Finite Group Databasebeta

← #6433308

Subgroups — #6433308

Order 512.

1918 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#36625C211yesmaximal, characteristic
2C2×#2296C211yesmaximal, characteristic
3#29974C218yesmaximal, complemented
4#29971C214yesmaximal, complemented
5#29970C214yesmaximal, complemented
6#23747C212yesmaximal, complemented
7#23670C214yesmaximal, complemented
8#23554C212yesmaximal, complemented
9#20845C212yesmaximal
10#20701C212yesmaximal
11#20626C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#428C2214yes—
17C2×#423C2212yes—
18C2×#407C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2855—22no—
22#2851C2214yes—
23#2848—22no—
24C4×C22⋊Q8—22no—
25C4×C22⋊Q8—22no—
26#2296—22no—
27#2296—22no—
28#2296C2212yes—
29#2296C2214yes—
30#2296C2214yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#385—22no—
34#2285C2211yescharacteristic
35#2280C2212yes—
36C2×C4×C4⋊C4—22no—
37#2101C2218yescomplemented
38#2100C2214yescomplemented
39#2099C2214yescomplemented
40#2088C2218yescomplemented
41#2088C2218yescomplemented
42#2087C2218yescomplemented
43#2086C2218yescomplemented
44#2059C2218yescomplemented
45#2058C2214yescomplemented
46#2057C2214yescomplemented
47#1854C2218yes—
48#1854C2218yes—
49#1825C2218yes—
50#1825C2218yes—
51#1797C2214yes—
52#1797C2214yes—
53#1736C2214yes—
54#1736C2214yes—
55#1736C2214yes—
56#1728C2212yes—
57C24×C4D412yes—
58C24×C4C2311yescharacteristic
59C22×M16C2314yes—
60#533—44no—
61C2×C4⋊Q8C2312yes—
62C2×#125C2312yes—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C4×Q8—22no—
67C2×#117—22no—
68C2×#117—24no—
69C2×#117—22no—
70C2×#117—44no—
71C2×#117Q818yescomplemented
72C2×#117C2312yes—
73C2×#117C2312yes—
74C2×#117C2314yes—
75C2×#117C2314yes—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42—44no—
85C22×C42D412yes—
86C22×C42D412yes—
87C22×C42D412yes—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2312yes—
91C22×C42C2312yes—
92#428—48no—
93C2×#107C2318yes—
94C2×#106C2318yes—
95#423—44no—
96C2×C4⋊C8C2318yes—
97#407—28no—
98C2×C22⋊C8C2318yes—
99#391—22no—
100#391—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—22no—
106#389—22no—
107#389C2314yes—
108#389C2314yes—
109#389C2314yes—
110#389C2314yes—
111#389C2314yes—
112#389C2314yes—
113#387—24no—
114#387—24no—
115#385—24no—
116#385—24no—
117#385—24no—
118#385—24no—
119#385—24no—
120#385—24no—
121#385—24no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#382C2312yes—
125#381—22no—
126#380—22no—
127#380—22no—
128#380—22no—
129#380C2314yes—
130#380C2314yes—
131#379—22no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—24no—
135C4×C4⋊C4—24no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4—22no—
138C4×C4⋊C4—22no—
139C4×C22⋊C4—24no—
140C2×#95—22no—
141C2×#95—22no—
142#340C2318yes—
143#340C2318yes—
144#340C2318yes—
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#336C2314yes—
149#336C2314yes—
150C25C2×D411yescharacteristic
151C22×Q8—22no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×Q814yes—
168C23×C4#4114yes—
169C23×C4C2412yes—
170C23×C4C2411yescharacteristic
171C2×M16—28no—
172C2×M16—28no—
173C2×M16—28no—
174C22×C8C2418yes—
175C4⋊Q8—44no—
176#125—42no—
177#125—42no—
178C22⋊Q8—48no—
179C22⋊Q8—48no—
180C22⋊Q8—48no—
181C4×Q8—48no—
182#117—216no—
183#117—28no—
184#117—28no—
185#117—28no—
186#117—48no—
187#117—48no—
188#117C2×Q8116yes—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4C2×Q818yes—
206C2×C4⋊C4C2414yes—
207C2×C4⋊C4C2414yes—
208C2×C4⋊C4C2412yes—
209C2×C4⋊C4C2412yes—
210C2×C4⋊C4C2414yes—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—44no—
219C2×C42—28no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—44no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42C2×D412yes—
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42C2411yescharacteristic
