Finite Group Databasebeta

← #399179

Subgroups — #399179

Order 512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#2095C211yesmaximal, characteristic
3C2×#1878C211yesmaximal, characteristic
4C2×#1878C212yesmaximal
5C2×#1863C212yesmaximal
6#15193C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic
8#11346C211yesmaximal, characteristic
9#11328C212yesmaximal
10#11067C212yesmaximal
11#10789C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C22×#107C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2212yes—
20Q8×C4⋊C4—24no—
21Q8×C4⋊C4—24no—
22C2×#391C2211yescharacteristic
23C2×#391C2212yes—
24C2×#389—22no—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#384—21no—
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31#2095—24no—
32#1878—28no—
33#1878—24no—
34#1863—28no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43#1445—22no—
44#1445—21no—
45#1444—21no—
46C4⋊#95C2211yescharacteristic
47C23×Q8C2311yescharacteristic
48C23×Q8C2311yescharacteristic
49C22×Q16—22no—
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—22no—
53C22×Q16Q814yes—
54C23×C8C2311yescharacteristic
55C23×C8C2312yes—
56C23×C8C2311yescharacteristic
57C2×C4×Q8—24no—
58C2×C4×Q8—24no—
59C2×C4×Q8—24no—
60C2×C4×Q8—24no—
61C2×C4×Q8—22no—
62C2×C4×Q8—24no—
63C2×C4×Q8—24no—
64C2×C4×Q8—42no—
65C2×C4×Q8—42no—
66C2×C4×Q8D414yes—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C42—21no—
80C22×C42—22no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—42no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—22no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—22no—
93C2×#103—24no—
94C2×#103—42no—
95C2×#103D418yes—
96#391—24no—
97#391—22no—
98#391—22no—
99#391—42no—
100#389—216no—
101#389—24no—
102#389—24no—
103#389—24no—
104#389—42no—
105#384—28no—
106#384—22no—
107#384D414yes—
108C4×C4⋊C4—28no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95D412yes—
120#340—24no—
121#340—24no—
122#340—22no—
123#340—24no—
124#336—24no—
125#336—24no—
126#336—22no—
127#336—22no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—21no—
142C22×Q8C2×D412yes—
143C22×Q8C2×D414yes—
144C22×Q8C2×Q812yes—
145C22×Q8C2×Q812yes—
146C22×Q8#4114yes—
147C22×Q8#4112yes—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C2×Q16—216no—
160C2×Q16—28no—
161C2×Q16—48no—
162C2×Q16—44no—
163C2×Q16—44no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8C2×D414yes—
180C22×C8C2×Q812yes—
181C4×Q8—216no—
182C4×Q8—28no—
183C4×Q8—28no—
184C4×Q8—48no—
185C4×Q8—48no—
186C4×Q8—48no—
187C4×Q8—48no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—81no—
220C2×C4⋊C4—81no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4Q1614yescomplemented
225C2×C4⋊C4C2×D414yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4#4114yes—
229C2×C42—28no—
230C2×C42—24no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—28no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244#107—48no—
245#103—216no—
246#103—48no—
247#103—44no—
248#103—44no—
249#95—28no—
250#95—24no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—42no—
255#95—42no—
256#95—42no—
257#95—81no—
258C24C22×D411yescharacteristic
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—24no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8C22⋊Q814yes—
291C2×Q8C22⋊Q818yes—
292C2×Q8#12318yes—
293C2×Q8#12314yes—
294C2×Q8#12714yes—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4#12012yes—
349C22×C4#12012yes—
350C22×C4C2×Q1614yes—
351C22×C4C2×Q1612yes—
352C22×C4#13712yes—
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D412yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×Q811yescharacteristic
357C22×C4C2×#4112yes—
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4112yes—
360C22×C4H(5,𝔽2)12yes—
361C22×C42−1+411yescharacteristic
362C22×C42−1+411yescharacteristic
363Q16—416no—
364C2×C8—24no—
365C2×C8—22no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—42no—
380C2×C8—42no—
381C4⋊C4—216no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—28no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—44no—
390C4⋊C4—48no—
391C4⋊C4—416no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—84no—
400C4⋊C4—88no—
401C4⋊C4—88no—
402C4⋊C4#12314yes—
403C42—216no—
404C42—22no—
405C42—22no—
406C42—24no—
407C42—416no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—42no—
412C23C2×#12011yescharacteristic
413C23#52512yes—
414C23D4212yes—
415C23#54811yescharacteristic
416C23D4×Q811yescharacteristic
417C23C22×Q1611yescharacteristic
418C23C2×#13711yescharacteristic
419C23#57512yes—
420C23#57611yescharacteristic
421C23#57711yescharacteristic
422C23#57812yes—
423Q8—24no—
424Q8—28no—
425Q8—28no—
426Q8—24no—
427Q8—24no—
428Q8—42no—
429Q8—44no—
430Q8—42no—
431Q8—81no—
432Q8—82no—
433Q8—81no—
434Q8#39918yes—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—28no—
442C2×C4—28no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4#45114yes—
493C2×C4C2×C22⋊Q812yes—
494C2×C4C2×C22⋊Q811yescharacteristic
495C2×C4C2×#12311yescharacteristic
496C2×C4C2×#12312yes—
497C2×C4C2×#12311yescharacteristic
498C2×C4#52512yes—
499C2×C4C2×#12711yescharacteristic
500C2×C4#53511yescharacteristic
501C2×C4#53611yescharacteristic
502C2×C4#53912yes—
503C2×C4#53912yes—
504C2×C4D4212yes—
505C2×C4#54712yes—
506C2×C4#54712yes—
507C2×C4#54811yescharacteristic
508C2×C4D4×Q811yescharacteristic
509C2×C4#55312yes—
510C2×C4#55412yes—
511C8—42no—
512C8—44no—
513C8—81no—
514C22#240911yescharacteristic
515C22#243311yescharacteristic
516C22#244312yes—
517C22#244912yes—
518C22#284911yescharacteristic
519C22C2×#45111yescharacteristic
520C22#300511yescharacteristic
521C22#301111yescharacteristic
522C22#301211yescharacteristic
523C22#310412yes—
524C22#311712yes—
525C22#312012yes—
526C22#312412yes—
527C22#326812yes—
528C22#327112yes—
529C22#327612yes—
530C22D4×Q1612yes—
531C22#336311yescharacteristic
532C22#337511yescharacteristic
533C22#337712yes—
534C22Q8×Q1611yescharacteristic
535C22#338711yescharacteristic
536C22#339012yes—
537C4—28no—
538C4—24no—
539C4—44no—
540C4—42no—
541C4—44no—
542C4—84no—
543C4—82no—
544C4—84no—
545C4—82no—
546C4—161no—
547C4C2×#39911yescharacteristic
548C4#244312yes—
549C4#244912yes—
550C4#248711yescharacteristic
551C4#250111yescharacteristic
552C4#262311yescharacteristic
553C2#1477711yescharacteristic
554C2#2433311yescharacteristic
555C2#2434411yescharacteristic
556C2#2576011yescharacteristic
557C2#2576511yescharacteristic
558C2#2617112yes—
559C2#2617712yes—
560C2#2638112yes—
561C2#2639512yes—
562C2#3301011yescharacteristic
563C2#3301711yescharacteristic
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
224. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#20952
C2×#18782
C2×#18632
#151932
#113572
#113462
#113282
#110672
#107892