Finite Group Databasebeta

← #6643212

Subgroups — #6643212

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#2494C211yesmaximal, characteristic
3#33149C212yesmaximal, complemented
4#31706C218yesmaximal, complemented
5#31446C218yesmaximal, complemented
6#29968C212yesmaximal, complemented
7#29302C212yesmaximal, complemented
8#27969C212yesmaximal, complemented
9#27601C212yesmaximal
10#27600C212yesmaximal
11#21906C211yesmaximal, characteristic
12C2×#543—21no—
13#2857C2212yes—
14#2857C2212yes—
15#2659—21no—
16#2659—21no—
17#2549—24no—
18#2494—24no—
19#2494C2218yes—
20#2440C2214yes—
21#2440C2214yes—
22C2×#401—21no—
23C2×#397—21no—
24#2315—21no—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#388—21no—
28C2×#387—21no—
29C2×#382C2211yescharacteristic
30C2×#380C2212yes—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35#2101C2218yescomplemented
36#2100C2214yescomplemented
37#2088C2218yescomplemented
38#2084C2214yescomplemented
39#2077C2218yes—
40#2075C2218yes—
41#2058C2214yescomplemented
42#2057C2214yescomplemented
43#1993C2214yescomplemented
44#1987C2214yescomplemented
45#1947C2214yescomplemented
46#1944C2218yescomplemented
47#1924C2218yes—
48#1921C2218yescomplemented
49#1853C2214yes—
50#1853C2214yes—
51#1852C2214yes—
52#1852C2214yes—
53#1836C2214yescomplemented
54#1824C2214yes—
55#1823C2214yes—
56#1823C2214yes—
57#1823C2214yes—
58#1797C2212yes—
59#1779C2212yes—
60#1749C2212yes—
61C24×C4C2311yescharacteristic
62#543—48no—
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2312yes—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—22no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×#107C2318yes—
86C2×#106C2318yes—
87C2×C4⋊C8C2314yes—
88C2×C22⋊C8C2314yes—
89C2×C22⋊C8C2314yes—
90C2×C22⋊C8C2314yes—
91C2×#97C2314yes—
92#401—28no—
93#398—24no—
94#398—24no—
95#397—28no—
96#395—24no—
97#395—24no—
98#391—22no—
99#391—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389C2314yes—
105#389C2314yes—
106#389C2314yes—
107#389C2314yes—
108#388—28no—
109#387—28no—
110#385—24no—
111#385—24no—
112#384C2312yes—
113#384C2312yes—
114#382—22no—
115#382C2314yes—
116#380—22no—
117#380—22no—
118#380—22no—
119#380—22no—
120#380—22no—
121#380—22no—
122#380C2314yes—
123#380C2314yes—
124#380C2314yes—
125#380C2314yes—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2314yes—
128C4×C22⋊C4D414yes—
129C4×C22⋊C4C2314yes—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95D411yescharacteristic
139C2×#95D411yescharacteristic
140#340C2318yes—
141#340C2318yes—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#336C2314yes—
147#336C2314yes—
148C25C2×D411yescharacteristic
149C22×Q8—22no—
150C22×Q8—22no—
151C23×C4—24no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D411yescharacteristic
166C23×C4C2412yes—
167C23×C4C2411yescharacteristic
168C22×C8C2418yes—
169C4⋊Q8—44no—
170C4⋊Q8—44no—
171C22⋊Q8—416no—
172C22⋊Q8—416no—
173C22⋊Q8—48no—
174C22⋊Q8—48no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4C2414yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2412yes—
202C2×C4⋊C4C2412yes—
203C2×C4⋊C4C2412yes—
204C2×C4⋊C4C2412yes—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—48no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4C2×D414yes—
215C2×C22⋊C4C2×D414yes—
216C2×C42—24no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—82no—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2412yes—
223C2×C42C2412yes—
224#107—48no—
225#106—48no—
226C4⋊C8—44no—
227C22⋊C8—28no—
228C22⋊C8—44no—
229C22⋊C8—44no—
230#97—44no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95C2×D414yes—
243#95C2×D414yes—
244C24—22no—
245C24—22no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24—41no—
250C24AGL(1,ℤ/8)14yes—
251C24C22×D412yes—
252C24C22×D411yescharacteristic
253C2×Q8—44no—
254C2×Q8—44no—
255C2×Q8—82no—
256C2×Q8—82no—
257C2×Q8—82no—
258C2×Q8—82no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—88no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4#13714yes—
307C22×C4C22×D412yes—
308C22×C4C22×D412yes—
309C22×C4C22×D412yes—
310C22×C4C22×D412yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D412yes—
313C22×C42−1+414yes—
314C22×C4C2511yescharacteristic
315C2×C8—28no—
316C2×C8—28no—
317C2×C8—28no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—28no—
322C4⋊C4—24no—
323C4⋊C4—28no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—48no—
329C4⋊C4—48no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—48no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4—82no—
337C4⋊C4—84no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C22⋊C4—28no—
341C22⋊C4—28no—
342C22⋊C4—48no—
343C22⋊C4—48no—
344C22⋊C4—48no—
345C22⋊C4—84no—
346C42—24no—
347C42—24no—
348C42—42no—
349C42—42no—
350C42—84no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—28no—
362C23—44no—
363C23—44no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23#53612yes—
369C23C2×AGL(1,ℤ/8)14yes—
370C23C2×AGL(1,ℤ/8)12yes—
371C23C2×#13712yes—
372C23C23×D411yescharacteristic
373Q8—164no—
374Q8—164no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—28no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—44no—
393C2×C4—48no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—84no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4—164no—
422C2×C4—164no—
423C2×C4#53612yes—
424C2×C4C2×#13714yes—
425C2×C4C2×2−1+412yes—
426C2×C4#58512yes—
427C2×C4#58512yes—
428C8—48no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—24no—
436C22—24no—
437C22—24no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—88no—
443C22#323814yes—
444C22C2×#53611yescharacteristic
445C22#348511yescharacteristic
446C22#348511yescharacteristic
447C22C22×AGL(1,ℤ/8)11yescharacteristic
448C22C22×#13711yescharacteristic
449C22#358511yescharacteristic
450C22#358611yescharacteristic
451C4—24no—
452C4—22no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—84no—
458C4—82no—
459C4—84no—
460C4—82no—
461C4—82no—
462C4—82no—
463C4#356414yes—
464C2—24no—
465C2—44no—
466C2#3697011yescharacteristic
467C2C2×#323811yescharacteristic
468C2#6087011yescharacteristic
469C2#6170411yescharacteristic
470C2#6170711yescharacteristic
471C2#6170811yescharacteristic
472C2#6172311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33149C2semidirect
4. #31706C2semidirect
5. #31446C2semidirect
6. #29968C2semidirect
7. #29302C2semidirect
8. #27969C2semidirect
35. #2101C22semidirect
36. #2100C22semidirect
37. #2088C22semidirect
38. #2084C22semidirect
41. #2058C22semidirect
42. #2057C22semidirect
43. #1993C22semidirect
44. #1987C22semidirect
45. #1947C22semidirect
46. #1944C22semidirect
48. #1921C22semidirect
53. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#24942
#331492
#317062
#314462
#299682
#293022
#279692
#276012
#276002
#219062