Finite Group Databasebeta

← #373715

Subgroups — #373715

Order 512.

1502 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#1854C212yesmaximal
3C2×#1818C211yesmaximal, characteristic
4#13795C211yesmaximal, characteristic
5#11324C211yesmaximal, characteristic, complemented
6#11323C211yesmaximal, characteristic, complemented
7#11268C211yesmaximal, characteristic, complemented
8#11268C211yesmaximal, characteristic, complemented
9#10908C212yesmaximal, complemented
10#10907C212yesmaximal, complemented
11#10730C212yesmaximal
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C22⋊C8C2212yes—
18#2314—28no—
19C2×#388—22no—
20C2×#387—22no—
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24C2×C4×C4⋊C4C2212yes—
25#1854—28no—
26#1818C418yescomplemented
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C4⋊#95—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C22⋊#95—22no—
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#117—24no—
47C2×#117—24no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—24no—
51C2×#117—42no—
52C2×#117C2×C414yes—
53C2×#117Q814yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—42no—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4D412yes—
70C22×C42—21no—
71C22×C42—22no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#107—24no—
75C2×#107—24no—
76C2×#107—42no—
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—42no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8C2×C418yes—
83#388—216no—
84#387—216no—
85#384—24no—
86#382—24no—
87#380—24no—
88#380D418yes—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—24no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—42no—
103#340—42no—
104#336C2×C414yescomplemented
105#336C2×C414yescomplemented
106#336C2×C418yescomplemented
107C25C2×D411yescharacteristic
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—24no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—82no—
124C23×C4—82no—
125C23×C4C22×C412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D414yes—
130C23×C4C2×Q812yes—
131C23×C4#4114yes—
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8C22×C414yes—
146C22×C8C22×C414yes—
147#117—216no—
148#117—216no—
149#117—48no—
150#117—48no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
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—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C4⋊C418yes—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×D414yes—
186C2×C4⋊C4#4112yes—
187C2×C4⋊C4#4112yes—
188C2×C4⋊C4#4112yes—
189C2×C4⋊C4#4112yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4C4⋊C418yes—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—24no—
204C2×C42—28no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—82no—
208C2×C42—82no—
209C2×C42C22×C412yes—
210C2×C42C22×C412yes—
211C2×C42C2×Q812yes—
212C2×C42C2×Q812yes—
213#107—48no—
214#106—48no—
215C22⋊C8—216no—
216#95—28no—
217#95—28no—
218#95—28no—
219#95—82no—
220#95—82no—
221#95—82no—
222#95—82no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24C2×C4⋊C412yes—
232C24#12012yes—
233C24#12312yes—
234C24C22×D411yescharacteristic
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—48no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—88no—
301C22×C4—88no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4C2×C4⋊C414yes—
305C22×C4C2×C4⋊C414yes—
306C22×C4C2×C4⋊C412yes—
307C22×C4C4×D418yes—
308C22×C4#12012yes—
309C22×C4C22⋊Q818yes—
310C22×C4#12312yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D412yes—
313C22×C4C22×Q811yescharacteristic
314C22×C4C2×#4112yes—
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4H(5,𝔽2)12yes—
320C22×C42−1+412yes—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C4⋊C4—216no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—48no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—88no—
354C22⋊C4—216no—
355C22⋊C4—48no—
356C22⋊C4—88no—
357C42—24no—
358C42—24no—
359C42—24no—
360C42—24no—
361C42—48no—
362C42—88no—
363C23—22no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—24no—
373C23—24no—
374C23—22no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23#38514yes—
386C23C22×C4⋊C411yescharacteristic
387C23C2×#12011yescharacteristic
388C23C2×#12311yescharacteristic
389C23#52511yescharacteristic
390C23#52511yescharacteristic
391C23#53511yescharacteristic
392C23#53611yescharacteristic
393C23#57612yes—
394C23#57712yes—
395C23#57712yes—
396C23#57812yes—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—28no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—816no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4#38514yes—
457C2×C4C2×C4×D412yes—
458C2×C4C4×#4111yescharacteristic
459C2×C4C4×#4111yescharacteristic
460C2×C4#52012yes—
461C2×C4C2×C22⋊Q812yes—
462C2×C4#53311yescharacteristic
463C2×C4#53311yescharacteristic
464C2×C4#54412yes—
465C2×C4#54612yes—
466C2×C4#54612yes—
467C2×C4#54712yes—
468C2×C4#54712yes—
469C2×C4#55112yes—
470C2×C4#55112yes—
471C2×C4#55212yes—
472C2×C4#55212yes—
473C8—44no—
474C8—44no—
475C22—24no—
476C22—28no—
477C22—44no—
478C22—44no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—44no—
484C22—42no—
485C22—42no—
486C22C2×#38511yescharacteristic
487C22#229611yescharacteristic
488C22#231711yescharacteristic
489C22#240411yescharacteristic
490C22#240611yescharacteristic
491C22#248211yescharacteristic
492C22#248411yescharacteristic
493C22#291212yes—
494C22#291312yes—
495C22#301212yes—
496C22#301312yes—
497C22#309612yes—
498C22#309712yes—
499C22#311812yes—
500C22#311912yes—
501C22#312012yes—
502C22#312112yes—
503C22#322011yescharacteristic
504C22#322111yescharacteristic
505C22#322211yescharacteristic
506C22#322311yescharacteristic
507C22#324312yes—
508C22#324412yes—
509C4—24no—
510C4—44no—
511C4—44no—
512C4—84no—
513C4—82no—
514C4—84no—
515C4—82no—
516C4—82no—
517C4—82no—
518C4#235912yes—
519C4#235912yes—
520C4#243812yes—
521C4#243812yes—
522C2—44no—
523C2#1412811yescharacteristic
524C2#2084912yes—
525C2#2136712yes—
526C2#2199011yescharacteristic
527C2#2199111yescharacteristic
528C2#2394111yescharacteristic
529C2#2394411yescharacteristic
530C2#2407512yes—
531C2#2759411yescharacteristic
532C2#2759611yescharacteristic
533C2#2761912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11324C2semidirect
6. #11323C2semidirect
7. #11268C2semidirect
8. #11268C2semidirect
9. #10908C2semidirect
10. #10907C2semidirect
26. #1818C4semidirect
104. #336C2×C4semidirect
105. #336C2×C4semidirect
106. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#18542
C2×#18182
#137952
#113242
#113232
#112682
#109082
#109072
#107302