Finite Group Databasebeta

← #472035

Subgroups — #472035

Order 512.

1500 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2092C212yesmaximal, complemented
2C2×#2084C211yesmaximal, characteristic, complemented
3C2×#2083C211yesmaximal, characteristic, complemented
4C2×#2041C211yesmaximal, characteristic
5C2×#2041C211yesmaximal, characteristic
6#14713C211yesmaximal, characteristic
7#14707C211yesmaximal, characteristic, complemented
8#11123C212yesmaximal, complemented
9#11121C212yesmaximal, complemented
10#11051C212yesmaximal, complemented
11#10795C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C2×#390—22no—
20C2×#390—22no—
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#389C2211yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C2×C43C2211yescharacteristic
29#2092—28no—
30#2084—24no—
31#2083—24no—
32#2041—24no—
33#2041—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39C2×#336C2211yescharacteristic
40#1436—22no—
41#1436—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C23×D4—21no—
45C23×D4—21no—
46C23×D4C2312yes—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#127—24no—
50C2×#127—22no—
51C2×#127—22no—
52C2×#127—42no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4—24no—
63C22×C22⋊C4—24no—
64C22×C42D412yes—
65C22×C42D412yes—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#107—24no—
70C2×#107—24no—
71C2×#107—42no—
72C2×#106—24no—
73C2×#106—24no—
74C2×#106—42no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—42no—
84C2×#102—42no—
85C2×#102—42no—
86C2×#102—42no—
87#390—24no—
88#390—24no—
89#390—42no—
90#390—42no—
91#390—48no—
92#390—48no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#384—22no—
98#384—22no—
99#382—22no—
100#382—42no—
101#382D414yescomplemented
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—21no—
108C43—42no—
109#340—24no—
110#340—24no—
111#336—24no—
112#336—42no—
113#336D414yes—
114#336D414yes—
115C25—22no—
116C22×D4—24no—
117C22×D4—24no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—42no—
132C22×D4—42no—
133C22×D4—44no—
134C22×D4—44no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4—44no—
138C22×D4—44no—
139C22×D4#4114yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
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—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8C2×D414yes—
163C22×C8C2×D414yes—
164#127—48no—
165#127—48no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C22⋊C4—48no—
201C2×C22⋊C4—48no—
202C2×C22⋊C4—48no—
203C2×C22⋊C4—48no—
204C2×C22⋊C4—48no—
205C2×C22⋊C4—48no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—22no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—22no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42#4112yes—
224#107—48no—
225#106—48no—
226#102—48no—
227#102—48no—
228#102—48no—
229#102—48no—
230#95—28no—
231#95—44no—
232#95—44no—
233#95—42no—
234#95—42no—
235#95—44no—
236C24—24no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—44no—
244C24—44no—
245C24—44no—
246C24—44no—
247C24C22×D411yescharacteristic
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—48no—
255C2×D4—48no—
256C2×D4—416no—
257C2×D4—416no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
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—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—84no—
281C2×D4—84no—
282C2×D4—88no—
283C2×D4—88no—
284C2×D4—84no—
285C2×D4—84no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4#12114yes—
330C22×C4#12712yes—
331C22×C4#12712yes—
332C22×C4#13514yes—
333C22×C4AGL(1,ℤ/8)14yes—
334C22×C4AGL(1,ℤ/8)14yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C22×D411yescharacteristic
338C22×C4C2×#4112yes—
339C22×C4C2×#4111yescharacteristic
340C22×C4H(5,𝔽2)12yes—
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C42−1+412yes—
344C22×C42−1+412yes—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—84no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C22⋊C4—816no—
380C22⋊C4—816no—
381C42—24no—
382C42—24no—
383C42—24no—
384C42—24no—
385C42—48no—
386C42—48no—
387C42—42no—
388C42—42no—
389C23—48no—
390C23—48no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—84no—
402C23—84no—
403C23—84no—
404C23—84no—
405C23—84no—
406C23—84no—
407C23—84no—
408C23—84no—
409C23—84no—
410C23—84no—
411C23—82no—
412C23C2×#12111yescharacteristic
413C23#52511yescharacteristic
414C23C2×#12711yescharacteristic
415C23#53411yescharacteristic
416C23#53511yescharacteristic
417C23#53611yescharacteristic
418C23#53611yescharacteristic
419C23C2×#13512yes—
420C23C2×AGL(1,ℤ/8)12yes—
421C23C2×AGL(1,ℤ/8)12yes—
422C23#57512yes—
423D4—44no—
424D4—44no—
425D4—44no—
426D4—44no—
427D4—816no—
428D4—816no—
429D4—82no—
430D4—82no—
431D4—82no—
432D4—82no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—84no—
476C2×C4—84no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4#49514yes—
481C2×C4#49614yes—
482C2×C4#53612yes—
483C2×C4#53612yes—
484C2×C4#53712yes—
485C2×C4#53912yes—
486C2×C4#53912yes—
487C2×C4#53912yes—
488C2×C4D4211yescharacteristic
489C2×C4D4211yescharacteristic
490C2×C4#54612yes—
491C2×C4#54612yes—
492C2×C4#54812yes—
493C2×C4#55011yescharacteristic
494C2×C4#55011yescharacteristic
495C2×C4#55212yes—
496C2×C4#56212yes—
497C2×C4#56212yes—
498C8—44no—
499C8—44no—
500C22—88no—
501C22—88no—
502C22—84no—
503C22—84no—
504C22—84no—
505C22—164no—
506C22—162no—
507C22—162no—
508C22—164no—
509C22—162no—
510C22—162no—
511C22#253611yescharacteristic
512C22#253911yescharacteristic
513C22#254111yescharacteristic
514C22#261811yescharacteristic
515C22#263311yescharacteristic
516C22#265811yescharacteristic
517C22#281611yescharacteristic
518C22#303512yes—
519C22#303812yes—
520C22#310512yes—
521C22#311412yes—
522C22C2×#49511yescharacteristic
523C22C2×#49611yescharacteristic
524C22#315111yescharacteristic
525C22#315311yescharacteristic
526C22#318812yes—
527C22#319412yes—
528C22#321212yes—
529C22#321712yes—
530C22#322812yes—
531C22#322812yes—
532C22#323412yes—
533C22#323512yes—
534C4—24no—
535C4—24no—
536C4—44no—
537C4—44no—
538C4—82no—
539C4—82no—
540C4—162no—
541C4—161no—
542C4—161no—
543C4#268112yes—
544C4#268112yes—
545C4#269412yes—
546C4#269412yes—
547C2—164no—
548C2#1681111yescharacteristic
549C2#2820712yes—
550C2#2830311yescharacteristic
551C2#2831211yescharacteristic
552C2#2843911yescharacteristic
553C2#2844811yescharacteristic
554C2#2975812yes—
555C2#3007712yes—
556C2#3027811yescharacteristic
557C2#3028311yescharacteristic
558C2#3277612yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
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×#2092C2semidirect
2. C2×#2084C2semidirect
3. C2×#2083C2semidirect
7. #14707C2semidirect
8. #11123C2semidirect
9. #11121C2semidirect
10. #11051C2semidirect
11. #10795C2semidirect
101. #382D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20922
C2×#20842
C2×#20832
C2×#20412
#147132
#147072
#111232
#111212
#110512
#107952