Finite Group Databasebeta

← #455425

Subgroups — #455425

Order 512.

1460 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×#2084C212yesmaximal, complemented
3C2×#1864C211yesmaximal, characteristic
4#15143C211yesmaximal, characteristic, complemented
5#13920C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic, complemented
7#11255C211yesmaximal, characteristic, complemented
8#11228C211yesmaximal, characteristic, complemented
9#11220C212yesmaximal, complemented
10#11189C212yesmaximal
11#11049C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#476—28no—
15C2×#476—24no—
16C22×#106C2211yescharacteristic
17C22×#106C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2211yescharacteristic
21C22×#102C2211yescharacteristic
22C2×#395—22no—
23C2×#390C2212yes—
24C2×#389C2211yescharacteristic
25C2×#389C2212yes—
26C2×#384C2212yes—
27C2×#380—21no—
28#2084—28no—
29#1864—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2212yes—
38#1444—21no—
39#1444—21no—
40#1439—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C22⋊#95—21no—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—42no—
55C2×C4×D4—24no—
56C2×C4×D4—22no—
57C2×C4×D4—42no—
58C2×C4×D4D414yes—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74#476—232no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—22no—
79C2×#106D418yes—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—22no—
82C2×#102—22no—
83C2×#102—24no—
84C2×#102—22no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—22no—
88C2×#102—42no—
89C2×#102Q814yescomplemented
90#395—48no—
91#390—24no—
92#390—42no—
93#389—24no—
94#389—22no—
95#389—22no—
96#389—42no—
97#384—24no—
98#384—42no—
99#380—44no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95D412yes—
107C2×#95D411yescharacteristic
108C2×#95D411yescharacteristic
109#340—24no—
110#340—24no—
111#340—22no—
112#340—42no—
113#340—42no—
114#336—24no—
115#336—22no—
116#336—42no—
117C25—21no—
118C22×Q8—44no—
119C22×Q8—44no—
120C22×Q8—42no—
121C22×Q8—81no—
122C22×Q8—81no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4C2×D412yes—
131C22×D4C2×Q812yes—
132C22×D4#4112yes—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—44no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—82no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8C2×D414yes—
164C22×C8C2×Q812yes—
165C4⋊Q8—28no—
166C4⋊Q8—48no—
167C4⋊Q8—44no—
168C4×D4—28no—
169C4×D4—44no—
170C4×D4—48no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C2×D414yes—
207C2×C4⋊C4C2×D414yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×Q812yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—82no—
217C2×C22⋊C4—82no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42C2×D412yes—
227C2×C42#4112yes—
228#106—216no—
229#106—44no—
230C4⋊C8—28no—
231#102—216no—
232#102—28no—
233#102—44no—
234#95—24no—
235#95—22no—
236#95—44no—
237#95—44no—
238#95—42no—
239#95—42no—
240#95—44no—
241#95—81no—
242#95SD1614yes—
243C24—22no—
244C24—22no—
245C24—22no—
246C24—42no—
247C24—42no—
248C24—41no—
249C24—42no—
250C24—41no—
251C24—41no—
252C24—42no—
253C24—41no—
254C24C22×D411yescharacteristic
255C2×Q8—48no—
256C2×Q8—84no—
257C2×Q8—84no—
258C2×Q8—88no—
259C2×Q8—88no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4C22⋊Q814yes—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—48no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—48no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—84no—
328C22×C4—88no—
329C22×C4—88no—
330C22×C4#12112yes—
331C22×C4#12112yes—
332C22×C4C2×SD1614yes—
333C22×C4C2×SD1612yes—
334C22×C4AGL(1,ℤ/8)14yes—
335C22×C4#13712yes—
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D412yes—
338C22×C4C22×Q811yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)12yes—
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C42−1+412yes—
344C22×C42−1+411yescharacteristic
345C22×C42−1+412yes—
346C2×C8—24no—
347C2×C8—22no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—41no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—41no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—42no—
362C2×C8—42no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—48no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—88no—
382C4⋊C4—88no—
383C4⋊C4—84no—
384C4⋊C4—88no—
385C4⋊C4#12014yes—
386C22⋊C4—48no—
387C22⋊C4—88no—
388C22⋊C4—88no—
389C42—22no—
390C42—22no—
391C42—24no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—42no—
396C23—24no—
397C23—42no—
398C23—42no—
399C23—44no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—44no—
404C23—44no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—44no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23C2×#12111yescharacteristic
420C23#53612yes—
421C23D4212yes—
422C23#54611yescharacteristic
423C23D4×Q811yescharacteristic
424C23C22×SD1611yescharacteristic
425C23C2×AGL(1,ℤ/8)12yes—
426C23C2×#13711yescharacteristic
427C23#57611yescharacteristic
428C23#57711yescharacteristic
429C23#57712yes—
430Q8—816no—
431D4—24no—
432D4—24no—
433D4—42no—
434D4—44no—
435D4—44no—
436D4—42no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—816no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—81no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4#46114yes—
491C2×C4C2×#12011yescharacteristic
492C2×C4C2×C22⋊Q811yescharacteristic
493C2×C4#53411yescharacteristic
494C2×C4#53611yescharacteristic
495C2×C4#53612yes—
496C2×C4#53611yescharacteristic
497C2×C4#53712yes—
498C2×C4#53711yescharacteristic
499C2×C4#53912yes—
500C2×C4#54112yes—
501C2×C4#54412yes—
502C2×C4#54612yes—
503C2×C4#54812yes—
504C2×C4D4×Q812yes—
505C2×C4#55211yescharacteristic
506C2×C4#55411yescharacteristic
507C2×C4#56212yes—
508C2×C4#56312yes—
509C8—44no—
510C8—42no—
511C8—81no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—82no—
516C22—82no—
517C22—84no—
518C22—84no—
519C22—82no—
520C22—82no—
521C22—84no—
522C22—84no—
523C22#241011yescharacteristic
524C22#241711yescharacteristic
525C22#262111yescharacteristic
526C22#267712yes—
527C22#268112yes—
528C22C2×#46111yescharacteristic
529C22#303811yescharacteristic
530C22#304211yescharacteristic
531C22#304511yescharacteristic
532C22#322712yes—
533C22#323812yes—
534C22#324112yes—
535C22#324412yes—
536C22#326512yes—
537C22#326612yes—
538C22#327412yes—
539C22D4×SD1612yes—
540C22#329511yescharacteristic
541C22#330612yes—
542C22#333111yescharacteristic
543C22Q8×SD1611yescharacteristic
544C22#338412yes—
545C22#338511yescharacteristic
546C4—24no—
547C4—44no—
548C4—42no—
549C4—44no—
550C4—44no—
551C4—84no—
552C4—82no—
553C4—161no—
554C4—162no—
555C4#240911yescharacteristic
556C4#261611yescharacteristic
557C4#262711yescharacteristic
558C4#269012yes—
559C4#270412yes—
560C4#281411yescharacteristic
561C2—84no—
562C2#1626411yescharacteristic
563C2#2435811yescharacteristic
564C2#2437011yescharacteristic
565C2#2481511yescharacteristic
566C2#2482911yescharacteristic
567C2#2986811yescharacteristic
568C2#2987411yescharacteristic
569C2#3095412yes—
570C2#3097012yes—
571C2#3111812yes—
572C2#3113612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2084C2semidirect
4. #15143C2semidirect
6. #11384C2semidirect
7. #11255C2semidirect
8. #11228C2semidirect
9. #11220C2semidirect
89. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#20842
C2×#18642
#151432
#139202
#113842
#112552
#112282
#112202
#111892
#110492