Finite Group Databasebeta

← #405513

Subgroups — #405513

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35090C211yesmaximal, characteristic
2#14705C212yesmaximal
3#13756C211yesmaximal, characteristic
4#11390C212yesmaximal
5#11344C212yesmaximal
6#11222C214yesmaximal
7#10736C212yesmaximal
8#10726C211yesmaximal, characteristic
9C2×#543C2211yescharacteristic
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C42C2211yescharacteristic
13#2855—22no—
14#2627—22no—
15#2438—28no—
16#2315C2211yescharacteristic
17#2315C2212yes—
18C2×#385C2214yes—
19C2×#381C2212yes—
20C2×#381C2212yes—
21C2×#342C2212yes—
22C2×#342C2214yes—
23C2×#342C2214yes—
24C2×#342C2212yes—
25#1436—28no—
26C4⋊#95—22no—
27C22⋊#95C2212yes—
28C22⋊#95C2214yes—
29C22⋊#95C2212yes—
30C4×#95—22no—
31C24×C4C2312yes—
32C24×C4C2311yescharacteristic
33#543—24no—
34#543—42no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—24no—
37C2×#117—24no—
38C2×#117—22no—
39C2×#117—24no—
40C2×#117—42no—
41C2×#117—42no—
42C2×#117—42no—
43C2×#117Q814yescomplemented
44C22×C4⋊C4—21no—
45C22×C4⋊C4—24no—
46C22×C4⋊C4D412yes—
47C22×C22⋊C4C2311yescharacteristic
48C22×C22⋊C4C2312yes—
49C22×C22⋊C4C2312yes—
50C22×C22⋊C4C2312yes—
51C22×C22⋊C4C2314yes—
52C22×C22⋊C4C2311yescharacteristic
53C22×C42—21no—
54C22×C42—21no—
55C22×C42—21no—
56C22×C42—24no—
57C22×C42—24no—
58C22×C42—42no—
59C22×C42D412yes—
60#400—24no—
61#398—28no—
62#395—24no—
63#391—24no—
64#389—24no—
65#387—28no—
66#385—24no—
67#385—24no—
68#385—24no—
69#385—28no—
70#385—44no—
71#381—22no—
72#381—22no—
73#381—24no—
74#381—24no—
75#381Q818yescomplemented
76#380—22no—
77#380—24no—
78C2×#95—24no—
79C2×#95—24no—
80C2×#95—24no—
81C2×#95—28no—
82C2×#95—24no—
83C2×#95D418yes—
84#342—24no—
85#342—44no—
86#342—44no—
87#342—42no—
88C25C2411yescharacteristic
89C22×Q8—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—24no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—24no—
100C23×C4—48no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D414yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D414yes—
105C23×C4C2×D412yes—
106C23×C4C2×D414yes—
107C23×C4C2×D412yes—
108C23×C4C2×Q812yes—
109C23×C4C2×Q814yes—
110C23×C4#4112yes—
111C23×C4#4114yes—
112C22⋊Q8—416no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115#117—48no—
116#117—416no—
117#117—48no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—48no—
129C2×C4⋊C4—416no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4C2×Q814yes—
152C2×C22⋊C4C2×Q818yes—
153C2×C22⋊C4#4114yes—
154C2×C42—24no—
155C2×C42—24no—
156C2×C42—24no—
157C2×C42—24no—
158C2×C42—24no—
159C2×C42—42no—
160C2×C42—416no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42—42no—
164C2×C42—42no—
165C2×C42—42no—
166C2×C42—42no—
167#95—24no—
168#95—28no—
169#95—28no—
170#95—28no—
171#95—216no—
172#95—48no—
173#95—48no—
174#95—48no—
175#95—416no—
176C24—44no—
177C24C22×D411yescharacteristic
178C24C22×D412yes—
179C24C22×Q811yescharacteristic
180C24C22×Q812yes—
181C24C2×#4111yescharacteristic
182C24C2×#4112yes—
183C24C2×#4112yes—
184C24H(5,𝔽2)12yes—
185C24H(5,𝔽2)12yes—
186C2×Q8—44no—
187C2×Q8—84no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—28no—
201C22×C4—28no—
202C22×C4—28no—
203C22×C4—28no—
204C22×C4—28no—
