Finite Group Databasebeta

← #6434023

Subgroups — #6434023

Order 512.

2230 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36624C212yesmaximal, complemented
2#30365C214yesmaximal, complemented
3#30324C214yesmaximal, complemented
4#30293C211yesmaximal, characteristic, complemented
5#30239C211yesmaximal, characteristic, complemented
6#30185C212yesmaximal, complemented
7#29732C214yesmaximal, complemented
8#23761C214yesmaximal, complemented
9#23573C214yesmaximal, complemented
10#21526C214yesmaximal, complemented
11C2×#1749C211yesmaximal, characteristic, complemented
12C2×#532C2212yes—
13C2×#525—22no—
14C23×C42C2211yescharacteristic
15C2×#466C2214yes—
16C2×#442C2214yescomplemented
17C22×C22⋊C8C2212yes—
18C22×#97C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2856C2212yes—
22#2856C2212yes—
23#2854—22no—
24#2854C2214yes—
25#2850—22no—
26#2850—22no—
27#2847—22no—
28#2847C2214yes—
29C4×#121—22no—
30C4×#121—22no—
31C4×#121C2214yes—
32C4×#121C2214yes—
33#2297—22no—
34#2285C2212yes—
35#2099C2214yescomplemented
36#2093C2214yescomplemented
37#2084C2214yescomplemented
38#2076C2218yes—
39#2071C2214yescomplemented
40#2058C2214yescomplemented
41#2041C2214yescomplemented
42#2034C2218yes—
43#2034C2218yes—
44#2023C2214yescomplemented
45#2011C2214yescomplemented
46#1999C2218yes—
47#1972C2214yescomplemented
48#1943C2214yescomplemented
49#1936C2218yescomplemented
50#1893C2218yes—
51#1807C2214yes—
52#1776C2214yescomplemented
53#1762C2218yescomplemented
54#1749C2214yes—
55#1749C2214yes—
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C22×#41—22no—
59C24×C4D412yes—
60C24×C4C2311yescharacteristic
61C22×D8C2314yes—
62C23×C8C2312yes—
63C23×C8C2312yes—
64#532—48no—
65C2×C4⋊Q8C2312yes—
66C2×#127C2312yes—
67C2×#124—22no—
68#525—416no—
69C2×#123—24no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×#121—22no—
73C2×#121—22no—
74C2×#121—22no—
75C2×#121—22no—
76C2×#121—22no—
77C2×#121C2314yes—
78C2×#121C2314yes—
79C2×#121C2314yes—
80C2×#120—24no—
81C2×C4×D4—24no—
82C2×C4×D4—24no—
83C2×C4×D4—22no—
84C2×C4×D4C2314yes—
85C2×#117—22no—
86C22×C42—44no—
87C22×C42D412yes—
88C22×C42D412yes—
89C22×C42D412yes—
90C22×C42C2311yescharacteristic
91C22×C42C2312yes—
92C22×C42C2312yes—
93C22×C42C2311yescharacteristic
94#466—216no—
95#442—48no—
96C2×#107C2314yes—
97C2×#106C2314yes—
98C2×#106C2314yes—
99C2×#102C2318yes—
100C2×#102C2318yes—
101C2×#102C2318yes—
102C2×C22⋊C8—28no—
103C2×C22⋊C8C2318yes—
104C2×#97—44no—
105C2×#97C2314yes—
106C2×#97C2314yes—
107#391—22no—
108#391—22no—
109#390—22no—
110#390—22no—
111#390—22no—
112#390—22no—
113#390C2314yes—
114#390C2314yes—
115#389—22no—
116#389—22no—
117#389C2314yes—
118#389C2314yes—
119#388—24no—
120#388—24no—
121#388—24no—
122#388—24no—
123#386—24no—
124#386—24no—
125#386—24no—
126#386—24no—
127#386—24no—
128#386—24no—
129#384C2312yes—
130#384C2312yes—
131#382—22no—
132#380—22no—
133#380—22no—
134#380—22no—
135#380C2314yes—
136#380C2314yes—
137#380C2314yes—
138C4×C4⋊C4—22no—
139C4×C4⋊C4C2314yes—
140C4×C22⋊C4—24no—
141C4×C22⋊C4—24no—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#336C2314yes—
146#336C2314yes—
147C25C2×D411yescharacteristic
148C2×#41—48no—
149C2×#41—48no—
150C2×#41—48no—
151C22×Q8—22no—
152C22×D4—24no—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4C2414yes—
158C23×C4—28no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—44no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4—42no—
165C23×C4—44no—
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176C2×D8—28no—
177C2×D8—44no—
178C2×D8—44no—
179C22×C8—24no—
180C22×C8—42no—
181C22×C8—42no—
182C22×C8—42no—
183C22×C8—42no—
184C22×C8#4118yescomplemented
185C22×C8C2414yes—
186C22×C8C2414yes—
187C4⋊Q8—44no—
188#127—44no—
189#124—48no—
190#123—416no—
191C22⋊Q8—48no—
192C22⋊Q8—48no—
193#121—216no—
194#121—44no—
195#121—44no—
196#121—48no—
197#121—48no—
198#121—48no—
199#121—48no—
200#121—48no—
201#121—48no—
202#120—416no—
203C4×D4—216no—
204C4×D4—48no—
205C4×D4—416no—
206C4×D4—416no—
207#117—48no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4C2414yes—
216C2×C4⋊C4C2414yes—
217C2×C4⋊C4C2414yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C42—28no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—44no—
