Finite Group Databasebeta

← #6435542

Subgroups — #6435542

Order 512.

2238 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3139C211yesmaximal, characteristic, complemented
2C2×#2920C211yesmaximal, characteristic
3#36625C211yesmaximal, characteristic
4C2×#2857C211yesmaximal, characteristic
5#29852C218yesmaximal, complemented
6#29845C214yesmaximal, complemented
7#29844C214yesmaximal, complemented
8#23619C214yesmaximal, complemented
9#23481C214yesmaximal
10#21322C211yesmaximal, characteristic, complemented
11#21202C211yesmaximal, characteristic, complemented
12#20800C211yesmaximal, characteristic, complemented
13C2×#533C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C23×C42C2211yescharacteristic
17#3139—28no—
18C2×#491C2214yescomplemented
19C2×#487C2214yescomplemented
20#2920—28no—
21C2×#432C2214yes—
22C2×#431C2214yes—
23C2×#428C2214yes—
24C2×#423C2212yes—
25C2×#419C2214yes—
26C2×#419C2214yes—
27C2×#407C2212yes—
28C2×C4×M16C2211yescharacteristic
29C2×C4×M16C2211yescharacteristic
30#2857—22no—
31#2857—22no—
32#2857C2212yes—
33#2857C2212yes—
34#2857C2214yes—
35#2857C2214yes—
36#2855—22no—
37#2851C2214yes—
38#2848—22no—
39C4×C22⋊Q8—22no—
40C4×C22⋊Q8—22no—
41#2296C2212yes—
42C2×#391C2212yes—
43C2×#389—22no—
44#2285C2211yescharacteristic
45C2×#384C2211yescharacteristic
46#2280C2212yes—
47C2×#380—22no—
48#2068C22116yescomplemented
49#2066C22116yescomplemented
50#2036C22116yes—
51#2035C22116yes—
52#1992C2214yescomplemented
53#1990C2214yescomplemented
54#1960C2214yescomplemented
55#1956C2212yescomplemented
56#1956C2212yescomplemented
57#1847C2214yescomplemented
58#1845C2212yescomplemented
59#1844C2212yescomplemented
60#1736C2214yes—
61C23×Q8C2312yes—
62C24×C4D412yes—
63C24×C4C2311yescharacteristic
64C22×M16C2312yes—
65C22×M16C2312yes—
66#533—24no—
67#533—24no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—24no—
70C2×C4⋊Q8—24no—
71C2×C4⋊Q8C2312yes—
72C2×C4⋊Q8C2312yes—
73C2×C4⋊Q8C2312yes—
74C2×#125C2312yes—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—24no—
79C2×C22⋊Q8—48no—
80C2×C22⋊Q8C2318yes—
81C2×C4×Q8—22no—
82C2×#117C2312yes—
83C2×#117C2312yes—
84C22×C4⋊C4—22no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—22no—
89C22×C42—24no—
90C22×C42—24no—
91C22×C42D412yes—
92C22×C42D412yes—
93C22×C42D412yes—
94C22×C42C2312yes—
95C22×C42C2312yes—
96C22×C42C2311yescharacteristic
97C22×C42C2311yescharacteristic
98#491—216no—
99#487—216no—
100#432—216no—
101#431—216no—
102#428—48no—
103C2×#107C2318yes—
104C2×#106C2318yes—
105#423—28no—
106C2×C4⋊C8C2318yes—
107#419—216no—
108#419—48no—
109C2×#103C23116yes—
110C2×#103C2318yes—
111C2×#103C2318yes—
112#407—28no—
113C2×C22⋊C8C2318yes—
114C4×M16—24no—
115C4×M16D418yes—
116C2×#97C2312yes—
117C2×#97C2312yes—
118C2×C4×C8C2312yes—
119C2×C4×C8C2312yes—
120#391—22no—
121#391—22no—
122#391—22no—
123#391—22no—
124#391C2318yes—
125#389—22no—
126#389—22no—
127#389—24no—
128#389—24no—
129#389—24no—
130#389—24no—
131#389C2314yes—
132#389C2314yes—
133#387—24no—
134#387—24no—
135#385—24no—
136#385—24no—
137#385—24no—
138#384—22no—
139#384C2311yescharacteristic
140#384C2312yes—
141#384C2311yescharacteristic
142#384C2312yes—
143#382C2312yes—
144#381—22no—
145#380—22no—
146#380—22no—
147#380—22no—
148#380—24no—
149#380—24no—
150#380—24no—
151#380—24no—
152#380C2314yes—
153#380C2314yes—
154#379—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—22no—
158C4×C22⋊C4—24no—
159C2×#95—24no—
160#340C23116yes—
161C25C2×D411yescharacteristic
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—28no—
167C22×Q8—28no—
168C22×Q8C2414yes—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—44no—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C2×M16—24no—
189C2×M16—24no—
190C2×M16—24no—
191C2×M16—24no—
192C2×M16—24no—
193C2×M16C2×D418yes—
194C22×C8C2414yes—
195C22×C8C2414yes—
196C4⋊Q8—24no—
197C4⋊Q8—24no—
198C4⋊Q8—24no—
199C4⋊Q8—24no—
200C4⋊Q8—24no—
201C4⋊Q8—24no—
202C4⋊Q8—48no—
203#125—24no—
204#125—24no—
205C22⋊Q8—416no—
206C22⋊Q8—48no—
207C22⋊Q8—416no—
208C22⋊Q8—48no—
209C22⋊Q8—48no—
210C4×Q8—48no—
211#117—24no—
212#117—24no—
213#117—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—48no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4C2414yes—
231C2×C4⋊C4C2414yes—