205C22×C4—216no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—28no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—28no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—28no—
220C22×C4—48no—
221C22×C4—416no—
222C22×C4—48no—
223C22×C4—48no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—48no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—48no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—48no—
239C22×C4—48no—
240C22×C4#12012yes—
241C22×C4#12012yes—
242C22×C4C22⋊Q818yes—
243C22×C4C22⋊Q812yes—
244C22×C4C22⋊Q812yes—
245C22×C4#12314yes—
246C22×C4C4⋊Q812yes—
247C22×C4C4⋊Q812yes—
248C22×C4C4⋊Q818yes—
249C4⋊C4—48no—
250C4⋊C4—48no—
251C4⋊C4—48no—
252C4⋊C4—44no—
253C4⋊C4—816no—
254C4⋊C4—84no—
255C4⋊C4—88no—
256C22⋊C4—28no—
257C22⋊C4—28no—
258C22⋊C4—28no—
259C22⋊C4—28no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—44no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—44no—
266C22⋊C4—44no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C42—48no—
271C42—48no—
272C42—48no—
273C42—48no—
274C23—28no—
275C23—24no—
276C23—42no—
277C23—42no—
278C23—44no—
279C23—42no—
280C23—44no—
281C23—42no—
282C23—44no—
283C23—44no—
284C23C2×#12011yescharacteristic
285C23C2×C22⋊Q811yescharacteristic
286C23C2×C22⋊Q812yes—
287C23C2×#12311yescharacteristic
288C23#52511yescharacteristic
289C23#52912yes—
290C23C2×C4⋊Q811yescharacteristic
291C23C2×C4⋊Q812yes—
292C23#53312yes—
293C23#53411yescharacteristic
294C23#53511yescharacteristic
295C23#53812yes—
296C23#54312yes—
297C23#54614yes—
298C23#54614yes—
299C23#55414yes—
300C23#55414yes—
301Q8—164no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—216no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—216no—
316C2×C4—216no—
317C2×C4—28no—
318C2×C4—24no—
319C2×C4—48no—
320C2×C4—44no—
321C2×C4—48no—
322C2×C4—416no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—416no—
328C2×C4—48no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—44no—
337C2×C4—48no—
338C2×C4—82no—
339C2×C4—84no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—88no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—82no—
348C2×C4—88no—
349C2×C4#39514yes—
350C2×C4#39514yes—
351C22—22no—
352C22—24no—
353C22—22no—
354C22—24no—
355C22—24no—
356C22—22no—
357C22—22no—
358C22—24no—
359C22—22no—
360C22—22no—
361C22—28no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—44no—
367C22—44no—
368C22C2×#39511yescharacteristic
369C22#240411yescharacteristic
370C22#240511yescharacteristic
371C22#245014yes—
372C22#250211yescharacteristic
373C22#254914yes—
374C22#258311yescharacteristic
375C22#262611yescharacteristic
376C22#285511yescharacteristic
377C22#302514yes—
378C4—24no—
379C4—24no—
380C4—48no—
381C4—44no—
382C4—84no—
383C4—88no—
384C4—84no—
385C4—82no—
386C4—84no—
387C2—21no—
388C2—24no—
389C2—21no—
390C2—21no—
391C2—21no—
392C2—24no—
393C2#1491411yescharacteristic
394C2#2356412yes—
395C2#2398612yes—
396C2#2405712yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, prime radical, cosocleC25C24
LCSG ▷ C25 ▷ 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
43. C2×#117Q8semidirect
75. #381Q8semidirect

Maximal subgroups

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

subgroupindex
#350902
#147052
#137562
#113902
#113442
#112222
#107362
#107262