Finite Group Databasebeta

← #6434024

Subgroups — #6434024

Order 512.

1834 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C212yesmaximal
2#30367C214yesmaximal, complemented
3#30326C214yesmaximal, complemented
4#30293C211yesmaximal, characteristic, complemented
5#30242C211yesmaximal, characteristic, complemented
6#30187C212yesmaximal, complemented
7#29733C214yesmaximal
8#23762C214yesmaximal, complemented
9#23577C214yesmaximal, complemented
10#21527C214yesmaximal, complemented
11C2×#1749C211yesmaximal, characteristic
12C2×#533C2212yes—
13C23×C42C2211yescharacteristic
14C2×#467C2214yes—
15C2×#441C2214yescomplemented
16C22×C22⋊C8C2212yes—
17C22×#97C2212yes—
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#2084C2214yescomplemented
38#2075C2218yes—
39#2072C2214yescomplemented
40#2058C2214yescomplemented
41#2043C2214yescomplemented
42#2035C2218yes—
43#2035C2218yes—
44#2024C2214yescomplemented
45#2016C2214yescomplemented
46#2001C2218yes—
47#1971C2214yescomplemented
48#1943C2214yescomplemented
49#1935C2218yescomplemented
50#1892C2218yes—
51#1807C2214yes—
52#1776C2214yescomplemented
53#1763C2218yescomplemented
54#1749C2214yes—
55#1749C2214yes—
56C2×#336C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C24×C4D412yes—
59C24×C4C2311yescharacteristic
60C22×Q16C2314yes—
61C23×C8C2312yes—
62C23×C8C2312yes—
63#533—48no—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×#125—22no—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8C2314yes—
71C2×C22⋊Q8C2314yes—
72C2×C22⋊Q8C2314yes—
73C2×C4×Q8—22no—
74C2×C4×Q8C2314yes—
75C2×#117—22no—
76C2×#117—22no—
77C22×C42—44no—
78C22×C42D412yes—
79C22×C42D412yes—
80C22×C42D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2312yes—
83C22×C42C2312yes—
84C22×C42C2311yescharacteristic
85#467—216no—
86#441—48no—
87C2×#107C2314yes—
88C2×#106C2314yes—
89C2×#106C2314yes—
90C2×#103C2318yes—
91C2×#103C2318yes—
92C2×#103C2318yes—
93C2×C22⋊C8—28no—
94C2×C22⋊C8C2318yes—
95C2×#97—44no—
96C2×#97C2314yes—
97C2×#97C2314yes—
98#391—22no—
99#391—22no—
100#391C2314yes—
101#391C2314yes—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389C2314yes—
109#389C2314yes—
110#387—24no—
111#387—24no—
112#387—24no—
113#387—24no—
114#385—24no—
115#385—24no—
116#385—24no—
117#385—24no—
118#385—24no—
119#385—24no—
120#384C2312yes—
121#384C2312yes—
122#382—22no—
123#381—24no—
124#380—22no—
125#380—22no—
126#380—22no—
127#380—22no—
128#380—22no—
129#380—22no—
130#380—22no—
131#380C2314yes—
132#380C2314yes—
133#380C2314yes—
134#379—24no—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—24no—
137C4×C4⋊C4—24no—
138C4×C4⋊C4C2314yes—
139C4×C22⋊C4—24no—
140C4×C22⋊C4—24no—
141#340C2318yes—
142#340C2318yes—
143#340C2318yes—
144#336C2314yes—
145#336C2314yes—
146C25C2×D411yescharacteristic
147C22×Q8—22no—
148C22×Q8C2414yes—
149C23×C4—28no—
150C23×C4—24no—
151C23×C4—44no—
152C23×C4—41no—
153C23×C4—41no—
154C23×C4—42no—
155C23×C4—44no—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2412yes—
165C23×C4C2411yescharacteristic
166C2×Q16—28no—
167C2×Q16—44no—
168C2×Q16—44no—
169C22×C8—24no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8#4118yes—
175C22×C8C2414yes—
176C22×C8C2414yes—
177C4⋊Q8—44no—
178C4⋊Q8—44no—
179#125—48no—
180C22⋊Q8—216no—
181C22⋊Q8—44no—
182C22⋊Q8—44no—
183C22⋊Q8—48no—
184C22⋊Q8—48no—
185C22⋊Q8—48no—
186C22⋊Q8—48no—
187C4×Q8—216no—
188C4×Q8—48no—
189#117—48no—