232C2×C4⋊C4C2412yes—
233C2×C4⋊C4C2412yes—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—48no—
240C2×C42—28no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—28no—
244C2×C42—28no—
245C2×C42—24no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D414yes—
259C2×C42#4114yes—
260C2×C42C2412yes—
261C2×C42C2411yescharacteristic
262C2×C42C2411yescharacteristic
263#107—48no—
264#106—48no—
265C4⋊C8—216no—
266#103—232no—
267#103—48no—
268#103—48no—
269C22⋊C8—216no—
270#97—24no—
271#97C2×D418yes—
272C4×C8—24no—
273C4×C8C2×D418yes—
274#95—28no—
275#95—28no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281C24—22no—
282C24—24no—
283C24—22no—
284C24—22no—
285C24—42no—
286C24C22×D412yes—
287C24C22×D411yescharacteristic
288C2×Q8—24no—
289C2×Q8—24no—
290C2×Q8—28no—
291C2×Q8—28no—
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—44no—
298C2×Q8—416no—
299C2×Q8—416no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—432no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—48no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4#12118yes—
340C22×C4#13718yes—
341C22×C4C22×D412yes—
342C22×C4C22×D412yes—
343C22×C4C22×D412yes—
344C22×C4C22×D412yes—
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D412yes—
347C22×C4C22×D412yes—
348C22×C4C2×#4112yes—
349C22×C4H(5,𝔽2)14yes—
350C22×C42−1+414yes—
351C22×C4C2511yescharacteristic
352M16—216no—
353M16—48no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8C22×D418yes—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—28no—
365C4⋊C4—48no—
366C4⋊C4—416no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—44no—
374C4⋊C4—48no—
375C4⋊C4—48no—
376C4⋊C4—44no—
377C22⋊C4—416no—
378C22⋊C4—48no—
379C22⋊C4—44no—
380C22⋊C4—44no—
381C22⋊C4—48no—
382C42—22no—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—22no—
387C42—24no—
388C42—22no—
389C42—22no—
390C42—44no—
391C42—44no—
392C42#12118yes—
393C42C22×D412yes—
394C42C22×D412yes—
395C42C2×#4112yes—
396C42C2×#4112yes—
397C23—22no—
398C23—22no—
399C23—28no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—48no—
407C23—48no—
408C23—44no—
409C23—44no—
410C23C2×#12112yes—
411C23#53512yes—
412C23#53612yes—
413C23C2×#13714yes—
414C23C23×D411yescharacteristic
415Q8—48no—
416Q8—48no—
417Q8—88no—
418Q8—88no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—216no—
424C2×C4—216no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4—48no—
434C2×C4—44no—
435C2×C4—48no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—48no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—48no—
451C2×C4—88no—
452C2×C4—88no—
453C2×C4C2×#12114yes—
454C2×C4C2×#12112yes—
455C2×C4C2×#12114yes—
456C2×C4#53512yes—
457C2×C4#53612yes—
458C2×C4#53718yes—
459C2×C4#54218yes—
460C2×C4C2×#13714yes—
461C2×C4C2×#13714yes—
462C2×C4#57514yes—
463C2×C4C23×D411yescharacteristic
464C2×C4C22×#4111yescharacteristic
465C2×C4C2×H(5,𝔽2)12yes—
466C2×C4C2×2−1+412yes—
467C8—28no—
468C8—44no—
469C22—24no—
470C22—22no—
471C22—22no—
472C22—24no—
473C22—24no—
474C22—24no—
475C22—24no—
476C22—416no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22#262014yes—
482C22C22×#12111yescharacteristic
483C22C2×#53511yescharacteristic
484C22C2×#53611yescharacteristic
485C22C22×#13712yes—
486C22C2×#57512yes—
487C4—24no—
488C4—24no—
489C4—42no—
490C4—42no—
491C4—84no—
492C4—84no—
493C4—82no—
494C4—82no—
495C4—84no—
496C4#262014yes—
497C4#305618yes—
498C4C2×#53712yes—
499C4C2×#54212yes—
500C2—24no—
501C2—44no—
502C2C2×#262011yescharacteristic
503C2C2×#305612yes—
504C2#6080011yescharacteristic
505C2#6080111yescharacteristic
506C2#6086512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3139C2semidirect
5. #29852C2semidirect
6. #29845C2semidirect
7. #29844C2semidirect
8. #23619C2semidirect
10. #21322C2semidirect
11. #21202C2semidirect
12. #20800C2semidirect
18. C2×#491C22semidirect
19. C2×#487C22semidirect
48. #2068C22semidirect
49. #2066C22semidirect
52. #1992C22semidirect
53. #1990C22semidirect
54. #1960C22semidirect
55. #1956C22semidirect
56. #1956C22semidirect
57. #1847C22semidirect
58. #1845C22semidirect
59. #1844C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31392
C2×#29202
#366252
C2×#28572
#298522
#298452
#298442
#236192
#234812
#213222
#212022
#208002