Finite Group Databasebeta

← #476023

Subgroups — #476023

Order 512.

1210 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×#2024C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3#14282C211yesmaximal, characteristic, complemented
4#14273C211yesmaximal, characteristic
5#11267C212yesmaximal, complemented
6#11077C212yesmaximal, complemented
7#11076C212yesmaximal
8#11052C211yesmaximal, characteristic, complemented
9#11051C211yesmaximal, characteristic
10#11009C212yesmaximal
11#10833C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#376C2211yescharacteristic
26#2024—24no—
27#2023—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
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—42no—
52C2×#124—42no—
53C2×#124Q814yescomplemented
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—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C4⋊C8—24no—
68C2×C4⋊C8—42no—
69C2×C4⋊C8—42no—
70C2×#103—24no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103—42no—
76C2×#102—24no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102—42no—
81C2×#102—42no—
82#391—22no—
83#391—22no—
84#391—24no—
85#390—22no—
86#390—22no—
87#390—24no—
88#384—42no—
89#382—42no—
90#376—42no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95D412yes—
101C2×#95D412yes—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#336—22no—
107#336—22no—
108#336—24no—
109C25—21no—
110C22×Q8—24no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—82no—
123C22×Q8—82no—
124C22×Q8C2×Q812yes—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4C2×Q812yes—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2411yescharacteristic
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—
158C22×C8—22no—
159C4⋊Q8—48no—
160C4⋊Q8—48no—
161#124—48no—
162#124—48no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
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—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C2×D414yes—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
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×C42C2×Q812yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42#4114yes—
211C4⋊C8—48no—
212#103—48no—
213#103—48no—
214#102—48no—
215#102—48no—
216#95—42no—
217#95—42no—
218#95—42no—
219#95—42no—
220#95—42no—
221#95—42no—
222#95—44no—
223#95—82no—
224#95—82no—
225#95—82no—
226#95—82no—
227C24—22no—
228C24—41no—
229C24—41no—
230C24—41no—
231C24—41no—
232C24—41no—
233C24—41no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24C22×D411yescharacteristic
239C2×Q8—24no—
240C2×Q8—22no—
241C2×Q8—24no—
242C2×Q8—22no—
243C2×Q8—48no—
244C2×Q8—42no—
245C2×Q8—41no—
246C2×Q8—42no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—84no—
257C2×Q8—84no—
258C2×Q8—88no—
259C2×Q8—88no—
260C2×D4—24no—
261C2×D4—22no—
262C2×D4—24no—
263C2×D4—22no—
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—
274C2×D4—42no—
275C2×D4—42no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
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—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
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—42no—
326C22×C4—42no—
327C22×C4#12112yes—
328C22×C4#12112yes—
329C22×C4C22⋊Q814yes—
330C22×C4#12414yes—
331C22×C4C22×D411yescharacteristic
332C22×C4C22×Q811yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4112yes—
336C22×C4H(5,𝔽2)12yes—
337C22×C4H(5,𝔽2)12yes—
338C22×C4H(5,𝔽2)11yescharacteristic
339C22×C4H(5,𝔽2)12yes—
340C22×C42−1+411yescharacteristic
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—
353C2×C8—42no—
354C2×C8—42no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—48no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—88no—
369C22⋊C4—416no—
370C22⋊C4—88no—
371C22⋊C4—84no—
372C22⋊C4—84no—
373C42—24no—
374C42—24no—
375C42—24no—
376C42—24no—
377C42—24no—
378C42—24no—
379C23—42no—
380C23—44no—
381C23—44no—
382C23—42no—
383C23—42no—
384C23—81no—
385C23—81no—
386C23—82no—
387C23—82no—
388C23—81no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23C2×#12111yescharacteristic
403C23C2×C22⋊Q811yescharacteristic
404C23C2×#12411yescharacteristic
405C23#53411yescharacteristic
406C23#53511yescharacteristic
407C23#53511yescharacteristic
408C23#53711yescharacteristic
409C23#57612yes—
410C23#57712yes—
411C23#57712yes—
412C23#57812yes—
413Q8—44no—
414Q8—44no—
415Q8—816no—
416Q8—82no—
417Q8—82no—
418D4—44no—
419D4—44no—
420D4—82no—
421D4—82no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—81no—
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—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4#53412yes—
478C2×C4#53812yes—
479C2×C4#53812yes—
480C2×C4#53812yes—
481C2×C4#53912yes—
482C2×C4#54312yes—
483C2×C4D4×Q811yescharacteristic
484C2×C4#55012yes—
485C2×C4#55011yescharacteristic
486C2×C4#55412yes—
487C2×C4#55411yescharacteristic
488C2×C4#55512yes—
489C2×C4#55611yescharacteristic
490C2×C4#55912yes—
491C2×C4H(3,𝔽4)12yes—
492C2×C4#56212yes—
493C8—82no—
494C8—82no—
495C22—84no—
496C22—82no—
497C22—82no—
498C22—82no—
499C22—84no—
500C22—161no—
501C22—161no—
502C22—162no—
503C22—161no—
504C22—161no—
505C22—162no—
506C22#253811yescharacteristic
507C22#254811yescharacteristic
508C22#256311yescharacteristic
509C22#261711yescharacteristic
510C22#262611yescharacteristic
511C22#263911yescharacteristic
512C22#281911yescharacteristic
513C22#304111yescharacteristic
514C22#304211yescharacteristic
515C22#304311yescharacteristic
516C22#304411yescharacteristic
517C22#308312yes—
518C22#308412yes—
519C22#313612yes—
520C22#313712yes—
521C22#319812yes—
522C22#320112yes—
523C22#322412yes—
524C22#322412yes—
525C22#322512yes—
526C22#322512yes—
527C22#326212yes—
528C22#326312yes—
529C4—44no—
530C4—42no—
531C4—42no—
532C4—44no—
533C4—84no—
534C4—82no—
535C4—162no—
536C4—161no—
537C4—161no—
538C4#273212yes—
539C4#273312yes—
540C4#280412yes—
541C4#280812yes—
542C2—162no—
543C2#1707711yescharacteristic
544C2#2828811yescharacteristic
545C2#2828911yescharacteristic
546C2#2875811yescharacteristic
547C2#2875911yescharacteristic
548C2#2915112yes—
549C2#2972711yescharacteristic
550C2#2972811yescharacteristic
551C2#2996112yes—
552C2#3019512yes—
553C2#3281712yes—
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. #11267C2semidirect
6. #11077C2semidirect
8. #11052C2semidirect
11. #10833C2semidirect
53. C2×#124Q8semidirect

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
#112672
#110772
#110762
#110522
#110512
#110092
#108332