Finite Group Databasebeta

← #473902

Subgroups — #473902

Order 512.

1182 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2024C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3#14282C211yesmaximal, characteristic, complemented
4#14273C211yesmaximal, characteristic
5#11357C212yesmaximal, complemented
6#11190C212yesmaximal, complemented
7#11189C212yesmaximal
8#11052C211yesmaximal, characteristic, complemented
9#11051C211yesmaximal, characteristic
10#11009C212yesmaximal
11#10796C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×#376C2211yescharacteristic
25#2024—24no—
26#2023—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35#1446—22no—
36#1445—22no—
37#1439—22no—
38#1439—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C23×Q8—21no—
42C23×Q8C2311yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—42no—
48C2×C4⋊Q8—42no—
49C2×C4⋊Q8D414yes—
50C2×#124—22no—
51C2×#124—22no—
52C2×#124—24no—
53C2×#124—42no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—42no—
70C2×#103—42no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—42no—
76C2×#102—42no—
77C2×#102—42no—
78C2×#102—42no—
79#391—22no—
80#391—22no—
81#391—42no—
82#390—22no—
83#390—22no—
84#390—42no—
85#384—24no—
86#382—42no—
87#376—42no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D412yes—
98C2×#95D412yes—
99#340—24no—
100#340—24no—
101#340—42no—
102#340—42no—
103#336—22no—
104#336—22no—
105#336—24no—
106#336—42no—
107C25—21no—
108C22×Q8—24no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8—82no—
121C22×Q8—82no—
122C22×Q8#4112yes—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4#4112yes—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2411yescharacteristic
143C22×C8—22no—
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—
157C4⋊Q8—48no—
158C4⋊Q8—48no—
159#124—48no—
160#124—48no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
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—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4C2×D414yes—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42C2×D412yes—
207C2×C42#4112yes—
208C2×C42#4114yes—
209#103—48no—
210#103—48no—
211#102—48no—
212#102—48no—
213#95—42no—
214#95—42no—
215#95—42no—
216#95—42no—
217#95—44no—
218#95—42no—
219#95—42no—
220#95—82no—
221#95—82no—
222#95—82no—
223#95—82no—
224C24—22no—
225C24—41no—
226C24—41no—
227C24—41no—
228C24—41no—
229C24—41no—
230C24—41no—
231C24—42no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24C22×D411yescharacteristic
236C2×Q8—24no—
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8—22no—
240C2×Q8—48no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×Q8—42no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—84no—
254C2×Q8—84no—
255C2×Q8—88no—
256C2×Q8—88no—
257C2×D4—24no—
258C2×D4—22no—
259C2×D4—24no—
260C2×D4—22no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
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—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—41no—
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—42no—
325C22×C4#12112yes—
326C22×C4#12112yes—
327C22×C4#12414yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4112yes—
333C22×C4H(5,𝔽2)12yes—
334C22×C4H(5,𝔽2)12yes—
335C22×C4H(5,𝔽2)12yes—
336C22×C42−1+411yescharacteristic
337C22×C42−1+411yescharacteristic
338C2×C8—42no—
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—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—48no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—88no—
366C22⋊C4—416no—
367C22⋊C4—84no—
368C22⋊C4—84no—
369C22⋊C4—88no—
370C42—24no—
371C42—24no—
372C42—24no—
373C42—24no—
374C42—42no—
375C42—42no—
376C23—42no—
377C23—44no—
378C23—44no—
379C23—42no—
380C23—42no—
381C23—81no—
382C23—81no—
383C23—82no—
384C23—82no—
385C23—81no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23C2×#12111yescharacteristic
400C23#52511yescharacteristic
401C23C2×#12411yescharacteristic
402C23#53411yescharacteristic
403C23#53511yescharacteristic
404C23#53511yescharacteristic
405C23#53611yescharacteristic
406C23#57612yes—
407C23#57712yes—
408C23#57712yes—
409C23#57812yes—
410Q8—44no—
411Q8—44no—
412Q8—816no—
413Q8—82no—
414Q8—82no—
415D4—44no—
416D4—44no—
417D4—82no—
418D4—82no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4#53412yes—
475C2×C4#53812yes—
476C2×C4#53812yes—
477C2×C4#53812yes—
478C2×C4#54212yes—
479C2×C4#54212yes—
480C2×C4#54811yescharacteristic
481C2×C4#54811yescharacteristic
482C2×C4#55012yes—
483C2×C4#55112yes—
484C2×C4#55112yes—
485C2×C4#55211yescharacteristic
486C2×C4#55211yescharacteristic
487C2×C4#55912yes—
488C2×C4#56212yes—
489C2×C4#56212yes—
490C8—82no—
491C8—82no—
492C22—82no—
493C22—82no—
494C22—84no—
495C22—84no—
496C22—82no—
497C22—162no—
498C22—162no—
499C22—161no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22#253611yescharacteristic
504C22#254811yescharacteristic
505C22#256111yescharacteristic
506C22#261711yescharacteristic
507C22#263111yescharacteristic
508C22#264011yescharacteristic
509C22#281611yescharacteristic
510C22#304111yescharacteristic
511C22#304211yescharacteristic
512C22#304311yescharacteristic
513C22#304411yescharacteristic
514C22#312412yes—
515C22#312512yes—
516C22#313612yes—
517C22#313712yes—
518C22#319812yes—
519C22#320112yes—
520C22#322412yes—
521C22#322412yes—
522C22#322512yes—
523C22#322512yes—
524C22#324312yes—
525C22#324412yes—
526C4—42no—
527C4—42no—
528C4—44no—
529C4—82no—
530C4—84no—
531C4—82no—
532C4—161no—
533C4—161no—
534C4—162no—
535C4#273012yes—
536C4#273012yes—
537C4#275112yes—
538C4#275112yes—
539C2—162no—
540C2#1688711yescharacteristic
541C2#2821111yescharacteristic
542C2#2821211yescharacteristic
543C2#2875811yescharacteristic
544C2#2875911yescharacteristic
545C2#2912012yes—
546C2#2972711yescharacteristic
547C2#2972811yescharacteristic
548C2#3005512yes—
549C2#3021112yes—
550C2#3278112yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2024C2semidirect
3. #14282C2semidirect
5. #11357C2semidirect
6. #11190C2semidirect
8. #11052C2semidirect
11. #10796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20242
C2×#20232
#142822
#142732
#113572
#111902
#111892
#110522
#110512
#110092
#107962