Finite Group Databasebeta

← #6435136

Subgroups — #6435136

Order 512.

2652 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3152C211yesmaximal, characteristic, complemented
2C2×#2920C211yesmaximal, characteristic, complemented
3#36624C211yesmaximal, characteristic
4C2×#2857C211yesmaximal, characteristic, complemented
5#29815C218yesmaximal, complemented
6#29808C214yesmaximal, complemented
7#29807C214yesmaximal, complemented
8#23547C214yesmaximal, complemented
9#23392C214yesmaximal, complemented
10#21202C211yesmaximal, characteristic, complemented
11#21200C211yesmaximal, characteristic, complemented
12#20764C211yesmaximal, characteristic, complemented
13C22×#137C2214yes—
14C2×#532C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C2×#525C2212yes—
17C22×C22⋊Q8C2212yes—
18C23×C42C2211yescharacteristic
19#3152—28no—
20C2×#497C2214yescomplemented
21C2×#494C2212yescomplemented
22C2×#492C2212yescomplemented
23#2920—28no—
24C2×#432C2214yes—
25C2×#431C2214yes—
26C2×#423C2212yes—
27C2×#417C2214yes—
28C2×#419C2214yes—
29C2×#407C2212yes—
30C2×C4×M16C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32#2857—22no—
33#2857—22no—
34#2857C2214yes—
35#2857C2214yes—
36#2857C2212yes—
37#2856C2212yes—
38#2854—22no—
39#2850C2214yes—
40#2847—22no—
41C4×#121—22no—
42C4×#121—22no—
43#2297C2212yes—
44C2×#391C2212yes—
45C2×#389—22no—
46#2285C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#380—22no—
49#2056C22116yescomplemented
50#2055C22116yescomplemented
51#2001C22116yes—
52#2000C22116yes—
53#1960C2214yescomplemented
54#1959C2214yescomplemented
55#1957C2212yescomplemented
56#1956C2214yescomplemented
57#1956C2212yescomplemented
58#1836C2214yescomplemented
59#1834C2214yescomplemented
60#1736C2214yes—
61C22×#41C2312yes—
62C23×Q8C2312yes—
63C24×C4D412yes—
64C24×C4C2311yescharacteristic
65C2×#137—216no—
66C2×#137—48no—
67C2×#137—48no—
68C22×Q16C2318yes—
69C22×SD16C2318yes—
70C22×M16C2312yes—
71C22×M16C2312yes—
72#532—24no—
73#532—24no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—24no—
77C2×C4⋊Q8C2312yes—
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2312yes—
80C2×#127C2311yescharacteristic
81C2×#124C2312yes—
82#525—48no—
83C2×#123—22no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—48no—
86C2×C22⋊Q8C2314yes—
87C2×C22⋊Q8C2318yes—
88C2×#121—22no—
89C2×#121—22no—
90C2×#121—22no—
91C2×#121C2314yes—
92C2×#120—22no—
93C2×C4×D4—24no—
94C2×C4×D4—22no—
95C2×#117C2312yes—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4C2312yes—
100C22×C22⋊C4—22no—
101C22×C42—24no—
102C22×C42—24no—
103C22×C42D412yes—
104C22×C42D412yes—
105C22×C42D412yes—
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#497—216no—
111#494—28no—
112#492—28no—
113#432—216no—
114#431—216no—
115#423—28no—
116C2×C4⋊C8C2318yes—
117#419—48no—
118#417—48no—
119C2×#103C23116yes—
120C2×#103C2318yes—
121C2×#102C2318yes—
122#407—28no—
123C2×C22⋊C8C2318yes—
124C4×M16—24no—
125C4×M16D418yescomplemented
126C2×#97C2312yes—
127C2×#97C2312yes—
128C2×C4×C8C2312yes—
129C2×C4×C8C2312yes—
130#391—22no—
131#391—22no—
132#391C2318yes—
133#391C2314yes—
134#390—22no—
135#390—22no—
136#390C2314yes—
137#389—22no—
138#389—22no—
139#389—24no—
140#389—24no—
141#389—24no—
142#389—24no—
143#388—24no—
144#388—24no—
145#386—24no—
146#386—24no—
147#386—24no—
148#384—22no—
149#384C2312yes—
150#384C2312yes—
151#384C2312yes—
152#382C2312yes—
153#380—22no—
154#380—22no—
155#380—22no—
156#380—24no—
157#380—24no—
158#380—24no—
159#380—24no—
160C4×C4⋊C4—22no—
161C4×C22⋊C4—22no—
162C4×C22⋊C4—22no—
163C2×#95—24no—
164#340C23116yes—
165C25C2×D411yescharacteristic
166C2×#41—24no—
167C2×#41—24no—
168C2×#41—28no—
169C2×#41—28no—
170C22×Q8—22no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—28no—
174C22×Q8—28no—
175C22×Q8C2412yes—
176C22×Q8C2414yes—
177C22×D4—24no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4C2412yes—
181C23×C4—28no—
182C23×C4—28no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—44no—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201#137—464no—
202C2×Q16—216no—
203C2×Q16—48no—
204C2×Q16—48no—
205C2×SD16—216no—
206C2×SD16—48no—
207C2×SD16—48no—
208C2×M16—24no—
209C2×M16—24no—
210C2×M16—24no—
211C2×M16—24no—
212C2×M16—24no—
213C2×M16C2×D418yes—
214C22×C8C2414yes—
215C22×C8C2414yes—
216C4⋊Q8—24no—
217C4⋊Q8—24no—
218C4⋊Q8—22no—
219C4⋊Q8—24no—
220C4⋊Q8—24no—
221C4⋊Q8—22no—
222C4⋊Q8—48no—
223#127—22no—
224#127—22no—
225#124—24no—
226#124—24no—
227#123—48no—
228C22⋊Q8—416no—
229C22⋊Q8—48no—
230C22⋊Q8—416no—
231#121—48no—
232#121—48no—
233#121—48no—
234#121—48no—
235#120—48no—
236C4×D4—416no—
237C4×D4—48no—
238#117—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—48no—
251C2×C4⋊C4—48no—
