Finite Group Databasebeta

← #415358

Subgroups — #415358

Order 512.

1386 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4#14160C211yesmaximal, characteristic
5#11384C212yesmaximal
6#11384C212yesmaximal
7#11293C212yesmaximal
8#11292C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2212yes—
14C22×#125C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17#2551—24no—
18#2551—24no—
19C2×#395—22no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#384C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1992—24no—
27#1989—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1441—21no—
41#1441—21no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×Q8—22no—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C2×C4⋊Q8—28no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—42no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—48no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125Q814yescomplemented
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×#97—22no—
85C2×#97—22no—
86C2×#97—22no—
87#395—216no—
88#389—28no—
89#389—24no—
90#389—24no—
91#384—22no—
92#384D414yes—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4D414yes—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D412yes—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—42no—
115#340—42no—
116#340Q818yescomplemented
117#336—42no—
118#336—42no—
119C22×Q8—28no—
120C22×Q8—44no—
121C22×Q8—44no—
122C22×Q8—44no—
123C22×Q8—44no—
124C22×Q8—82no—
125C22×Q8—82no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8C2×Q814yes—
154C4⋊Q8—48no—
155C4⋊Q8—48no—
156C4⋊Q8—432no—
157C4⋊Q8—48no—
158C4⋊Q8—48no—
159#125—28no—
160#125—28no—
161#125—44no—
162#125—44no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
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—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—81no—
207C2×C4⋊C4—81no—
208C2×C4⋊C4C2×D414yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×Q812yes—
212C2×C4⋊C4C2×Q812yes—
213C2×C4⋊C4C2×Q814yes—
214C2×C4⋊C4C2×Q814yes—
215C2×C4⋊C4#4114yes—
216C2×C42—24no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—44no—
224C2×C42—42no—
225C2×C42—81no—
226C2×C42—81no—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×Q812yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232C2×C42#4112yes—
233C4⋊C8—44no—
234#97—44no—
235#95—28no—
236#95—24no—
237#95—24no—
238#95—42no—
239#95—44no—
240#95—44no—
241#95—82no—
242#95—82no—
243C24C22×D411yescharacteristic
244C2×Q8—416no—
245C2×Q8—416no—
246C2×Q8—88no—
247C2×Q8—88no—
248C2×Q8—88no—
249C2×Q8—88no—
250C2×Q8—88no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
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—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4#12012yes—
306C22×C4#12012yes—
307C22×C4C22⋊Q812yes—
308C22×C4C22⋊Q812yes—
309C22×C4#12412yes—
310C22×C4#12412yes—
311C22×C4AGL(1,ℤ/8)14yes—
312C22×C4#13714yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×Q811yescharacteristic
316C22×C4C22×Q812yes—
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4112yes—
321C22×C4H(5,𝔽2)12yes—
322C22×C42−1+412yes—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C4⋊C4—28no—
341C4⋊C4—28no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—48no—
355C4⋊C4—416no—
356C4⋊C4—416no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—84no—
368C4⋊C4#12414yes—
369C4⋊C4C4⋊Q814yes—
370C42—48no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—84no—
376C42C22⋊Q814yes—
377C42C22⋊Q814yes—
378C42#12414yes—
379C42#12414yes—
380C23C2×#12011yescharacteristic
381C23C2×C22⋊Q811yescharacteristic
382C23C2×#12411yescharacteristic
383C23#54612yes—
384C23D4×Q812yes—
385C23C2×AGL(1,ℤ/8)12yes—
386C23C2×#13712yes—
387C23#57611yescharacteristic
388C23#57711yescharacteristic
389C23#57711yescharacteristic
390C23#57811yescharacteristic
391Q8—832no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4#39314yes—
453C2×C4#39314yes—
454C2×C4C2×C22⋊Q811yescharacteristic
455C2×C4C2×C22⋊Q811yescharacteristic
456C2×C4C2×#12411yescharacteristic
457C2×C4C2×#12411yescharacteristic
458C2×C4C2×#12411yescharacteristic
459C2×C4C2×C4⋊Q811yescharacteristic
460C2×C4#53312yes—
461C2×C4#53312yes—
462C2×C4#53511yescharacteristic
463C2×C4#53611yescharacteristic
464C2×C4#54212yes—
465C2×C4#54212yes—
466C2×C4#54612yes—
467C2×C4D4×Q812yes—
468C2×C4#55412yes—
469C2×C4#55612yes—
470C2×C4#55712yes—
471C2×C4Q8212yes—
472C8—44no—
473C8—81no—
474C8—81no—
475C22C2×#39311yescharacteristic
476C22#243411yescharacteristic
477C22#243511yescharacteristic
478C22#245112yes—
479C22#249212yes—
480C22#300311yescharacteristic
481C22#300411yescharacteristic
482C22#301211yescharacteristic
483C22#301311yescharacteristic
484C22#307912yes—
485C22#308311yescharacteristic
486C22#308411yescharacteristic
487C22#313611yescharacteristic
488C22#313711yescharacteristic
489C22#314112yes—
490C22#331012yes—
491C22#331212yes—
492C22#331812yes—
493C22#332012yes—
494C22#338412yes—
495C22#338512yes—
496C22#338612yes—
497C22#338712yes—
498C4—24no—
499C4—24no—
500C4—44no—
501C4—44no—
502C4—44no—
503C4—44no—
504C4—84no—
505C4—82no—
506C4—82no—
507C4—84no—
508C4#245112yes—
509C4#249212yes—
510C4#256711yescharacteristic
511C4#256811yescharacteristic
512C4#264011yescharacteristic
513C4#286211yescharacteristic
514C2#1519111yescharacteristic
515C2#2345211yescharacteristic
516C2#2345811yescharacteristic
517C2#2584411yescharacteristic
518C2#2584711yescharacteristic
519C2#2589111yescharacteristic
520C2#2589411yescharacteristic
521C2#2645612yes—
522C2#2645812yes—
523C2#2782512yes—
524C2#2782712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
62. C2×#125Q8semidirect
116. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#19922
C2×#19892
#141602
#113842
#112932
#112922
#108492
#108352
#107032