—22no—
102#382—22no—
103#382—22no—
104#382—22no—
105#382D414yes—
106#376—22no—
107#376—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D411yescharacteristic
118C2×#95D411yescharacteristic
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—24no—
129#336—24no—
130#336Q818yescomplemented
131C22×Q8—24no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—82no—
135C22×Q8—82no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8C2×Q814yes—
162C22×C8C2×Q814yes—
163C4⋊Q8—28no—
164C4⋊Q8—28no—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167#125—28no—
168#125—28no—
169#125—44no—
170#125—44no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D414yes—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×Q814yes—
237C2×C42#4114yes—
238C2×C42#4112yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241#97—44no—
242C4×C8—44no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—24no—
252#95—24no—
253#95—44no—
254#95—82no—
255#95—82no—
256C24C22×D411yescharacteristic
257C2×Q8—48no—
258C2×Q8—84no—
259C2×Q8—84no—
260C2×Q8—88no—
261C2×Q8—88no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4#12012yes—
317C22×C4#12012yes—
318C22×C4#12112yes—
319C22×C4#12112yes—
320C22×C4#12412yes—
321C22×C4#12412yes—
322C22×C4#12412yes—
323C22×C4#12412yes—
324C22×C4C4⋊Q812yes—
325C22×C4C4⋊Q812yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×Q812yes—
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4112yes—
334C22×C42−1+412yes—
335C22×C42−1+412yes—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—28no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—44no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—84no—
374C4⋊C4—84no—
375C4⋊C4—88no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C42—24no—
381C42—24no—
382C42—24no—
383C42—24no—
384C42#12114yes—
385C42#12114yes—
386C42#12314yes—
387C42#12314yes—
388C23C2×#12011yescharacteristic
389C23C2×#12111yescharacteristic
390C23C2×#12411yescharacteristic
391C23C2×#12411yescharacteristic
392C23C2×C4⋊Q811yescharacteristic
393C23#53611yescharacteristic
394C23#53611yescharacteristic
395C23#57612yes—
396C23#57712yes—
397C23#57712yes—
398C23#57812yes—
399Q8—816no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4#39414yes—
463C2×C4#39414yes—
464C2×C4C2×#12111yescharacteristic
465C2×C4C2×#12111yescharacteristic
466C2×C4C2×#12311yescharacteristic
467C2×C4C2×#12311yescharacteristic
468C2×C4#53312yes—
469C2×C4#53312yes—
470C2×C4#54112yes—
471C2×C4#54112yes—
472C2×C4#54712yes—
473C2×C4#54812yes—
474C2×C4#54812yes—
475C2×C4D4×Q812yes—
476C2×C4D4×Q812yes—
477C2×C4D4×Q812yes—
478C2×C4#55612yes—
479C2×C4#55712yes—
480C8—44no—
481C8—44no—
482C22C2×#39411yescharacteristic
483C22#243411yescharacteristic
484C22#243611yescharacteristic
485C22#254611yescharacteristic
486C22#254711yescharacteristic
487C22#264211yescharacteristic
488C22#286211yescharacteristic
489C22#300811yescharacteristic
490C22#300911yescharacteristic
491C22#301011yescharacteristic
492C22#301111yescharacteristic
493C22#304512yes—
494C22#304612yes—
495C22#313612yes—
496C22#313612yes—
497C22#313712yes—
498C22#313712yes—
499C22#316312yes—
500C22#316412yes—
501C22#324112yes—
502C22#324212yes—
503C22#324312yes—
504C22#324412yes—
505C4—24no—
506C4—24no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—84no—
512C4—82no—
513C4—82no—
514C4—82no—
515C4—82no—
516C4—162no—
517C4#242912yes—
518C4#242912yes—
519C4#244612yes—
520C4#244612yes—
521C2#1518811yescharacteristic
522C2#2350911yescharacteristic
523C2#2351011yescharacteristic
524C2#2582011yescharacteristic
525C2#2582111yescharacteristic
526C2#2591411yescharacteristic
527C2#2591511yescharacteristic
528C2#2869812yes—
529C2#2873012yes—
530C2#3023212yes—
531C2#3329212yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
130. #336Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26602
C2×#19932
C2×#19922
C2×#19912
C2×#19902
#142772
#112682
#112672
#112082
#112072
#106942