Finite Group Databasebeta

← #411259

Subgroups — #411259

Order 512.

1178 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2072C211yesmaximal, characteristic, complemented
2C2×#2071C211yesmaximal, characteristic
3#14273C211yesmaximal, characteristic
4#13779C211yesmaximal, characteristic, complemented
5#11267C212yesmaximal, complemented
6#11052C211yesmaximal, characteristic, complemented
7#11051C211yesmaximal, characteristic
8#11009C212yesmaximal
9#10932C212yesmaximal, complemented
10#10931C212yesmaximal
11#10796C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4C2212yes—
25C2×#376C2211yescharacteristic
26#2072—24no—
27#2071—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1439—22no—
37#1439—22no—
38#1436—22no—
39#1436—22no—
40C4⋊#95C2212yes—
41C23×Q8C2311yescharacteristic
42C23×D4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×#124—22no—
46C2×#124—22no—
47C2×#124—24no—
48C2×#124—42no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C42—22no—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—42no—
68C2×#103—42no—
69C2×#103—42no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—42no—
74C2×#102—42no—
75C2×#102—42no—
76#391—22no—
77#391—22no—
78#391—24no—
79#390—22no—
80#390—22no—
81#390—24no—
82#384—22no—
83#384—42no—
84#384D414yes—
85#382—24no—
86C4×C4⋊C4—42no—
87C4×C4⋊C4—42no—
88#376—24no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—21no—
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—24no—
107C25—21no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—21no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8#4112yes—
118C22×D4—21no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4#4112yes—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2411yescharacteristic
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
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—
152#124—48no—
153#124—48no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
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—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4C2×D414yes—
180C2×C4⋊C4#4114yes—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4—82no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—44no—
200C2×C42—82no—
201C2×C42—82no—
202C2×C42C2×D412yes—
203C2×C42#4112yes—
204C2×C42#4114yes—
205C2×C42#4114yes—
206#103—48no—
207#103—48no—
208#102—48no—
209#102—48no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—44no—
215#95—42no—
216#95—42no—
217#95—44no—
218#95—82no—
219#95—82no—
220C24—22no—
221C24—41no—
222C24—41no—
223C24—41no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24—42no—
228C24—42no—
229C24—42no—
230C24—42no—
231C24C22×D411yescharacteristic
232C2×Q8—24no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—22no—
236C2×Q8—42no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—24no—
251C2×D4—22no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—42no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
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—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4#12112yes—
316C22×C4#12112yes—
317C22×C4#12314yes—
318C22×C4#12414yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4112yes—
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4112yes—
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4112yes—
326C22×C4H(5,𝔽2)11yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C4H(5,𝔽2)12yes—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
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—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C4⋊C4—88no—
358C22⋊C4—416no—
359C22⋊C4—88no—
360C22⋊C4—84no—
361C22⋊C4—84no—
362C42—24no—
363C42—24no—
364C42—24no—
365C42—24no—
366C42—24no—
367C42—24no—
368C42—88no—
369C23—44no—
370C23—44no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—82no—
375C23—82no—
376C23—81no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—81no—
382C23—82no—
383C23—82no—
384C23—82no—
385C23—82no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23C2×#12111yescharacteristic
393C23C2×#12311yescharacteristic
394C23#52511yescharacteristic
395C23#52511yescharacteristic
396C23C2×#12411yescharacteristic
397C23#53511yescharacteristic
398C23#53511yescharacteristic
399C23#57612yes—
400C23#57712yes—
401C23#57712yes—
402C23#57812yes—
403Q8—44no—
404Q8—44no—
405Q8—82no—
406Q8—82no—
407D4—44no—
408D4—44no—
409D4—82no—
410D4—82no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
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—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4#52912yes—
467C2×C4#52912yes—
468C2×C4#52912yes—
469C2×C4#53212yes—
470C2×C4#53812yes—
471C2×C4#53812yes—
472C2×C4#54611yescharacteristic
473C2×C4#54611yescharacteristic
474C2×C4#55012yes—
475C2×C4#55112yes—
476C2×C4#55112yes—
477C2×C4#55311yescharacteristic
478C2×C4#55311yescharacteristic
479C2×C4#55312yes—
480C2×C4#55312yes—
481C2×C4#55912yes—
482C8—82no—
483C8—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—84no—
488C22—84no—
489C22—162no—
490C22—161no—
491C22—161no—
492C22—161no—
493C22—161no—
494C22—162no—
495C22#239811yescharacteristic
496C22#248211yescharacteristic
497C22#250411yescharacteristic
498C22#253511yescharacteristic
499C22#254811yescharacteristic
500C22#263111yescharacteristic
501C22#285411yescharacteristic
502C22#304111yescharacteristic
503C22#304211yescharacteristic
504C22#304311yescharacteristic
505C22#304411yescharacteristic
506C22#309812yes—
507C22#309912yes—
508C22#311912yes—
509C22#312012yes—
510C22#312412yes—
511C22#312512yes—
512C22#313612yes—
513C22#313712yes—
514C22#322412yes—
515C22#322412yes—
516C22#322512yes—
517C22#322512yes—
518C4—44no—
519C4—42no—
520C4—42no—
521C4—44no—
522C4—84no—
523C4—82no—
524C4—84no—
525C4—161no—
526C4—161no—
527C4#255412yes—
528C4#255412yes—
529C4#259712yes—
530C4#259712yes—
531C2—162no—
532C2#1511211yescharacteristic
533C2#2383212yes—
534C2#2759312yes—
535C2#2798412yes—
536C2#2816711yescharacteristic
537C2#2816811yescharacteristic
538C2#2875811yescharacteristic
539C2#2875911yescharacteristic
540C2#3005512yes—
541C2#3312011yescharacteristic
542C2#3312111yescharacteristic
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. #13779C2semidirect
5. #11267C2semidirect
6. #11052C2semidirect
9. #10932C2semidirect
11. #10796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20722
C2×#20712
#142732
#137792
#112672
#110522
#110512
#110092
#109322
#109312
#107962