Finite Group Databasebeta

← #6458914

Subgroups — #6458914

Order 512.

1754 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34052C211yesmaximal, characteristic
2C2×#2314C211yesmaximal, characteristic
3#27623C218yesmaximal, complemented
4#27622C218yesmaximal, complemented
5#23745C212yesmaximal, complemented
6#23659C212yesmaximal, complemented
7#21033C214yesmaximal, complemented
8#20846C212yesmaximal
9#20702C212yesmaximal
10#20683C211yesmaximal, characteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C2×#428C2214yes—
15C2×#407C2212yes—
16#2317—22no—
17#2317C2214yes—
18#2316C2214yes—
19#2315—22no—
20#2314—22no—
21#2314—22no—
22#2314C2214yes—
23#2314C2214yes—
24#2296—22no—
25#2296C2212yes—
26#2296C2212yes—
27C2×#388—22no—
28C2×#387—22no—
29#2285—22no—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2212yes—
34#2100C2218yescomplemented
35#2099C2218yescomplemented
36#2085C2218yescomplemented
37#2085C2218yescomplemented
38#1921C2218yescomplemented
39#1920C2218yescomplemented
40#1919C2218yescomplemented
41#1918C2218yescomplemented
42#1854C2218yes—
43#1853C2214yes—
44#1852C2214yes—
45#1825C2218yes—
46#1824C2214yes—
47#1823C2214yes—
48#1818C2214yes—
49#1818C2214yes—
50#1736C2214yes—
51#1736C2214yes—
52#1736C2214yes—
53#1728C2212yes—
54#1728C2212yes—
55C24×C4C2311yescharacteristic
56C22×M16C2314yes—
57C2×#117—21no—
58C2×#117—21no—
59C2×#117—28no—
60C2×#117—28no—
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—28no—
64C2×#117D412yes—
65C2×#117D412yes—
66C2×#117C2312yes—
67C2×#117C2312yes—
68C2×#117C2312yes—
69C2×#117C2312yes—
70C2×#117C2312yes—
71C2×#117C2312yes—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D412yes—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42—22no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#428—216no—
88C2×#107C2318yes—
89C2×#106C2318yes—
90#407—28no—
91C2×C22⋊C8C2314yes—
92C2×C22⋊C8C2314yes—
93#389—24no—
94#389—24no—
95#389C2314yes—
96#389C2314yes—
97#389C2314yes—
98#389C2314yes—
99#388—28no—
100#388—28no—
101#387—28no—
102#387—28no—
103#385—24no—
104#385—24no—
105#385—24no—
106#385—24no—
107#384—24no—
108#384—22no—
109#384C2314yes—
110#382—24no—
111#382—22no—
112#382C2314yes—
113#380—22no—
114#380—22no—
115#380—24no—
116#380—24no—
117#380C2314yes—
118#380C2314yes—
119C4×C4⋊C4—24no—
120C4×C4⋊C4—24no—
121C4×C4⋊C4—24no—
122C4×C4⋊C4C2318yes—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126#340C2318yes—
127#340C2318yes—
128#340C2318yes—
129#340C2318yes—
130#340C2318yes—
131#340C2318yes—
132#336C2314yes—
133#336C2314yes—
134#336C2318yes—
135C25C2×D411yescharacteristic
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C23×C4#4118yes—
148C23×C4C2411yescharacteristic
149C23×C4C2411yescharacteristic
150C23×C4C2411yescharacteristic
151C2×M16—28no—
152C2×M16—28no—
153C2×M16—28no—
154C22×C8C2418yes—
155#117—28no—
156#117—28no—
157#117—28no—
158#117—28no—
159#117—24no—
160#117—28no—
161#117—28no—
162#117—28no—
163#117—28no—
164#117—28no—
165#117—44no—
166#117—44no—
167#117#4118yes—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—84no—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2414yes—
186C2×C4⋊C4C2414yes—
187C2×C4⋊C4C2414yes—
188C2×C4⋊C4C2414yes—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—84no—