252C2×C4⋊C4—48no—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2414yes—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—48no—
263C2×C42—28no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—28no—
267C2×C42—28no—
268C2×C42—24no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D412yes—
276C2×C42C2×D412yes—
277C2×C42C2×D412yes—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D414yes—
281C2×C42#4114yes—
282C2×C42C2412yes—
283C2×C42C2411yescharacteristic
284C2×C42C2411yescharacteristic
285C4⋊C8—216no—
286#103—48no—
287#103—416no—
288#102—48no—
289C22⋊C8—216no—
290#97—24no—
291#97C2×D418yes—
292C4×C8—24no—
293C4×C8C2×D418yes—
294#95—28no—
295#95—28no—
296#95—24no—
297#95—24no—
298#95—24no—
299C24—22no—
300C24—24no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—42no—
305C24C22×D412yes—
306C24C22×D411yescharacteristic
307#41—416no—
308#41—416no—
309C2×Q8—22no—
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—28no—
314C2×Q8—24no—
315C2×Q8—28no—
316C2×Q8—24no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—416no—
321C2×Q8—416no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—432no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4#12118yes—
377C22×C4#13718yescomplemented
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D412yes—
384C22×C4C22×D412yes—
385C22×C4C2×#4112yes—
386C22×C4H(5,𝔽2)14yes—
387C22×C4H(5,𝔽2)14yes—
388C22×C4C2511yescharacteristic
389Q16—432no—
390SD16—432no—
391M16—216no—
392M16—48no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8C22×D418yes—
399C4⋊C4—28no—
400C4⋊C4—28no—
401C4⋊C4—48no—
402C4⋊C4—416no—
403C4⋊C4—48no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C22⋊C4—48no—
412C22⋊C4—48no—
413C22⋊C4—416no—
414C22⋊C4—48no—
415C22⋊C4—44no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C42—22no—
419C42—24no—
420C42—24no—
421C42—24no—
422C42—22no—
423C42—24no—
424C42—22no—
425C42—22no—
426C42—44no—
427C42#12118yes—
428C42C22×D412yes—
429C42C22×D412yes—
430C42C2×#4112yes—
431C42C2×#4112yes—
432C23—22no—
433C23—22no—
434C23—28no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—48no—
442C23—48no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—44no—
450C23C2×#12112yes—
451C23#53512yes—
452C23#53512yes—
453C23C2×#13714yes—
454C23C23×D411yescharacteristic
455Q8—48no—
456Q8—44no—
457Q8—48no—
458Q8—44no—
459Q8—88no—
460D4—44no—
461D4—44no—
462D4—816no—
463D4—88no—
464D4—88no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—216no—
470C2×C4—216no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—48no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—44no—
481C2×C4—48no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—48no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—48no—
496C2×C4—88no—
497C2×C4—88no—
498C2×C4—88no—
499C2×C4C2×#12114yes—
500C2×C4C2×#12112yes—
501C2×C4C2×#12114yes—
502C2×C4#53512yes—
503C2×C4#53512yes—
504C2×C4#53518yes—
505C2×C4#54018yes—
506C2×C4C2×#13714yes—
507C2×C4C2×#13714yes—
508C2×C4#57514yes—
509C2×C4C23×D411yescharacteristic
510C2×C4C22×#4111yescharacteristic
511C2×C4C2×H(5,𝔽2)12yes—
512C2×C4C2×H(5,𝔽2)12yes—
513C8—28no—
514C8—44no—
515C22—24no—
516C22—22no—
517C22—22no—
518C22—24no—
519C22—24no—
520C22—24no—
521C22—24no—
522C22—416no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—84no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—84no—
532C22#261914yes—
533C22C22×#12111yescharacteristic
534C22C2×#53511yescharacteristic
535C22C2×#53511yescharacteristic
536C22C22×#13712yes—
537C22C2×#57512yes—
538C4—24no—
539C4—24no—
540C4—42no—
541C4—42no—
542C4—82no—
543C4—84no—
544C4—84no—
545C4—84no—
546C4#261914yes—
547C4#305618yes—
548C4C2×#53512yes—
549C4C2×#54012yes—
550C2—24no—
551C2—44no—
552C2—162no—
553C2C2×#261911yescharacteristic
554C2C2×#305612yes—
555C2#6080011yescharacteristic
556C2#6080111yescharacteristic
557C2#6080412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36624C2
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×#3152C2semidirect
2. C2×#2920C2semidirect
4. C2×#2857C2semidirect
5. #29815C2semidirect
6. #29808C2semidirect
7. #29807C2semidirect
8. #23547C2semidirect
9. #23392C2semidirect
10. #21202C2semidirect
11. #21200C2semidirect
12. #20764C2semidirect
20. C2×#497C22semidirect
21. C2×#494C22semidirect
22. C2×#492C22semidirect
49. #2056C22semidirect
50. #2055C22semidirect
53. #1960C22semidirect
54. #1959C22semidirect
55. #1957C22semidirect
56. #1956C22semidirect
57. #1956C22semidirect
58. #1836C22semidirect
59. #1834C22semidirect
125. C4×M16D4semidirect
377. C22×C4#137semidirect

Maximal subgroups

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

subgroupindex
C2×#31522
C2×#29202
#366242
C2×#28572
#298152
#298082
#298072
#235472
#233922
#212022
#212002
#207642