Finite Group Databasebeta

← #475812

Subgroups — #475812

Order 512.

1174 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2072C211yesmaximal, characteristic, complemented
2C2×#2071C211yesmaximal, characteristic
3C2×#2042C212yesmaximal
4#14276C211yesmaximal, characteristic, complemented
5#14273C211yesmaximal, characteristic
6#11207C212yesmaximal, complemented
7#11077C212yesmaximal, complemented
8#11076C212yesmaximal
9#11052C211yesmaximal, characteristic, complemented
10#11051C211yesmaximal, characteristic
11#10833C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#2072—24no—
28#2071—24no—
29#2042—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1441—22no—
38#1441—22no—
39#1439—22no—
40#1439—22no—
41C4⋊#95C2212yes—
42C23×Q8C2311yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#124—22no—
47C2×#124—42no—
48C2×#124—42no—
49C2×#124D414yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×C4⋊C8—24no—
66C2×C4⋊C8—42no—
67C2×C4⋊C8—42no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102—42no—
80#391—22no—
81#391—22no—
82#391—24no—
83#390—22no—
84#390—22no—
85#390—24no—
86#389—24no—
87#389—24no—
88#384—22no—
89#384—42no—
90#384D414yes—
91#382—42no—
92#376—42no—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95D412yes—
102C2×#95D412yes—
103#340—24no—
104#340—24no—
105#340—42no—
106#340—42no—
107#336—22no—
108#336—22no—
109#336—24no—
110C25—21no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—21no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8C2×D412yes—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4C2×D412yes—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
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—
155#124—48no—
156#124—48no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
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—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42#4112yes—
207C2×C42#4112yes—
208C2×C42#4114yes—
209C4⋊C8—48no—
210#103—48no—
211#103—48no—
212#102—48no—
213#102—48no—
214#95—42no—
215#95—42no—
216#95—42no—
217#95—42no—
218#95—44no—
219#95—42no—
220#95—42no—
221#95—44no—
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—22no—
239C2×Q8—24no—
240C2×Q8—42no—
241C2×Q8—41no—
242C2×Q8—41no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×D4—24no—
253C2×D4—22no—
254C2×D4—24no—
255C2×D4—22no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
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—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4#12114yes—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4#12414yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4112yes—
327C22×C4C2×#4111yescharacteristic
328C22×C4H(5,𝔽2)12yes—
329C22×C4H(5,𝔽2)12yes—
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C42−1+412yes—
332C22×C42−1+411yescharacteristic
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
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—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—84no—
360C4⋊C4—84no—
361C4⋊C4—88no—
362C4⋊C4—88no—
363C22⋊C4—416no—
364C22⋊C4—88no—
365C22⋊C4—84no—
366C22⋊C4—84no—
367C42—24no—
368C42—24no—
369C42—24no—
370C42—24no—
371C42—24no—
372C42—24no—
373C23—42no—
374C23—44no—
375C23—44no—
376C23—42no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23—82no—
381C23—82no—
382C23—81no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23C2×#12111yescharacteristic
397C23C2×#12111yescharacteristic
398C23C2×#12411yescharacteristic
399C23#53411yescharacteristic
400C23#53511yescharacteristic
401C23#53611yescharacteristic
402C23#53711yescharacteristic
403C23#57612yes—
404C23#57712yes—
405C23#57712yes—
406C23#57812yes—
407Q8—44no—
408Q8—44no—
409Q8—82no—
410Q8—82no—
411D4—44no—
412D4—44no—
413D4—82no—
414D4—82no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—42no—
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—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4#53412yes—
471C2×C4#53712yes—
472C2×C4#53812yes—
473C2×C4#53812yes—
474C2×C4#53912yes—
475C2×C4#53912yes—
476C2×C4D4211yescharacteristic
477C2×C4#54712yes—
478C2×C4#54711yescharacteristic
479C2×C4#54711yescharacteristic
480C2×C4#55012yes—
481C2×C4#55511yescharacteristic
482C2×C4#55612yes—
483C2×C4#55912yes—
484C2×C4#56212yes—
485C2×C4#56312yes—
486C8—82no—
487C8—82no—
488C22—84no—
489C22—84no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—162no—
494C22—161no—
495C22—161no—
496C22—161no—
497C22—161no—
498C22—162no—
499C22#253711yescharacteristic
500C22#254711yescharacteristic
501C22#254811yescharacteristic
502C22#261711yescharacteristic
503C22#261811yescharacteristic
504C22#263911yescharacteristic
505C22#282011yescharacteristic
506C22#304111yescharacteristic
507C22#304211yescharacteristic
508C22#304311yescharacteristic
509C22#304411yescharacteristic
510C22#304512yes—
511C22#304612yes—
512C22#313612yes—
513C22#313712yes—
514C22#320312yes—
515C22#320412yes—
516C22#322412yes—
517C22#322512yes—
518C22#324312yes—
519C22#324412yes—
520C22#325712yes—
521C22#325812yes—
522C4—44no—
523C4—42no—
524C4—42no—
525C4—44no—
526C4—84no—
527C4—82no—
528C4—162no—
529C4—161no—
530C4—161no—
531C4#268212yes—
532C4#272412yes—
533C4#272712yes—
534C4#280812yes—
535C2—162no—
536C2#1706911yescharacteristic
537C2#2825711yescharacteristic
538C2#2825811yescharacteristic
539C2#2873012yes—
540C2#2875811yescharacteristic
541C2#2875911yescharacteristic
542C2#2972912yes—
543C2#2976711yescharacteristic
544C2#2976811yescharacteristic
545C2#3019112yes—
546C2#3283312yes—
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×#2072C2semidirect
4. #14276C2semidirect
6. #11207C2semidirect
7. #11077C2semidirect
9. #11052C2semidirect
11. #10833C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20722
C2×#20712
C2×#20422
#142762
#142732
#112072
#110772
#110762
#110522
#110512
#108332