Finite Group Databasebeta

← #6433975

Subgroups — #6433975

Order 512.

1866 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C212yesmaximal
2#30358C214yesmaximal, complemented
3#30318C214yesmaximal, complemented
4#30291C211yesmaximal, characteristic, complemented
5#30242C211yesmaximal, characteristic, complemented
6#30187C212yesmaximal, complemented
7#29731C214yesmaximal
8#23757C214yesmaximal, complemented
9#23571C214yesmaximal, complemented
10C4×#467C214yesmaximal, complemented
11C2×C4×C22⋊C8C211yesmaximal, characteristic
12C2×#533C2212yes—
13C23×C42C2211yescharacteristic
14C2×#467C2214yes—
15C2×C4×Q16C2214yescomplemented
16C22×C22⋊C8C2212yes—
17C22×C4×C8C2212yes—
18#2857C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2857C2212yes—
22#2855—22no—
23#2855C2214yes—
24#2851—22no—
25#2851—22no—
26#2848—22no—
27#2848C2214yes—
28C4×C22⋊Q8—22no—
29C4×C22⋊Q8—22no—
30C4×C22⋊Q8C2214yes—
31C4×C22⋊Q8C2214yes—
32#2296—22no—
33#2285C2212yes—
34#2280—22no—
35#2100C2214yescomplemented
36#2095C2214yescomplemented
37#2083C2214yescomplemented
38#2076C2218yes—
39#2072C2214yescomplemented
40#2057C2214yescomplemented
41#2043C2214yescomplemented
42#2036C2218yes—
43#2036C2218yes—
44#2024C2214yescomplemented
45#2016C2214yescomplemented
46#2001C2218yes—
47#1969C2214yescomplemented
48#1942C2214yescomplemented
49#1933C2218yescomplemented
50#1892C2218yes—
51#1808C2214yes—
52C4×#107C2214yescomplemented
53C4×#103C2218yescomplemented
54C4×C22⋊C8C2214yes—
55C4×C22⋊C8C2214yes—
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C24×C4D411yescharacteristic
59C24×C4D411yescharacteristic
60C24×C4C2311yescharacteristic
61C22×Q16C2314yes—
62C23×C8C2312yes—
63C23×C8C2312yes—
64#533—48no—
65C2×C4⋊Q8C2312yes—
66C2×C4⋊Q8C2312yes—
67C2×#125—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8C2314yes—
72C2×C22⋊Q8C2314yes—
73C2×C22⋊Q8C2314yes—
74C2×C4×Q8—22no—
75C2×C4×Q8C2314yes—
76C2×#117—22no—
77C2×#117—22no—
78C22×C42—44no—
79C22×C42D411yescharacteristic
80C22×C42D412yes—
81C22×C42D412yes—
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2312yes—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87#467—216no—
88C4×Q16—48no—
89C2×#107C2314yes—
90C2×#107C2314yes—
91C2×#107C2314yes—
92C2×#103C2318yes—
93C2×#103C2318yes—
94C2×#103C2318yes—
95C2×C22⋊C8—28no—
96C2×C22⋊C8C2318yes—
97C2×C4×C8—44no—
98C2×C4×C8C2314yes—
99C2×C4×C8C2314yes—
100#391—22no—
101#391—22no—
102#391C2314yes—
103#391C2314yes—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389C2314yes—
111#389C2314yes—
112#387—24no—
113#387—24no—
114#387—24no—
115#387—24no—
116#385—24no—
117#385—24no—
118#385—24no—
119#385—24no—
120#385—24no—
121#385—24no—
122#384C2312yes—
123#384C2312yes—
124#382—22no—
125#381—24no—
126#380—22no—
127#380—22no—
128#380—22no—
129#380—22no—
130#380—22no—
131#380—22no—
132#380—22no—
133#380C2314yes—
134#380C2314yes—
135#380C2314yes—
136#379—24no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—24no—
139C4×C4⋊C4—24no—
140C4×C4⋊C4C2314yes—
141C4×C22⋊C4—24no—
142C4×C22⋊C4—24no—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#336C2314yes—
147#336C2314yes—
148C25C2×D411yescharacteristic
149C22×Q8—22no—
150C22×Q8C2414yes—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—41no—
158C23×C4—41no—
159C23×C4—42no—
160C23×C4Q1618yescomplemented
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2412yes—
172C23×C4C2411yescharacteristic
173C2×Q16—28no—
174C2×Q16—44no—
175C2×Q16—44no—
176C22×C8—24no—
177C22×C8—42no—
178C22×C8—42no—
179C22×C8—42no—
180C22×C8—42no—
181C22×C8#4118yes—
182C22×C8C2414yes—
183C22×C8C2414yes—
184C4⋊Q8—44no—
185C4⋊Q8—44no—
186#125—48no—
187C22⋊Q8—216no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190C22⋊Q8—48no—
191C22⋊Q8—48no—
192C22⋊Q8—48no—
193C22⋊Q8—48no—
194C4×Q8—216no—
195C4×Q8—48no—
196#117—48no—
197#117—48no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4C2414yes—
212C2×C4⋊C4C2414yes—
213C2×C4⋊C4C2414yes—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C42—28no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—44no—
225C2×C42—44no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—44no—
