Finite Group Databasebeta

← #403803

Subgroups — #403803

Order 512.

1446 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic, complemented
2C2×#2042C212yesmaximal
3C2×#2026C211yesmaximal, characteristic, complemented
4C2×#2025C211yesmaximal, characteristic
5#14708C211yesmaximal, characteristic
6#11208C212yesmaximal, complemented
7#11052C211yesmaximal, characteristic, complemented
8#11051C211yesmaximal, characteristic
9#10932C212yesmaximal, complemented
10#10931C212yesmaximal
11#10795C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C4×#125—28no—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389C2212yes—
24C2×#387—22no—
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C4×C4⋊C4—22no—
28C2×C4×C4⋊C4C2212yes—
29C2×C43C2211yescharacteristic
30#2042—28no—
31#2026—24no—
32#2025—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1437—22no—
42#1437—22no—
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#125—24no—
48C2×#125—24no—
49C2×#125—22no—
50C2×#125D414yes—
51C2×#124—22no—
52C2×#124—22no—
53C2×#124—24no—
54C2×#124—42no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C42—22no—
66C22×C42D412yes—
67C22×C42D412yes—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—42no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—42no—
83#391—22no—
84#391—22no—
85#391—24no—
86#390—22no—
87#390—22no—
88#390—24no—
89#389—24no—
90#389D418yes—
91#387—216no—
92#382—24no—
93#382—42no—
94C4×C4⋊C4—216no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—24no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C43—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109#336—22no—
110#336—22no—
111#336—24no—
112#336—24no—
113C25—21no—
114C22×Q8—21no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8#4112yes—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4#4112yes—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
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×C8—22no—
158#125—216no—
159#125—216no—
160#124—48no—
161#124—48no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×D414yes—
186C2×C4⋊C4C2×D414yes—
187C2×C4⋊C4#4114yes—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—44no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—28no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—44no—
214C2×C42—44no—
215C2×C42C2×D414yes—
216C2×C42C2×D412yes—
217C2×C42#4112yes—
218C2×C42#4114yes—
219#103—48no—
220#103—48no—
221#102—48no—
222#102—48no—
223#95—28no—
224#95—28no—
225#95—28no—
226#95—44no—
227#95—44no—
228#95—42no—
229#95—42no—
230C24—22no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24—41no—
235C24—41no—
236C24—41no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24C22×D411yescharacteristic
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—42no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4#12014yes—
325C22×C4#12114yes—
326C22×C4#12412yes—
327C22×C4#12412yes—
328C22×C4#13514yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D412yes—
331C22×C4C2×#4112yes—
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4H(5,𝔽2)12yes—
337C22×C42−1+411yescharacteristic
338C22×C42−1+411yescharacteristic
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
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—232no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—416no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4#12118yes—
367C22⋊C4—416no—
368C22⋊C4—84no—
369C22⋊C4—88no—
370C22⋊C4—84no—
371C42—24no—
372C42—24no—
373C42—24no—
374C42—24no—
375C42—24no—
376C42—24no—
377C42—48no—
378C42—416no—
379C42—48no—
380C23—44no—
381C23—44no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—82no—
386C23—81no—
387C23—82no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—81no—
403C23C2×#12011yescharacteristic
404C23C2×#12111yescharacteristic
405C23#52511yescharacteristic
406C23#52511yescharacteristic
407C23C2×#12411yescharacteristic
408C23#53511yescharacteristic
409C23#53611yescharacteristic
410C23C2×#13512yes—
411C23#57512yes—
412C23#57512yes—
413C23#57512yes—
414Q8—44no—
415Q8—44no—
416Q8—82no—
417Q8—82no—
418D4—44no—
419D4—44no—
420D4—82no—
421D4—82no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—82no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4#48814yes—
477C2×C4C2×#12112yes—
478C2×C4#52512yes—
479C2×C4#52912yes—
480C2×C4#52912yes—
481C2×C4#54212yes—
482C2×C4#54212yes—
483C2×C4#54612yes—
484C2×C4#54612yes—
485C2×C4#54811yescharacteristic
486C2×C4#54811yescharacteristic
487C2×C4#54812yes—
488C2×C4#54812yes—
489C2×C4#55012yes—
490C2×C4#55211yescharacteristic
491C2×C4#55211yescharacteristic
492C2×C4#55912yes—
493C8—82no—
494C8—82no—
495C22—82no—
496C22—82no—
497C22—82no—
498C22—84no—
499C22—84no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22—162no—
505C22—162no—
506C22#239811yescharacteristic
507C22#240411yescharacteristic
508C22#243611yescharacteristic
509C22#253611yescharacteristic
510C22#254811yescharacteristic
511C22#263211yescharacteristic
512C22#284711yescharacteristic
513C22#300212yes—
514C22#300512yes—
515C22#303512yes—
516C22#304012yes—
517C22#310212yes—
518C22#310212yes—
519C22#311112yes—
520C22#311112yes—
521C22C2×#48811yescharacteristic
522C22#313511yescharacteristic
523C22#314011yescharacteristic
524C22#314211yescharacteristic
525C22#321212yes—
526C22#321912yes—
527C22#323212yes—
528C22#324012yes—
529C4—24no—
530C4—42no—
531C4—44no—
532C4—44no—
533C4—48no—
534C4—82no—
535C4—84no—
536C4—161no—
537C4—161no—
538C4#242612yes—
539C4#242612yes—
540C4#245812yes—
541C4#245812yes—
542C2—162no—
543C2#1488311yescharacteristic
544C2#2381811yescharacteristic
545C2#2382611yescharacteristic
546C2#2393812yes—
547C2#2590711yescharacteristic
548C2#2590911yescharacteristic
549C2#2820812yes—
550C2#2875211yescharacteristic
551C2#2875711yescharacteristic
552C2#3006512yes—
553C2#3291012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C4×#125C2semidirect
3. C2×#2026C2semidirect
6. #11208C2semidirect
7. #11052C2semidirect
9. #10932C2semidirect
11. #10795C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1252
C2×#20422
C2×#20262
C2×#20252
#147082
#112082
#110522
#110512
#109322
#109312
#107952