240C2×C42C2411yescharacteristic
241C2×C42C2412yes—
242#107—48no—
243#106—48no—
244C4⋊C8—48no—
245C22⋊C8—216no—
246#95—24no—
247#95—24no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255C24—22no—
256C24—22no—
257C24—42no—
258C24—42no—
259C24C22⋊Q814yes—
260C24C22×D412yes—
261C24C22×D411yescharacteristic
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—44no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—216no—
270C22×C4—216no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—28no—
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—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—24no—
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—44no—
298C22×C4—44no—
299C22×C4—432no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4C22⋊Q814yes—
304C22×C4C22⋊Q818yes—
305C22×C4C22×D412yes—
306C22×C4C22×D412yes—
307C22×C4C22×D412yes—
308C22×C4C22×D412yes—
309C22×C4C22×D412yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×Q814yes—
312C22×C4C22×Q812yes—
313C22×C4C2×#4114yes—
314C22×C4C2×#4112yes—
315C22×C4H(5,𝔽2)14yes—
316C22×C42−1+414yes—
317C22×C4C2511yescharacteristic
318M16—416no—
319C2×C8—28no—
320C2×C8—28no—
321C2×C8—28no—
322C4⋊C4—28no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C4⋊C4—48no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—48no—
335C4⋊C4—44no—
336C4⋊C4—48no—
337C4⋊C4—48no—
338C4⋊C4—44no—
339C4⋊C4—48no—
340C4⋊C4C22×Q818yes—
341C22⋊C4—216no—
342C22⋊C4—48no—
343C22⋊C4—48no—
344C22⋊C4—48no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C42—216no—
348C42—22no—
349C42—22no—
350C42—24no—
351C42—24no—
352C42—22no—
353C42—22no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—48no—
359C42—44no—
360C42—44no—
361C23—22no—
362C23—22no—
363C23—28no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—48no—
371C23—48no—
372C23—44no—
373C23—44no—
374C23C2×C22⋊Q814yes—
375C23C2×C22⋊Q812yes—
376C23#53412yes—
377C23#53612yes—
378C23C23×D411yescharacteristic
379Q8—88no—
380C2×C4—28no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—216no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—48no—
400C2×C4—48no—
401C2×C4—48no—
402C2×C4—416no—
403C2×C4—48no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—88no—
417C2×C4C2×C22⋊Q814yes—
418C2×C4C2×C22⋊Q812yes—
419C2×C4#53412yes—
420C2×C4#53612yes—
421C2×C4#53918yes—
422C2×C4#54418yes—
423C2×C4#57514yes—
424C2×C4#57514yes—
425C2×C4C23×Q811yescharacteristic
426C2×C4C22×#4111yescharacteristic
427C2×C4C2×H(5,𝔽2)12yes—
428C2×C4C2×2−1+412yes—
429C8—48no—
430C22—24no—
431C22—24no—
432C22—22no—
433C22—22no—
434C22—24no—
435C22—24no—
436C22—24no—
437C22—416no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22#262714yes—
443C22C22×C22⋊Q811yescharacteristic
444C22C2×#53411yescharacteristic
445C22C2×#53611yescharacteristic
446C22C2×#57512yes—
447C22C2×#57512yes—
448C4—24no—
449C4—24no—
450C4—24no—
451C4—48no—
452C4—42no—
453C4—42no—
454C4—84no—
455C4—84no—
456C4—82no—
457C4—82no—
458C4#262714yes—
459C4C2×#53912yes—
460C4C2×#54412yes—
461C2—24no—
462C2—44no—
463C2C2×#262711yescharacteristic
464C2#6048412yes—
465C2#6075212yes—
466C2#6086211yescharacteristic
467C2#6086311yescharacteristic
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
3. #29974C2semidirect
4. #29971C2semidirect
5. #29970C2semidirect
6. #23747C2semidirect
7. #23670C2semidirect
8. #23554C2semidirect
37. #2101C22semidirect
38. #2100C22semidirect
39. #2099C22semidirect
40. #2088C22semidirect
41. #2088C22semidirect
42. #2087C22semidirect
43. #2086C22semidirect
44. #2059C22semidirect
45. #2058C22semidirect
46. #2057C22semidirect
71. C2×#117Q8semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×#22962
#299742
#299712
#299702
#237472
#236702
#235542
#208452
#207012
#206262