Finite Group Databasebeta

← #6528501

Subgroups — #6528501

Order 512.

1560 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2565C211yesmaximal, characteristic
3#29111C212yesmaximal
4#29105C212yesmaximal
5#27969C212yesmaximal
6#27960C212yesmaximal
7#27827C218yesmaximal, complemented
8#27741C218yesmaximal, complemented
9#21838C212yesmaximal
10#20740C212yesmaximal
11#20534C211yesmaximal, characteristic
12C2×#544—21no—
13C22×#125C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C2×#432C2212yes—
16#2857C2212yes—
17#2857C2212yes—
18#2659—21no—
19#2659—21no—
20#2565—24no—
21#2565C2218yes—
22#2552C2214yes—
23#2551C2214yes—
24#2550—24no—
25C2×#398—21no—
26C2×#398—21no—
27#2316—21no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#387—21no—
31C2×#387—21no—
32#2285C2212yes—
33C2×#384C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#2101C2218yes—
37#2099C2218yes—
38#2072C2214yes—
39#2072C2214yes—
40#2024C2214yes—
41#2024C2214yes—
42#1993C2214yes—
43#1992C2214yes—
44#1992C2214yes—
45#1990C2214yes—
46#1989C2218yes—
47#1935C2218yes—
48#1933C2218yes—
49#1836C2214yes—
50#1834C2214yes—
51#1824C2214yes—
52#1823C2214yes—
53#1823C2214yes—
54#1803C2214yes—
55#1803C2214yes—
56#1787C2214yes—
57#1771C2218yes—
58#1763C2218yes—
59#1749C2212yes—
60C4×C22⋊C8C2212yes—
61C23×Q8C2311yescharacteristic
62#544—48no—
63C2×C4⋊Q8—24no—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×#125—24no—
67C2×#125—22no—
68C2×#125—28no—
69C2×#125C2314yes—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—28no—
74C2×C4×Q8C2314yes—
75C2×#117—22no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#432—28no—
91C2×C4⋊C8C2314yes—
92C2×#103C2318yes—
93C2×#103C2318yes—
94C2×C22⋊C8C2314yes—
95C2×#97C2314yes—
96C2×#97C2314yes—
97C2×C4×C8C2314yes—
98#398—28no—
99#398—28no—
100#398—24no—
101#398—24no—
102#395—24no—
103#395—24no—
104#391—22no—
105#391—22no—
106#391—22no—
107#391—22no—
108#391C2314yes—
109#391C2314yes—
110#389—24no—
111#389—24no—
112#389—22no—
113#389—22no—
114#389C2314yes—
115#389C2314yes—
116#387—28no—
117#387—28no—
118#384—22no—
119#384C2314yes—
120#384C2312yes—
121#384C2312yes—
122#384C2312yes—
123#384C2312yes—
124#382—22no—
125#382—22no—
126#382C2314yes—
127#380—22no—
128#380—22no—
129#380—22no—
130#380C2314yes—
131C4×C4⋊C4—22no—
132C4×C4⋊C4D414yes—
133C4×C4⋊C4C2314yes—
134C4×C4⋊C4C2314yes—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95—21no—
140C2×#95—21no—
141C2×#95D411yescharacteristic
142C2×#95D411yescharacteristic
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#336C2314yes—
150#336C2314yes—
151C22×Q8—24no—
152C22×Q8—21no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—24no—
156C22×Q8—42no—
157C22×Q8C2412yes—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169C22×C8C2418yes—
170C4⋊Q8—44no—
171C4⋊Q8—44no—
172C4⋊Q8—416no—
173#125—28no—
174#125—28no—
175#125—416no—
176#125—48no—
177C22⋊Q8—48no—
178C22⋊Q8—48no—
179C4×Q8—28no—
180C4×Q8—28no—
181C4×Q8—48no—
182#117—48no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4C2412yes—
216C2×C4⋊C4C2412yes—
217C2×C4⋊C4C2412yes—
218C2×C4⋊C4C2414yes—
219C2×C4⋊C4C2414yes—
220C2×C22⋊C4—24no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—44no—
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42#4118yes—
230C2×C42C2412yes—
231C2×C42C2412yes—
232C2×C42C2412yes—
233C4⋊C8—28no—
234#103—216no—
235#103—48no—
236C22⋊C8—28no—
237#97—28no—
238#97—44no—
239C4×C8—44no—
240#95—24no—
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#95C2×D414yes—
251#95C2×D414yes—
252C24C22×D411yescharacteristic
253C2×Q8—24no—
254C2×Q8—22no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—416no—
258C2×Q8—42no—
259C2×Q8—44no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—82no—
263C2×Q8—82no—
264C2×Q8—82no—
265C2×Q8—82no—
266C2×Q82−1+414yes—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
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—42no—
300C22×C4—42no—
301C22×C4#12414yes—
302C22×C4#12414yes—
303C22×C4AGL(1,ℤ/8)14yes—
304C22×C4#13714yes—
305C22×C4C22×D412yes—
306C22×C4C22×D412yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D412yes—
309C22×C4C22×D412yes—
310C22×C4C22×D412yes—
311C22×C4C22×D412yes—
312C22×C4C2×#4114yes—
313C22×C4C2511yescharacteristic
314C2×C8—28no—
315C2×C8—28no—
316C2×C8—28no—
317C4⋊C4—28no—
318C4⋊C4—28no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—28no—
323C4⋊C4—28no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—48no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—48no—
332C4⋊C4—48no—
333C4⋊C4—48no—
334C4⋊C4—48no—
335C4⋊C4—44no—
336C4⋊C4—48no—
337C4⋊C4—48no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C22⋊C4—48no—
345C42—24no—
346C42—24no—
347C42—42no—
348C42—42no—
349C42—48no—
350C42—48no—
351C42#12418yes—
352C42C2×#4114yes—
353C42C2×#4114yes—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23C2×#12412yes—
359C23C2×AGL(1,ℤ/8)12yes—
360C23C2×#13712yes—
361C23C23×D411yescharacteristic
362Q8—28no—
363Q8—48no—
364Q8—44no—
365Q8—164no—
366Q8—164no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—28no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—28no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4C2×#12414yes—
413C2×C4C2×#12412yes—
414C2×C4C2×#12414yes—
415C2×C4C2×AGL(1,ℤ/8)14yes—
416C2×C4C2×#13714yes—
417C2×C4C22×#4112yes—
418C2×C4C2×2−1+412yes—
419C2×C4#58512yes—
420C8—48no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22#314114yes—
428C22C22×#12411yescharacteristic
429C22#348511yescharacteristic
430C22#348611yescharacteristic
431C22C22×AGL(1,ℤ/8)11yescharacteristic
432C22C22×#13711yescharacteristic
433C22#358611yescharacteristic
434C22#358611yescharacteristic
435C4—24no—
436C4—24no—
437C4—22no—
438C4—24no—
439C4—24no—
440C4—42no—
441C4—42no—
442C4—44no—
443C4—84no—
444C4—84no—
445C4—82no—
446C4—82no—
447C4—82no—
448C4#349514yes—
449C2—24no—
450C2#3564411yescharacteristic
451C2C2×#314111yescharacteristic
452C2#6062011yescharacteristic
453C2#6170411yescharacteristic
454C2#6170511yescharacteristic
455C2#6182811yescharacteristic
456C2#6182911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #27827C2semidirect
8. #27741C2semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#25652
#291112
#291052
#279692
#279602
#278272
#277412
#218382
#207402
#205342