Finite Group Databasebeta

← #6477781

Subgroups — #6477781

Order 512.

1756 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3#33151C214yesmaximal, complemented
4#29972C214yesmaximal, complemented
5#25998C218yesmaximal, complemented
6#25996C218yesmaximal, complemented
7#21940C211yesmaximal, characteristic
8#21938C211yesmaximal, characteristic
9#21930C211yesmaximal, characteristic
10C2×#1823C211yesmaximal, characteristic
11C2×#1823C211yesmaximal, characteristic
12C22×#117—21no—
13C23×C4⋊C4C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2851C2214yes—
17#2646—22no—
18#2569—22no—
19#2569—22no—
20#2442C2214yes—
21#2442C2214yes—
22#2344—22no—
23#2344—22no—
24#2344C2214yes—
25#2344C2214yes—
26#2315—21no—
27#2315—21no—
28C2×#389C2212yes—
29C2×#387—22no—
30C2×#381—21no—
31C2×#380C2211yescharacteristic
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C22⋊C4C2211yescharacteristic
36#2101C2218yescomplemented
37#2088C2218yescomplemented
38#2087C2218yescomplemented
39#2059C2218yescomplemented
40#1989C2218yescomplemented
41#1949C2218yescomplemented
42#1921C2218yescomplemented
43#1919C2218yescomplemented
44#1853C2214yes—
45#1853C2214yes—
46#1852C2214yes—
47#1852C2214yes—
48#1850C2214yes—
49#1850C2214yes—
50#1823C2212yes—
51#1823C2212yes—
52#1823C2212yes—
53#1823C2212yes—
54#1823C2214yes—
55#1823C2214yes—
56#1808C2214yes—
57#1807C2214yes—
58#1797C2214yes—
59C2×#340C2214yes—
60C2×#340C2214yes—
61C24×C4—21no—
62C24×C4C2311yescharacteristic
63C23×C8C2312yes—
64C23×C8C2312yes—
65C2×#125C2314yes—
66C2×#117—24no—
67C2×#117—24no—
68C2×#117—48no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4Q818yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4D411yescharacteristic
86C22×C22⋊C4D411yescharacteristic
87C22×C42—22no—
88C22×C42C2312yes—
89C2×#107C2318yes—
90C2×#106C2318yes—
91C2×C4⋊C8C2318yes—
92C2×C22⋊C8—24no—
93C2×C22⋊C8—24no—
94C2×C22⋊C8C2312yes—
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2312yes—
97C2×C22⋊C8C2312yes—
98#400—24no—
99#400—24no—
100#398—24no—
101#398—24no—
102#389—24no—
103#389C2318yes—
104#389C2314yes—
105#389C2314yes—
106#387—24no—
107#387—24no—
108#387—28no—
109#387—28no—
110#385—24no—
111#385—24no—
112#385—24no—
113#385—24no—
114#382C2314yes—
115#381—24no—
116#381—24no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—24no—
124#380—24no—
125#380C2312yes—
126#380C2312yes—
127#380C2312yes—
128#380C2312yes—
129#380C2312yes—
130#380C2312yes—
131C4×C4⋊C4—24no—
132C4×C4⋊C4C2318yes—
133C4×C22⋊C4D412yes—
134C4×C22⋊C4D412yes—
135C4×C22⋊C4C2312yes—
136C4×C22⋊C4C2312yes—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—22no—
140C2×#95—21no—
141C2×#95—22no—
142C2×#95D412yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2318yes—
150C25C2×D411yescharacteristic
151C23×C4—28no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×Q814yes—
168C23×C4#4114yes—
169C23×C4C2411yescharacteristic
170C23×C4C2411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—24no—
173C22×C8—24no—
174C22×C8—44no—
175C22×C8—44no—
176C22×C8C2414yes—
177C22×C8C2414yes—
178#125—48no—
179#117—416no—
180#117—416no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4C2×Q814yes—
205C2×C4⋊C4C2×Q814yes—
206C2×C4⋊C4C2×Q818yes—
207C2×C4⋊C4C2×Q818yes—
208C2×C4⋊C4C2414yes—
209C2×C4⋊C4C2414yes—
210C2×C4⋊C4C2414yes—
211C2×C4⋊C4C2412yes—
212C2×C4⋊C4C2412yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4C2×D412yes—
225C2×C22⋊C4C2×D412yes—
226C2×C22⋊C4C2×D412yes—
227C2×C22⋊C4C2×D412yes—
228C2×C42—28no—
229C2×C42—44no—
230C2×C42—48no—
231C2×C42C2×D414yes—
232C2×C42C2414yes—
