Finite Group Databasebeta

← #373727

Subgroups — #373727

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34177C211yesmaximal, characteristic
2C2×#1820C211yesmaximal, characteristic
3#14705C212yesmaximal
4#13756C211yesmaximal, characteristic
5#11344C212yesmaximal
6#11287C212yesmaximal
7#10918C214yesmaximal
8#10736C212yesmaximal
9C22×#117C2211yescharacteristic
10C22×#117C2211yescharacteristic
11C23×C42C2211yescharacteristic
12#2343—28no—
13#2315C2211yescharacteristic
14#2315C2211yescharacteristic
15#2313—22no—
16C2×#385C2214yes—
17#2283—22no—
18C2×#381C2212yes—
19C2×#381C2212yes—
20#2273C2212yes—
21#1820C418yescomplemented
22C2×#342C2214yes—
23C2×#342C2214yes—
24C2×#342C2212yes—
25C2×#342C2212yes—
26#1436—28no—
27C4⋊#95—22no—
28C22⋊#95C2212yes—
29C22⋊#95C2214yes—
30C22⋊#95C2212yes—
31C4×#95—22no—
32C24×C4C2312yes—
33C24×C4C2311yescharacteristic
34C2×#117—24no—
35C2×#117—24no—
36C2×#117—24no—
37C2×#117—24no—
38C2×#117—42no—
39C2×#117C2×C414yes—
40C2×#117Q814yes—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—24no—
43C22×C4⋊C4D412yes—
44C22×C22⋊C4C2312yes—
45C22×C22⋊C4C2312yes—
46C22×C22⋊C4C2314yes—
47C22×C22⋊C4C2312yes—
48C22×C22⋊C4C2311yescharacteristic
49C22×C22⋊C4C2311yescharacteristic
50C22×C42—21no—
51C22×C42—21no—
52C22×C42—21no—
53C22×C42—24no—
54C22×C42—24no—
55C22×C42—42no—
56C22×C42D412yes—
57#387—28no—
58#387—28no—
59#385—24no—
60#385—24no—
61#385—28no—
62#385—44no—
63#383—24no—
64#382—24no—
65#381—22no—
66#381—24no—
67#381—22no—
68#381—24no—
69#381C2×C418yes—
70#380—22no—
71#380—22no—
72C4×C4⋊C4—24no—
73C4×C22⋊C4—24no—
74C4×C22⋊C4—24no—
75C4×C22⋊C4—24no—
76#376—24no—
77C2×#95—24no—
78C2×#95—24no—
79C2×#95—24no—
80C2×#95—28no—
81C2×#95—24no—
82C2×#95D418yes—
83#342—28no—
84#342—44no—
85#342C2×C418yes—
86#342C2×C418yes—
87C25C2411yescharacteristic
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—24no—
93C23×C4—22no—
94C23×C4—24no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—24no—
98C23×C4—48no—
99C23×C4C22×C412yes—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D414yes—
103C23×C4C2×D414yes—
104C23×C4C2×D414yes—
105C23×C4C2×D412yes—
106C23×C4C2×D412yes—
107C23×C4C2×Q812yes—
108C23×C4#4114yes—
109C23×C4#4114yes—
110#117—216no—
111#117—216no—
112#117—416no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—48no—
121C2×C4⋊C4—416no—
122C2×C4⋊C4C4⋊C418yescomplemented
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4C22×C418yes—
144C2×C22⋊C4C22×C412yes—
145C2×C22⋊C4C22×C412yes—
146C2×C22⋊C4C2×Q812yes—
147C2×C22⋊C4C2×Q812yes—
148C2×C22⋊C4#4114yes—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—24no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—24no—
155C2×C42—28no—
156C2×C42—416no—
157C2×C42—42no—
158C2×C42—42no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42C4⋊C418yescomplemented
164#95—24no—
165#95—24no—
166#95—28no—
167#95—216no—
168#95—28no—
169#95—28no—
170#95—48no—
171#95—48no—
172#95—48no—
173#95—416no—
174C24—44no—
175C24C23×C411yescharacteristic
176C24C22×D411yescharacteristic
177C24C22×D412yes—
178C24C22×Q811yescharacteristic
179C24C2×#4112yes—
180C24C2×#4112yes—
181C24C2×#4112yes—
182C24H(5,𝔽2)12yes—
183C242−1+412yes—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—24no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—24no—