229C2×C42—42no—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D412yes—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2412yes—
238C2×C42C2411yescharacteristic
239C2×C42C2411yescharacteristic
240#107—28no—
241#107—44no—
242#107—44no—
243#103—216no—
244#103—48no—
245#103—48no—
246C22⋊C8—216no—
247C4×C8—28no—
248C4×C8—44no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—24no—
259C24—22no—
260C24—22no—
261C24—41no—
262C24—41no—
263C24—41no—
264C24—41no—
265C24C2×Q1614yes—
266C24C22×D412yes—
267C24C22×D411yescharacteristic
268C2×Q8—28no—
269C2×Q8—28no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—44no—
273C2×Q8—44no—
274C2×Q8—44no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—416no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—416no—
313C22×C4—48no—
314C22×C4—48no—
315C22×C4C2×Q1618yes—
316C22×C4C2×Q1614yes—
317C22×C4C2×Q1618yes—
318C22×C4C2×Q1614yes—
319C22×C4#13514yes—
320C22×C4C22×D412yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C22×D412yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C2×#4114yes—
329C22×C4C2511yescharacteristic
330Q16—416no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—48no—
337C2×C8—48no—
338C2×C8C2×#4118yes—
339C4⋊C4—28no—
340C4⋊C4—28no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—44no—
346C4⋊C4—44no—
347C4⋊C4—48no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—48no—
351C4⋊C4—48no—
352C4⋊C4—48no—
353C4⋊C4—44no—
354C4⋊C4—48no—
355C4⋊C4—48no—
356C22⋊C4—48no—
357C22⋊C4—48no—
358C22⋊C4—48no—
359C22⋊C4—48no—
360C22⋊C4—48no—
361C22⋊C4—48no—
362C42—22no—
363C42—22no—
364C42—24no—
365C42—24no—
366C42—22no—
367C42—24no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—48no—
372C42—48no—
373C42H(5,𝔽2)14yes—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—28no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23#53212yes—
392C23C22×Q1614yes—
393C23C22×Q1612yes—
394C23C2×#13512yes—
395C23C23×D411yescharacteristic
396Q8—48no—
397Q8—48no—
398Q8—88no—
399C2×C4—28no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—28no—
413C2×C4—216no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—44no—
420C2×C4—48no—
421C2×C4—44no—
422C2×C4—48no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4#53212yes—
441C2×C4#53412yes—
442C2×C4#53412yes—
443C2×C4C22×Q1612yes—
444C2×C4C22×Q1614yes—
445C2×C4C22×Q1612yes—
446C2×C4C2×#13514yes—
447C2×C4#57512yes—
448C2×C4#57512yes—
449C2×C4C22×#4112yes—
450C2×C4C2×H(5,𝔽2)12yes—
451C2×C4#58511yescharacteristic
452C2×C4#58511yescharacteristic
453C8—28no—
454C8—44no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—416no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22#317014yes—
470C22C2×#53211yescharacteristic
471C22C2×#53411yescharacteristic
472C22#344911yescharacteristic
473C22C23×Q1611yescharacteristic
474C22C22×#13511yescharacteristic
475C22C2×#57511yescharacteristic
476C22C2×#57511yescharacteristic
477C4—22no—
478C4—24no—
479C4—22no—
480C4—22no—
481C4—22no—
482C4—24no—
483C4—42no—
484C4—84no—
485C4—84no—
486C4—84no—
487C4—84no—
488C4#318914yes—
489C4#345912yes—
490C4#346012yes—
491C2—24no—
492C2—44no—
493C2#3457511yescharacteristic
494C2C2×#317011yescharacteristic
495C2#6067611yescharacteristic
496C2C2×#318911yescharacteristic
497C2#6073411yescharacteristic
498C2#6094311yescharacteristic
499C2#6094811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #30358C2semidirect
3. #30318C2semidirect
4. #30291C2semidirect
5. #30242C2semidirect
6. #30187C2semidirect
8. #23757C2semidirect
9. #23571C2semidirect
10. C4×#467C2semidirect
15. C2×C4×Q16C22semidirect
35. #2100C22semidirect
36. #2095C22semidirect
37. #2083C22semidirect
39. #2072C22semidirect
40. #2057C22semidirect
41. #2043C22semidirect
44. #2024C22semidirect
45. #2016C22semidirect
47. #1969C22semidirect
48. #1942C22semidirect
49. #1933C22semidirect
52. C4×#107C22semidirect
53. C4×#103C22semidirect
160. C23×C4Q16semidirect

Maximal subgroups

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

subgroupindex
#366252
#303582
#303182
#302912
#302422
#301872
#297312
#237572
#235712
C4×#4672
C2×C4×C22⋊C82