233#107—48no—
234#106—48no—
235C4⋊C8—48no—
236C22⋊C8—24no—
237C22⋊C8—24no—
238C22⋊C8—24no—
239C22⋊C8—24no—
240#95—28no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—28no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95C2×D414yes—
253#95C2×D414yes—
254C24—42no—
255C24—42no—
256C24C22⋊Q814yes—
257C24#13514yes—
258C24AGL(1,ℤ/8)14yes—
259C24C22×D411yescharacteristic
260C24C22×D411yescharacteristic
261C24C22×D411yescharacteristic
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—28no—
267C22×C4—28no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—48no—
277C22×C4—48no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—432no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4C22⋊Q814yes—
304C22×C4C22⋊Q814yes—
305C22×C4C22⋊Q814yes—
306C22×C4C22×D414yes—
307C22×C4C22×D414yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×Q812yes—
312C22×C4C22×Q812yes—
313C22×C4C22×Q812yes—
314C22×C4C2×#4112yes—
315C22×C4C2×#4112yes—
316C22×C4C2×#4112yes—
317C22×C4C2511yescharacteristic
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—28no—
321C2×C8—28no—
322C2×C8—84no—
323C2×C8—84no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—28no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—28no—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C4⋊C4—44no—
334C4⋊C4—416no—
335C4⋊C4—84no—
336C4⋊C4—84no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—84no—
340C4⋊C4—84no—
341C4⋊C4C22×Q814yes—
342C4⋊C4C22×Q814yes—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C42—28no—
354C42—44no—
355C42—416no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—44no—
363C23—44no—
364C23—48no—
365C23—48no—
366C23#478116yes—
367C23C2×C22⋊Q812yes—
368C23C2×C22⋊Q812yes—
369C23C2×C22⋊Q812yes—
370C23C2×#13512yes—
371C23C2×#13512yes—
372C23C2×#13512yes—
373C23C2×AGL(1,ℤ/8)12yes—
374C23C2×AGL(1,ℤ/8)12yes—
375C23C2×AGL(1,ℤ/8)12yes—
376C23C23×D411yescharacteristic
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—48no—
400C2×C4—416no—
401C2×C4—416no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
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—44no—
415C2×C4—84no—
416C2×C4—88no—
417C2×C4—164no—
418C2×C4—164no—
419C2×C4C2×C22⋊Q812yes—
420C2×C4C2×C22⋊Q812yes—
421C2×C4C2×C22⋊Q812yes—
422C2×C4C23×Q811yescharacteristic
423C2×C4C22×#4111yescharacteristic
424C2×C4#58514yes—
425C8—44no—
426C8—44no—
427C22—24no—
428C22—24no—
429C22—24no—
430C22—24no—
431C22—24no—
432C22—24no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—416no—
440C22C2×#47814yes—
441C22C2×#47814yes—
442C22C2×#47814yes—
443C22C22×C22⋊Q811yescharacteristic
444C22#348512yes—
445C22C22×#13511yescharacteristic
446C22C22×AGL(1,ℤ/8)11yescharacteristic
447C22#358612yes—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—44no—
452C4—44no—
453C4—44no—
454C4—44no—
455C4—84no—
456C4—84no—
457C4—84no—
458C4#345812yes—
459C4#349112yes—
460C2—24no—
461C2—24no—
462C2—24no—
463C2#3509011yescharacteristic
464C2C22×#47811yescharacteristic
465C2#6049111yescharacteristic
466C2#6169612yes—
467C2#6170612yes—
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. #33151C2semidirect
4. #29972C2semidirect
5. #25998C2semidirect
6. #25996C2semidirect
36. #2101C22semidirect
37. #2088C22semidirect
38. #2087C22semidirect
39. #2059C22semidirect
40. #1989C22semidirect
41. #1949C22semidirect
42. #1921C22semidirect
43. #1919C22semidirect

Maximal subgroups

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

subgroupindex
#350202
C2×#23442
#331512
#299722
#259982
#259962
#219402
#219382
#219302
C2×#18232