Finite Group Databasebeta

← #468065

Subgroups — #468065

Order 512.

1350 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic
3C2×#1878C211yesmaximal, characteristic
4#15193C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic
6#11346C211yesmaximal, characteristic
7#11207C212yesmaximal
8#11067C212yesmaximal
9#11050C212yesmaximal
10#11024C212yesmaximal
11#10789C211yesmaximal, characteristic
12C23×Q16C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19#2567—24no—
20#2567—24no—
21C2×#395—22no—
22C2×#391C2212yes—
23C2×#391C2211yescharacteristic
24C2×#391C2212yes—
25C2×#389—21no—
26C2×#389—21no—
27C2×#389C2212yes—
28C2×#384C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2024—24no—
31#1878—24no—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1445—22no—
42#1445—21no—
43#1444—21no—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×Q8C2311yescharacteristic
48C22×Q16—22no—
49C22×Q16—22no—
50C22×Q16—42no—
51C22×Q16—42no—
52C22×Q16Q814yescomplemented
53C23×C8C2312yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8D414yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—42no—
62C2×C4×Q8—42no—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C42—21no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78C2×#103—22no—
79C2×#103—24no—
80C2×#103—22no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88#395—216no—
89#391—24no—
90#391—24no—
91#391—24no—
92#391—22no—
93#391—22no—
94#391—42no—
95#389—28no—
96#389—28no—
97#389—24no—
98#389—42no—
99#384—24no—
100#384—24no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95D412yes—
113#340—22no—
114#340—24no—
115#340—42no—
116#340—42no—
117#340—42no—
118#336—22no—
119#336—22no—
120#336—42no—
121#336—42no—
122C22×Q8—24no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—21no—
139C22×Q8—44no—
140C22×Q8—81no—
141C22×Q8—82no—
142C22×Q8—81no—
143C22×Q8C2×D412yes—
144C22×Q8C2×Q812yes—
145C22×Q8C2×Q812yes—
146C22×Q8#4112yes—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C2×Q16—28no—
159C2×Q16—48no—
160C2×Q16—44no—
161C2×Q16—44no—
162C2×Q16—48no—
163C22×C8—21no—
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—22no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—22no—
179C22×C8C2×Q812yes—
180C4⋊Q8—48no—
181C4⋊Q8—44no—
182C4⋊Q8—44no—
183C4×Q8—44no—
184C4×Q8—44no—
185C4×Q8—48no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
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—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4Q1614yescomplemented
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4#4114yes—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—44no—
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236#103—48no—
237#103—44no—
238#103—48no—
239#103—44no—
240#95—28no—
241#95—28no—
242#95—24no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—42no—
247#95—42no—
248#95—42no—
249#95—81no—
250C24C22×D411yescharacteristic
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—48no—
267C2×Q8—42no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—41no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—42no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—84no—
288C2×Q8—84no—
289C2×Q8—88no—
290C2×Q8—88no—
291C2×Q8C22⋊Q814yes—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4#12012yes—
345C22×C4#12012yes—
346C22×C4C2×Q1614yes—
347C22×C4C2×Q1612yes—
348C22×C4#13714yes—
349C22×C4#13712yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×Q811yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4112yes—
355C22×C4H(5,𝔽2)12yes—
356C22×C4H(5,𝔽2)12yes—
357C22×C4H(5,𝔽2)12yes—
358C22×C42−1+411yescharacteristic
359C22×C42−1+411yescharacteristic
360Q16—416no—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—48no—
387C4⋊C4—48no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—42no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C4⋊C4—88no—
399C42—22no—
400C42—22no—
401C42—24no—
402C42—48no—
403C42—42no—
404C42—42no—
405C42—42no—
406C42—42no—
407C23C2×#12011yescharacteristic
408C23#53512yes—
409C23#54612yes—
410C23#54811yescharacteristic
411C23D4×Q811yescharacteristic
412C23C22×Q1611yescharacteristic
413C23C2×#13711yescharacteristic
414C23C2×#13712yes—
415C23#57611yescharacteristic
416C23#57712yes—
417C23#57711yescharacteristic
418Q8—24no—
419Q8—24no—
420Q8—42no—
421Q8—42no—
422Q8—44no—
423Q8—44no—
424Q8—42no—
425Q8—42no—
426Q8—44no—
427Q8—816no—
428Q8—81no—
429Q8—81no—
430Q8—82no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4#45114yes—
489C2×C4C2×C22⋊Q811yescharacteristic
490C2×C4#53211yescharacteristic
491C2×C4#53411yescharacteristic
492C2×C4#53411yescharacteristic
493C2×C4#53412yes—
494C2×C4#53511yescharacteristic
495C2×C4#53611yescharacteristic
496C2×C4#53812yes—
497C2×C4#53912yes—
498C2×C4#54212yes—
499C2×C4#54312yes—
500C2×C4#54811yescharacteristic
501C2×C4D4×Q811yescharacteristic
502C2×C4#55012yes—
503C2×C4#55112yes—
504C2×C4#55412yes—
505C2×C4#56212yes—
506C2×C4#56212yes—
507C8—42no—
508C8—81no—
509C8—82no—
510C22#240911yescharacteristic
511C22#243111yescharacteristic
512C22#261311yescharacteristic
513C22#269212yes—
514C22#269612yes—
515C22C2×#45111yescharacteristic
516C22#300411yescharacteristic
517C22#300911yescharacteristic
518C22#301211yescharacteristic
519C22#320912yes—
520C22#321812yes—
521C22#322212yes—
522C22#322412yes—
523C22#329912yes—
524C22#331412yes—
525C22#331612yes—
526C22#333112yes—
527C22#336311yescharacteristic
528C22#336912yes—
529C22#337511yescharacteristic
530C22Q8×Q1611yescharacteristic
531C22#338711yescharacteristic
532C22#338712yes—
533C4—24no—
534C4—42no—
535C4—44no—
536C4—44no—
537C4—44no—
538C4—84no—
539C4—82no—
540C4—84no—
541C4—82no—
542C4—161no—
543C4#256411yescharacteristic
544C4#262611yescharacteristic
545C4#266311yescharacteristic
546C4#273012yes—
547C4#273212yes—
548C4#281611yescharacteristic
549C2#1660011yescharacteristic
550C2#2433311yescharacteristic
551C2#2434511yescharacteristic
552C2#2557311yescharacteristic
553C2#2557911yescharacteristic
554C2#2962711yescharacteristic
555C2#2963211yescharacteristic
556C2#3152412yes—
557C2#3153612yes—
558C2#3164012yes—
559C2#3165012yes—
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
52. C22×Q16Q8semidirect
220. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#20242
C2×#18782
#151932
#113842
#113462
#112072
#110672
#110502
#110242
#107892