233C2×C42—44no—
234C2×C42—44no—
235C2×C42—44no—
236C2×C42—42no—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2412yes—
245C2×C42C2411yescharacteristic
246C2×C42C2411yescharacteristic
247#107—28no—
248#106—44no—
249#106—44no—
250#102—216no—
251#102—48no—
252#102—48no—
253C22⋊C8—216no—
254#97—28no—
255#97—44no—
256#95—24no—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262C24—22no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—41no—
267C24—41no—
268C24C22×D412yes—
269C24C22×D411yescharacteristic
270#41—832no—
271C2×Q8—44no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×D4—28no—
275C2×D4—28no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—48no—
283C2×D4—48no—
284C2×D4—48no—
285C2×D4—48no—
286C2×D4—48no—
287C2×D4—48no—
288C2×D4—48no—
289C2×D4—48no—
290C2×D4—48no—
291C2×D4—44no—
292C2×D4—44no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—432no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—48no—
331C22×C4—48no—
332C22×C4AGL(1,ℤ/8)18yes—
333C22×C4C22×D412yes—
334C22×C4C22×D412yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C22×D412yes—
339C22×C4C2×#4114yes—
340C22×C4C2511yescharacteristic
341D8—416no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—48no—
348C2×C8—48no—
349C2×C8C2×#4118yes—
350C4⋊C4—28no—
351C4⋊C4—28no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—44no—
355C4⋊C4—48no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—48no—
360C4⋊C4—44no—
361C22⋊C4—48no—
362C22⋊C4—48no—
363C22⋊C4—48no—
364C22⋊C4—48no—
365C22⋊C4—48no—
366C22⋊C4—48no—
367C22⋊C4—48no—
368C22⋊C4—48no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—48no—
372C42—24no—
373C42—24no—
374C42—24no—
375C42—22no—
376C42—24no—
377C42—41no—
378C42—42no—
379C42—41no—
380C42—42no—
381C42—48no—
382C42H(5,𝔽2)14yes—
383C23—21no—
384C23—21no—
385C23—22no—
386C23—28no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—48no—
394C23—48no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23#53212yes—
405C23C2×AGL(1,ℤ/8)14yes—
406C23#57514yes—
407C23C23×D411yescharacteristic
408Q8—88no—
409D4—48no—
410D4—48no—
411D4—88no—
412D4—88no—
413D4—816no—
414D4—816no—
415D4—816no—
416C2×C4—216no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—48no—
434C2×C4—48no—
435C2×C4—48no—
436C2×C4—48no—
437C2×C4—48no—
438C2×C4—48no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—816no—
452C2×C4#53212yes—
453C2×C4#53414yes—
454C2×C4C2×AGL(1,ℤ/8)14yes—
455C2×C4C2×AGL(1,ℤ/8)14yes—
456C2×C4C22×#4112yes—
457C2×C4C2×H(5,𝔽2)12yes—
458C2×C4#58511yescharacteristic
459C2×C4#58511yescharacteristic
460C8—28no—
461C8—44no—
462C22—22no—
463C22—22no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—416no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—84no—
475C22—84no—
476C22—84no—
477C22—84no—
478C22—84no—
479C22—84no—
480C22—84no—
481C22#318414yes—
482C22C2×#53211yescharacteristic
483C22C2×#53411yescharacteristic
484C22#344911yescharacteristic
485C22C22×AGL(1,ℤ/8)12yes—
486C22C2×#57512yes—
487C4—22no—
488C4—24no—
489C4—24no—
490C4—24no—
491C4—42no—
492C4—42no—
493C4—84no—
494C4—84no—
495C4—84no—
496C4#345912yes—
497C4#346012yes—
498C2—24no—
499C2—44no—
500C2—164no—
501C2#3457511yescharacteristic
502C2#6068811yescharacteristic
503C2C2×#318411yescharacteristic
504C2#6075012yes—
505C2#6096212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
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. #36624C2semidirect
2. #30365C2semidirect
3. #30324C2semidirect
4. #30293C2semidirect
5. #30239C2semidirect
6. #30185C2semidirect
7. #29732C2semidirect
8. #23761C2semidirect
9. #23573C2semidirect
10. #21526C2semidirect
11. C2×#1749C2semidirect
16. C2×#442C22semidirect
35. #2099C22semidirect
36. #2093C22semidirect
37. #2084C22semidirect
39. #2071C22semidirect
40. #2058C22semidirect
41. #2041C22semidirect
44. #2023C22semidirect
45. #2011C22semidirect
47. #1972C22semidirect
48. #1943C22semidirect
49. #1936C22semidirect
52. #1776C22semidirect
53. #1762C22semidirect
184. C22×C8#41semidirect

Maximal subgroups

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

subgroupindex
#366242
#303652
#303242
#302932
#302392
#301852
#297322
#237612
#235732
#215262
C2×#17492