Finite Group Databasebeta

← #365477

Subgroups — #365477

Order 512.

1314 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2370C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1947C211yesmaximal, characteristic
5#13910C211yesmaximal, characteristic
6#11328C212yesmaximal
7#11328C212yesmaximal
8#10885C212yesmaximal
9#10884C212yesmaximal
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16#2370—24no—
17#2370—24no—
18C2×#389—21no—
19C2×#389C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#387—21no—
24C2×#387—21no—
25C2×#384—21no—
26C2×#384C2211yescharacteristic
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1992—24no—
34#1989C418yescomplemented
35#1947—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1444—21no—
46#1444—21no—
47C4⋊#95—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125C2×C414yes—
54C2×#125D414yes—
55C2×#125Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C42—21no—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C2×#107—24no—
80C2×#107—42no—
81C2×#107—42no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8C2×C414yes—
85C2×#97—22no—
86C2×#97—22no—
87C2×#97—22no—
88#389—28no—
89#389—22no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—24no—
94#389—24no—
95#389—42no—
96#389—42no—
97#387—28no—
98#387—28no—
99#384—22no—
100#384—22no—
101#384—44no—
102C4×C4⋊C4—28no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2×C414yes—
112C4×C4⋊C4D414yes—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95D411yescharacteristic
122C2×#95D411yescharacteristic
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340C2×C418yes—
128#340C2×C418yes—
129#340Q818yes—
130#336—42no—
131#336—42no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
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×C8C22×C414yes—
158C22×C8C2×Q814yes—
159#125—28no—
160#125—28no—
161#125—28no—
162#125C4⋊C4116yescomplemented
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C414yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×Q812yes—
219C2×C4⋊C4C2×Q812yes—
220C2×C4⋊C4C2×Q814yes—
221C2×C4⋊C4C2×Q814yes—
222C2×C4⋊C4#4114yes—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—82no—
240C2×C42—82no—
241C2×C42C22×C412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×Q812yes—
245C2×C42#4112yes—
246C2×C42#4112yes—
247#107—48no—
248C4⋊C8—28no—
249#97—44no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—22no—
257#95—22no—
258#95—44no—
259#95—44no—
260C24C22×D411yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4C4×D412yes—
322C22×C4C4×D412yes—
323C22×C4#12012yes—
324C22×C4#12012yes—
325C22×C4C22⋊Q812yes—
326C22×C4C22⋊Q812yes—
327C22×C4AGL(1,ℤ/8)12yes—
328C22×C4#13712yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×Q811yescharacteristic
333C22×C4C22×Q812yes—
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4C2×#4111yescharacteristic
337C22×C4H(5,𝔽2)12yes—
338C22×C42−1+412yes—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—216no—
365C4⋊C4—216no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—48no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—88no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4—88no—
381C4⋊C4C2×C4⋊C414yes—
382C4⋊C4C2×C4⋊C414yes—
383C4⋊C4C4×Q818yes—
384C4⋊C4C4⋊Q814yes—
385C42—48no—
386C42—48no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—42no—
392C42—42no—
393C42—84no—
394C42—84no—
395C42C2×C4⋊C414yes—
396C42C4×D414yes—
397C42C22⋊Q814yes—
398C42#12314yes—
399C23C2×C4×D411yescharacteristic
400C23C2×#12011yescharacteristic
401C23C2×C22⋊Q811yescharacteristic
402C23#54612yes—
403C23D4×Q812yes—
404C23C2×AGL(1,ℤ/8)11yescharacteristic
405C23C2×#13711yescharacteristic
406C23#57511yescharacteristic
407C23#57511yescharacteristic
408C23#57612yes—
409C23#57812yes—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—28no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4#38514yes—
477C2×C4#38514yes—
478C2×C4C22×C4⋊C411yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×C4×Q812yes—
481C2×C4#52012yes—
482C2×C4C2×C22⋊Q811yescharacteristic
483C2×C4C2×#12311yescharacteristic
484C2×C4C2×C4⋊Q811yescharacteristic
485C2×C4#53211yescharacteristic
486C2×C4#53312yes—
487C2×C4#53511yescharacteristic
488C2×C4#53611yescharacteristic
489C2×C4#54212yes—
490C2×C4D4×Q812yes—
491C2×C4#55012yes—
492C2×C4#55212yes—
493C2×C4#55412yes—
494C2×C4#55712yes—
495C2×C4Q8212yes—
496C8—44no—
497C8—44no—
498C22C2×#38511yescharacteristic
499C22#236012yes—
500C22#243111yescharacteristic
501C22#243411yescharacteristic
502C22#245112yes—
503C22C4×AGL(1,ℤ/8)11yescharacteristic
504C22C4×#13711yescharacteristic
505C22#295112yes—
506C22#300611yescharacteristic
507C22#300711yescharacteristic
508C22#300811yescharacteristic
509C22#301111yescharacteristic
510C22#307911yescharacteristic
511C22#308011yescharacteristic
512C22#308312yes—
513C22#330912yes—
514C22#331212yes—
515C22#331512yes—
516C22#331812yes—
517C22#338512yes—
518C22#338612yes—
519C22#338912yes—
520C22#339012yes—
521C4—24no—
522C4—24no—
523C4—44no—
524C4—44no—
525C4—48no—
526C4—42no—
527C4—42no—
528C4—44no—
529C4—81no—
530C4—82no—
531C4—81no—
532C4—162no—
533C4C4×C4⋊Q811yescharacteristic
534C4#232611yescharacteristic
535C4Q8×C4⋊C412yes—
536C4#242312yes—
537C4#256511yescharacteristic
538C4#256711yescharacteristic
539C2#1397611yescharacteristic
540C2#2087511yescharacteristic
541C2#2088111yescharacteristic
542C2#2297412yes—
543C2#2298012yes—
544C2#2558111yescharacteristic
545C2#2558411yescharacteristic
546C2#2584911yescharacteristic
547C2#2585011yescharacteristic
548C2#2645612yes—
549C2#2645912yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
34. #1989C4semidirect
162. #125C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23702
C2×#19922
C2×#19892
C2×#19472
#139102
#113282
#108852
#108842
#108352
#107032