199C2×C22⋊C4C2×D414yes—
200C2×C22⋊C4C2×D414yes—
201C2×C42—28no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—28no—
209C2×C42—44no—
210C2×C42—84no—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2412yes—
214C2×C42C2412yes—
215#107—216no—
216#106—216no—
217C22⋊C8—28no—
218C22⋊C8—28no—
219#95—28no—
220#95—24no—
221#95—24no—
222#95—28no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227C24—22no—
228C24—22no—
229C24—42no—
230C24—42no—
231C24#12314yes—
232C24C22×D411yescharacteristic
233C24C22×D411yescharacteristic
234C24C22×D411yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—232no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—28no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4#12314yes—
269C22×C4#12314yes—
270C22×C4#12314yes—
271C22×C4C22×D414yes—
272C22×C4C22×D414yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C22×C4C2×#4114yes—
277C22×C4C2×#4114yes—
278C22×C4C2×#4114yes—
279C22×C4C2×#4114yes—
280C22×C42−1+414yes—
281C22×C4C2511yescharacteristic
282M16—416no—
283C2×C8—28no—
284C2×C8—28no—
285C2×C8—28no—
286C4⋊C4—28no—
287C4⋊C4—28no—
288C4⋊C4—28no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—84no—
297C4⋊C4—84no—
298C4⋊C4—84no—
299C4⋊C4—84no—
300C4⋊C4—88no—
301C4⋊C4C2×#4118yes—
302C22⋊C4—28no—
303C22⋊C4—28no—
304C22⋊C4—28no—
305C22⋊C4—28no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—84no—
309C22⋊C4—84no—
310C42—28no—
311C42—24no—
312C42—24no—
313C42—28no—
314C42—24no—
315C42—24no—
316C42—44no—
317C42—44no—
318C42—88no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—28no—
330C23—48no—
331C23—84no—
332C23—82no—
333C23—82no—
334C23C2×#12312yes—
335C23C2×#12312yes—
336C23C2×#12312yes—
337C23#52512yes—
338C23#53612yes—
339C23C23×D411yescharacteristic
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—28no—
356C2×C4—48no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—88no—
370C2×C4—88no—
371C2×C4—88no—
372C2×C4—88no—
373C2×C4C2×#12312yes—
374C2×C4C2×#12312yes—
375C2×C4C2×#12312yes—
376C2×C4#52512yes—
377C2×C4#53612yes—
378C2×C4#552116yes—
379C2×C4C22×#4112yes—
380C2×C4C22×#4112yes—
381C2×C4C2×2−1+412yes—
382C8—48no—
383C22—24no—
384C22—24no—
385C22—22no—
386C22—22no—
387C22—24no—
388C22—24no—
389C22—22no—
390C22—22no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—88no—
398C22#248414yes—
399C22C22×#12311yescharacteristic
400C22C2×#52511yescharacteristic
401C22C2×#53611yescharacteristic
402C22#358512yes—
403C22#358612yes—
404C4—22no—
405C4—22no—
406C4—44no—
407C4—44no—
408C4—48no—
409C4—84no—
410C4—84no—
411C4—84no—
412C4#248414yes—
413C4C2×#55214yes—
414C2—24no—
415C2—42no—
416C2—42no—
417C2C2×#248411yescharacteristic
418C2#6051712yes—
419C2#6056712yes—
420C2#6086811yescharacteristic
421C2#6086911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27623C2semidirect
4. #27622C2semidirect
5. #23745C2semidirect
6. #23659C2semidirect
7. #21033C2semidirect
34. #2100C22semidirect
35. #2099C22semidirect
36. #2085C22semidirect
37. #2085C22semidirect
38. #1921C22semidirect
39. #1920C22semidirect
40. #1919C22semidirect
41. #1918C22semidirect

Maximal subgroups

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

subgroupindex
#340522
C2×#23142
#276232
#276222
#237452
#236592
#210332
#208462
#207022
#206832