Finite Group Databasebeta

← #6435652

Subgroups — #6435652

Order 512.

1576 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34634C211yesmaximal, characteristic
2C2×#2314C211yesmaximal, characteristic
3#29949C214yesmaximal, complemented
4#29948C214yesmaximal, complemented
5#29945C218yesmaximal, complemented
6#23746C212yesmaximal, complemented
7#23658C212yesmaximal, complemented
8#23642C214yesmaximal
9#21312C212yesmaximal, complemented
10#21170C212yesmaximal, complemented
11#20627C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C2×#428C2214yes—
15C2×#423C2212yes—
16C2×#407C2212yes—
17#2629C2212yes—
18#2629C2212yes—
19#2628C2214yes—
20#2627—22no—
21#2626—22no—
22#2396—24no—
23#2314—24no—
24#2314C2214yes—
25#2314C2214yes—
26#2296C2212yes—
27C2×#388—22no—
28C2×#387—22no—
29#2285C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32#2280C2212yes—
33C2×#380C2212yes—
34C2×C4×C4⋊C4C2212yes—
35#2101C2218yescomplemented
36#2101C2218yescomplemented
37#2084C2218yescomplemented
38#2083C2218yescomplemented
39#2077C2218yes—
40#2077C2218yes—
41#2076C2218yes—
42#2075C2218yes—
43#1989C2218yescomplemented
44#1988C2214yescomplemented
45#1987C2214yescomplemented
46#1949C2218yescomplemented
47#1948C2214yescomplemented
48#1947C2214yescomplemented
49#1798C2214yes—
50#1798C2214yes—
51#1736C2214yes—
52#1736C2214yes—
53#1736C2214yes—
54#1728C2212yes—
55C24×C4C2311yescharacteristic
56C22×M16C2314yes—
57#533—28no—
58C2×C4⋊Q8C2312yes—
59C2×#125C2312yes—
60C2×C22⋊Q8—22no—
61C2×C4×Q8—22no—
62C2×#117—28no—
63C2×#117D414yes—
64C2×#117C2312yes—
65C2×#117C2312yes—
66C2×#117C2312yes—
67C2×#117C2312yes—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—22no—
73C22×C22⋊C4D412yes—
74C22×C42—22no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#428—48no—
78C2×#107C2318yes—
79C2×#106C2318yes—
80#423—28no—
81C2×C4⋊C8C2314yes—
82C2×C4⋊C8C2314yes—
83#407—44no—
84C2×C22⋊C8C2314yes—
85C2×C22⋊C8C2314yes—
86#400—24no—
87#399—24no—
88#398—24no—
89#398—24no—
90#396—24no—
91#396—24no—
92#395—24no—
93#393—24no—
94#389C2314yes—
95#389C2314yes—
96#388—28no—
97#388—28no—
98#387—28no—
99#387—28no—
100#385—24no—
101#384—22no—
102#384C2314yes—
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#382—22no—
106#382C2314yes—
107#382C2312yes—
108#381—22no—
109#380—24no—
110#380—22no—
111#380C2318yes—
112#380C2314yes—
113#380C2314yes—
114#379—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4C2314yes—
121C4×C4⋊C4C2314yes—
122C4×C22⋊C4—24no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95—22no—
126#340C2318yes—
127#340C2318yes—
128#340C2318yes—
129#340C2318yes—
130#340C2318yes—
131#340C2318yes—
132#336C2318yes—
133C25C2×D411yescharacteristic
134C22×Q8—22no—
135C23×C4—24no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2412yes—
146C23×C4C2411yescharacteristic
147C2×M16—28no—
148C2×M16—28no—
149C2×M16—28no—
150C22×C8C2418yes—
151C4⋊Q8—28no—
152#125—28no—
153C22⋊Q8—48no—
154C4×Q8—48no—
155#117—216no—
156#117—28no—
157#117—28no—
158#117—28no—
159#117—24no—
160#117—24no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—84no—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2×D414yes—
177C2×C4⋊C4C2414yes—
178C2×C4⋊C4C2414yes—
179C2×C4⋊C4C2414yes—
180C2×C4⋊C4C2412yes—
181C2×C4⋊C4C2412yes—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—84no—
189C2×C22⋊C4C2×D414yes—
