Finite Group Databasebeta

← #204647

Subgroups — #204647

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35011C211yesmaximal, characteristic
2#20524C211yesmaximal, characteristic
3C2×#1511C212yesmaximal
4C2×#1508C212yesmaximal
5#13953C211yesmaximal, characteristic
6#9683C214yesmaximal, complemented
7#9679C214yesmaximal, complemented
8C22×C22⋊Q8C2212yes—
9C2×#407C2212yes—
10#2857C2212yes—
11#2857C2212yes—
12#2659—22no—
13#2583—24no—
14#2502C414yescomplemented
15#2434C414yescomplemented
16#2315—22no—
17C2×#381C2211yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20#1753—24no—
21#1749C2212yes—
22C4×C22⋊C8C2212yes—
23#1511—28no—
24#1508—28no—
25C2×#328C2218yes—
26C2×#328C2218yes—
27#1439—22no—
28#1436—22no—
29C4⋊#95C2212yes—
30#1434C2212yes—
31C23×Q8C2×C412yes—
32C24×C4C2311yescharacteristic
33C22×M16—22no—
34C2×C4⋊Q8C2×C412yes—
35C2×C4⋊Q8C2×C412yes—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—24no—
39C2×C22⋊Q8—24no—
40C2×C22⋊Q8—24no—
41C2×C22⋊Q8—24no—
42C2×C22⋊Q8C2×C414yes—
43C2×C22⋊Q8C2314yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4C2×C412yes—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4D412yes—
51C22×C42—21no—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54#407—24no—
55#407D418yes—
56C2×C22⋊C8—22no—
57C2×C22⋊C8C2314yes—
58C2×#97—22no—
59C2×C4×C8—22no—
60#398—28no—
61#395—28no—
62#391C2×C414yes—
63#391C2×C414yes—
64#389—24no—
65#389—24no—
66#385—28no—
67#385—28no—
68#384C2×C412yes—
69#384C2×C412yes—
70#381—24no—
71#380—24no—
72#380—24no—
73C4×C22⋊C4—24no—
74C4×C22⋊C4C2×C414yes—
75C4×C22⋊C4D414yes—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95D412yes—
81C2×#95C2312yes—
82C2×#95C2312yes—
83#336—22no—
84#336—22no—
85#328—216no—
86#328—216no—
87C25C2×D411yescharacteristic
88C22×Q8—22no—
89C22×Q8—24no—
90C22×Q8—24no—
91C22×Q8—82no—
92C22×Q8—82no—
93C22×Q8C22×C414yes—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—81no—
99C23×C4—81no—
100C23×C4C22×C412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D414yes—
106C23×C4#4114yes—
107C23×C4C2411yescharacteristic
108C2×M16—28no—
109C2×M16—82no—
110C2×M16—82no—
111C22×C8—24no—
112C4⋊Q8—82no—
113C4⋊Q8—82no—
114C22⋊Q8—28no—
115C22⋊Q8—28no—
116C22⋊Q8—24no—
117C22⋊Q8—24no—
118C22⋊Q8—24no—
119C22⋊Q8—24no—
120C22⋊Q8—44no—
121C22⋊Q8—44no—
122C22⋊Q8—44no—
123C22⋊Q8—44no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—44no—
134C2×C4⋊C4—44no—
135C2×C4⋊C4—44no—
136C2×C4⋊C4—44no—
137C2×C4⋊C4—82no—
138C2×C4⋊C4C22×C414yes—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—44no—
144C2×C22⋊C4—44no—
145C2×C22⋊C4—44no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—81no—
149C2×C22⋊C4—81no—
150C2×C22⋊C4C22⋊C418yes—
151C2×C42—42no—
152C2×C42—81no—
153C2×C42—81no—
154C2×C42—81no—
155C2×C42—81no—
156C2×C42C22×C412yes—
157C2×C42C22×C412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C22⋊C8—28no—
161C22⋊C8—24no—
162C22⋊C8C2×D418yes—
163#97—82no—
164C4×C8—82no—
165#95—28no—
166#95—28no—
167#95—24no—
168#95—24no—
169#95—44no—
170#95—42no—
171#95—44no—
172#95—82no—
173#95C22⋊C418yes—
174C24—22no—
175C24—22no—
176C24—42no—
177C24—41no—
178C24—41no—
179C24C2×C22⋊C412yes—
180C24#12412yes—
181C24#12712yes—
182C24C22×D411yescharacteristic
183C2×Q8—24no—
184C2×Q8—24no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—44no—
188C2×Q8—82no—
189C2×Q8—82no—
190C2×Q8—88no—
191C2×Q8—88no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4—84no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4C2×C22⋊C414yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C414yes—
232C22×C4C4×D414yes—
233C22×C4C4×D414yes—
234C22×C4#12114yes—
235C22×C4#12114yes—
236C22×C4#12412yes—
237C22×C4#12712yes—
238C22×C4#13714yes—
239C22×C4#13714yes—
240C22×C4C23×C411yescharacteristic
241C22×C4C22×D411yescharacteristic
242C22×C4C22×D412yes—
243C22×C4C2×#4112yes—
244M16—88no—
245C2×C8—28no—
246C2×C8—44no—
247C2×C8—44no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—48no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C4⋊C4—84no—
257C4⋊C4—84no—
258C4⋊C4—88no—
259C22⋊C4—216no—
260C22⋊C4—48no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—48no—
264C22⋊C4—84no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—42no—
269C42—84no—
270C23—21no—
271C23—21no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—24no—
276C23—24no—
277C23—22no—
278C23—22no—
279C23—24no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—82no—
285C23—82no—
286C23—82no—
287C23—82no—
288C23#39014yes—
289C23C22×C22⋊C411yescharacteristic
290C23C2×#12411yescharacteristic
291C23C2×#12711yescharacteristic
292C23C2×#13712yes—
293C23C2×#13712yes—
294Q8—84no—
295Q8—84no—
296Q8—88no—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—48no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—88no—
332C2×C4—88no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4#39014yes—
342C2×C4#41918yes—
343C2×C4C2×C4×D412yes—
344C2×C4C2×#12112yes—
345C8—48no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—21no—
353C22—21no—
354C22—24no—
355C22—24no—
356C22—24no—
357C22—44no—
358C22—44no—
359C22—84no—
360C22—82no—
361C22—82no—
362C22C2×#39011yescharacteristic
363C22#288311yescharacteristic
364C22#288311yescharacteristic
365C22C2×#41912yes—
366C22#290112yes—
367C22#313911yescharacteristic
368C22#313911yescharacteristic
369C22#315212yes—
370C4—28no—
371C4—22no—
372C4—22no—
373C4—24no—
374C4—44no—
375C4—84no—
376C4—84no—
377C4—84no—
378C4—162no—
379C2—22no—
380C2—21no—
381C2—21no—
382C2—44no—
383C2#1762112yes—
384C2#1762712yes—
385C2#2120211yescharacteristic
386C2#2121311yescharacteristic
387C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9683C2semidirect
7. #9679C2semidirect
14. #2502C4semidirect
15. #2434C4semidirect

Maximal subgroups

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

subgroupindex
#350112
#205242
C2×#15112
C2×#15082
#139532
#96832
#96792