196C22×C4—28no—
197C22×C4—28no—
198C22×C4—28no—
199C22×C4—28no—
200C22×C4—28no—
201C22×C4—216no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—28no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—28no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—28no—
215C22×C4—48no—
216C22×C4—416no—
217C22×C4—48no—
218C22×C4—48no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—48no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—48no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—48no—
234C22×C4—48no—
235C22×C4C2×C4⋊C412yes—
236C22×C4C2×C4⋊C412yes—
237C22×C4C2×C4⋊C414yes—
238C22×C4C2×C4⋊C414yes—
239C22×C4C4×D418yes—
240C22×C4#12012yes—
241C22×C4#12012yes—
242C22×C4C22⋊Q818yes—
243C22×C4#12312yes—
244C22×C4#12312yes—
245C4⋊C4—216no—
246C4⋊C4—48no—
247C4⋊C4—816no—
248C4⋊C4—84no—
249C4⋊C4—84no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—28no—
255C22⋊C4—28no—
256C22⋊C4—28no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C42—28no—
267C42—28no—
268C42—48no—
269C42—48no—
270C42—48no—
271C42—88no—
272C23—24no—
273C23—28no—
274C23—42no—
275C23—42no—
276C23—44no—
277C23—42no—
278C23—42no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23C22×C4⋊C411yescharacteristic
283C23C2×C4×D412yes—
284C23C4×#4112yes—
285C23#52012yes—
286C23C2×#12011yescharacteristic
287C23C2×C22⋊Q812yes—
288C23C2×#12311yescharacteristic
289C23#52511yescharacteristic
290C23#52511yescharacteristic
291C23#53312yes—
292C23#53511yescharacteristic
293C23#53611yescharacteristic
294C23#54412yes—
295C23#54614yes—
296C23#54714yes—
297C23#55114yes—
298C23#55214yes—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—216no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—216no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—216no—
314C2×C4—28no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—48no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—416no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—48no—
328C2×C4—416no—
329C2×C4—48no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—48no—
334C2×C4—48no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—84no—
338C2×C4—88no—
339C2×C4—84no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—88no—
345C2×C4—84no—
346C2×C4#38514yes—
347C2×C4#38514yes—
348C22—22no—
349C22—28no—
350C22—22no—
351C22—22no—
352C22—24no—
353C22—22no—
354C22—24no—
355C22—24no—
356C22—22no—
357C22—22no—
358C22—24no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—44no—
364C22—44no—
365C22C2×#38511yescharacteristic
366C22#229611yescharacteristic
367C22#231711yescharacteristic
368C22#235914yes—
369C22#240411yescharacteristic
370C22#240611yescharacteristic
371C22#243814yes—
372C22#248211yescharacteristic
373C22#248411yescharacteristic
374C22#302514yes—
375C4—24no—
376C4—24no—
377C4—44no—
378C4—44no—
379C4—48no—
380C4—84no—
381C4—88no—
382C4—84no—
383C4—84no—
384C2—21no—
385C2—24no—
386C2—21no—
387C2—24no—
388C2—21no—
389C2—21no—
390C2#1412811yescharacteristic
391C2#2088812yes—
392C2#2398612yes—
393C2#2411312yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1820C4semidirect
122. C2×C4⋊C4C4⋊C4semidirect
163. C2×C42C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#341772
C2×#18202
#147052
#137562
#113442
#112872
#109182
#107362