190C2×C22⋊C4C2×D414yes—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—84no—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42#4118yes—
200C2×C42C2412yes—
201C2×C42C2411yescharacteristic
202C2×C42C2411yescharacteristic
203#107—48no—
204#106—48no—
205C4⋊C8—28no—
206C4⋊C8—28no—
207C22⋊C8—44no—
208C22⋊C8—44no—
209#95—24no—
210#95—24no—
211#95—28no—
212#95—28no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24—42no—
222C24—42no—
223C24C22×D412yes—
224C24C22×D411yescharacteristic
225C2×Q8—44no—
226C2×Q8—82no—
227C2×Q8—82no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4#12318yes—
257C22×C4C22×D412yes—
258C22×C4C22×D414yes—
259C22×C4C22×D412yes—
260C22×C4C22×D412yes—
261C22×C4C22×D411yescharacteristic
262C22×C4C2×#4114yes—
263C22×C4H(5,𝔽2)12yes—
264C22×C42−1+414yes—
265C22×C42−1+412yes—
266C22×C4C2511yescharacteristic
267M16—416no—
268C2×C8—28no—
269C2×C8—28no—
270C2×C8—28no—
271C4⋊C4—28no—
272C4⋊C4—28no—
273C4⋊C4—28no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—28no—
278C4⋊C4—24no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—48no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—82no—
286C4⋊C4—82no—
287C4⋊C4—84no—
288C4⋊C4—84no—
289C4⋊C4—88no—
290C22⋊C4—28no—
291C22⋊C4—28no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—84no—
295C22⋊C4—84no—
296C22⋊C4—88no—
297C42—24no—
298C42—24no—
299C42—44no—
300C42—44no—
301C42—84no—
302C42—84no—
303C42#12318yes—
304C42C2×#4114yes—
305C42C2×#4114yes—
306C23—22no—
307C23—22no—
308C23—24no—
309C23—24no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—48no—
314C23—44no—
315C23—84no—
316C23—82no—
317C23—82no—
318C23C2×#12312yes—
319C23#53612yes—
320C23#53712yes—
321C23C23×D411yescharacteristic
322Q8—88no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—48no—
334C2×C4—48no—
335C2×C4—44no—
336C2×C4—48no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—88no—
353C2×C4—88no—
354C2×C4C2×#12314yes—
355C2×C4C2×#12314yes—
356C2×C4C2×#12312yes—
357C2×C4#53612yes—
358C2×C4#53712yes—
359C2×C4#54118yes—
360C2×C4#54218yes—
361C2×C4C22×#4112yes—
362C2×C4C2×H(5,𝔽2)11yescharacteristic
363C2×C4C2×2−1+411yescharacteristic
364C2×C4C2×2−1+412yes—
365C8—48no—
366C22—24no—
367C22—24no—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—44no—
372C22—44no—
373C22—42no—
374C22—42no—
375C22—88no—
376C22#262514yes—
377C22C22×#12311yescharacteristic
378C22C2×#53611yescharacteristic
379C22C2×#53711yescharacteristic
380C22#358512yes—
381C22#358612yes—
382C4—24no—
383C4—24no—
384C4—42no—
385C4—44no—
386C4—84no—
387C4—84no—
388C4—82no—
389C4—84no—
390C4—82no—
391C4#262514yes—
392C4C2×#54112yes—
393C4C2×#54212yes—
394C2—24no—
395C2—82no—
396C2C2×#262511yescharacteristic
397C2#6051712yes—
398C2#6087012yes—
399C2#6092011yescharacteristic
400C2#6092111yescharacteristic
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. #29949C2semidirect
4. #29948C2semidirect
5. #29945C2semidirect
6. #23746C2semidirect
7. #23658C2semidirect
9. #21312C2semidirect
10. #21170C2semidirect
35. #2101C22semidirect
36. #2101C22semidirect
37. #2084C22semidirect
38. #2083C22semidirect
43. #1989C22semidirect
44. #1988C22semidirect
45. #1987C22semidirect
46. #1949C22semidirect
47. #1948C22semidirect
48. #1947C22semidirect

Maximal subgroups

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

subgroupindex
#346342
C2×#23142
#299492
#299482
#299452
#237462
#236582
#236422
#213122
#211702
#206272