190#117—48no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2414yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C42—28no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—24no—
217C2×C42—44no—
218C2×C42—44no—
219C2×C42—44no—
220C2×C42—44no—
221C2×C42—42no—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2412yes—
230C2×C42C2411yescharacteristic
231C2×C42C2411yescharacteristic
232#107—28no—
233#106—44no—
234#106—44no—
235#103—216no—
236#103—48no—
237#103—48no—
238C22⋊C8—216no—
239#97—28no—
240#97—44no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251C24—22no—
252C24—24no—
253C24—24no—
254C24—41no—
255C24—41no—
256C24C22×D412yes—
257C24C22×D411yescharacteristic
258C2×Q8—28no—
259C2×Q8—28no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—44no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—432no—
300C22×C4—48no—
301C22×C4—48no—
302C22×C4#13718yes—
303C22×C4C22×D412yes—
304C22×C4C22×D412yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D412yes—
307C22×C4C22×D412yes—
308C22×C4C22×D412yes—
309C22×C4C2×#4114yes—
310C22×C4C2511yescharacteristic
311Q16—416no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—48no—
318C2×C8—48no—
319C2×C8C2×#4118yes—
320C4⋊C4—28no—
321C4⋊C4—28no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—48no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—48no—
332C4⋊C4—48no—
333C4⋊C4—48no—
334C4⋊C4—48no—
335C4⋊C4—44no—
336C4⋊C4—48no—
337C22⋊C4—48no—
338C22⋊C4—48no—
339C22⋊C4—48no—
340C22⋊C4—48no—
341C22⋊C4—48no—
342C22⋊C4—48no—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—22no—
347C42—24no—
348C42—41no—
349C42—42no—
350C42—41no—
351C42—42no—
352C42—48no—
353C42—48no—
354C42H(5,𝔽2)14yes—
355C23—21no—
356C23—21no—
357C23—22no—
358C23—28no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—48no—
366C23—48no—
367C23—44no—
368C23—44no—
369C23#53212yes—
370C23C2×#13714yes—
371C23#57514yes—
372C23C23×D411yescharacteristic
373Q8—48no—
374Q8—48no—
375Q8—88no—
376C2×C4—216no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4#53212yes—
415C2×C4#53414yes—
416C2×C4C2×#13714yes—
417C2×C4C2×#13714yes—
418C2×C4C22×#4112yes—
419C2×C4C2×H(5,𝔽2)12yes—
420C2×C4#58511yescharacteristic
421C2×C4#58511yescharacteristic
422C8—28no—
423C8—44no—
424C22—22no—
425C22—22no—
426C22—24no—
427C22—24no—
428C22—24no—
429C22—24no—
430C22—24no—
431C22—416no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22#318514yes—
437C22C2×#53211yescharacteristic
438C22C2×#53411yescharacteristic
439C22#344911yescharacteristic
440C22C22×#13712yes—
441C22C2×#57512yes—
442C4—22no—
443C4—24no—
444C4—24no—
445C4—24no—
446C4—42no—
447C4—42no—
448C4—84no—
449C4—84no—
450C4—84no—
451C4—84no—
452C4#345912yes—
453C4#346012yes—
454C2—24no—
455C2—44no—
456C2#3457511yescharacteristic
457C2#6068811yescharacteristic
458C2C2×#318511yescharacteristic
459C2#6075112yes—
460C2#6096312yes—
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
2. #30367C2semidirect
3. #30326C2semidirect
4. #30293C2semidirect
5. #30242C2semidirect
6. #30187C2semidirect
8. #23762C2semidirect
9. #23577C2semidirect
10. #21527C2semidirect
15. C2×#441C22semidirect
35. #2100C22semidirect
36. #2095C22semidirect
37. #2084C22semidirect
39. #2072C22semidirect
40. #2058C22semidirect
41. #2043C22semidirect
44. #2024C22semidirect
45. #2016C22semidirect
47. #1971C22semidirect
48. #1943C22semidirect
49. #1935C22semidirect
52. #1776C22semidirect
53. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#366252
#303672
#303262
#302932
#302422
#301872
#297332
#237622
#235772
#215272
C2×#17492