Finite Group Databasebeta

← #241181

Subgroups — #241181

Order 512.

964 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 964 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35020C211yesmaximal, characteristic
2#21942C212yesmaximal
3#14295C211yesmaximal, characteristic
4#10572C214yesmaximal, complemented
5#10571C214yesmaximal, complemented
6#8015C212yesmaximal
7#7911C211yesmaximal, characteristic
8C2×#407C2212yes—
9#2851—22no—
10#2646C2212yes—
11#2646C2212yes—
12#2569—24no—
13#2442—22no—
14#2442—22no—
15#2315—22no—
16C2×#381C2211yescharacteristic
17C2×#380C2212yes—
18C2×#380C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20#1853—22no—
21#1853C2214yes—
22#1852—22no—
23#1852C2214yes—
24#1809—24no—
25#1444C2212yes—
26#1442C2212yes—
27#1439—22no—
28#1436—22no—
29#1294C2218yes—
30#1280C2212yes—
31#1280C2212yes—
32C24×C4C2311yescharacteristic
33C22×M16—22no—
34C2×#125—22no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4D412yes—
37C22×C4⋊C4D412yes—
38C22×C4⋊C4C2312yes—
39C22×C22⋊C4—22no—
40C22×C22⋊C4—21no—
41C22×C22⋊C4—21no—
42C22×C22⋊C4D412yes—
43C22×C42D412yes—
44C2×#107—24no—
45C2×#106—24no—
46#407—24no—
47#407Q818yes—
48C2×C22⋊C8—22no—
49C2×C22⋊C8C2314yes—
50#400—24no—
51#400—24no—
52#398—24no—
53#398—24no—
54#389—22no—
55#389—22no—
56#387—24no—
57#387—24no—
58#385—24no—
59#385—24no—
60#385—24no—
61#385—24no—
62#382—22no—
63#381—22no—
64#381Q814yes—
65#380—24no—
66#380—24no—
67#380—22no—
68#380—22no—
69#380—22no—
70#380—22no—
71#380D414yes—
72#380C2314yes—
73C4×C22⋊C4—22no—
74C4×C22⋊C4—22no—
75C2×#95—22no—
76C2×#95—21no—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95C2311yescharacteristic
82C2×#95C2311yescharacteristic
83C2×#95C2312yes—
84#340—24no—
85#340—24no—
86C25C2×D411yescharacteristic
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—82no—
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×Q814yes—
98C23×C4C2×Q812yes—
99C23×C4#4112yes—
100C23×C4C2411yescharacteristic
101C2×M16—28no—
102C2×M16—82no—
103C2×M16—82no—
104C22×C8—24no—
105#125—84no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—82no—
116C2×C4⋊C4—82no—
117C2×C4⋊C4—82no—
118C2×C4⋊C4—82no—
119C2×C4⋊C4—82no—
120C2×C4⋊C4C2×D414yes—
121C2×C4⋊C4C2×D414yes—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—44no—
130C2×C22⋊C4—44no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—82no—
133C2×C22⋊C4—81no—
134C2×C22⋊C4—81no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42—82no—
138#107—84no—
139#106—84no—
140C22⋊C8—28no—
141C22⋊C8—24no—
142C22⋊C8C2×Q818yes—
143#95—24no—
144#95—24no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—22no—
151#95—22no—
152#95—22no—
153#95—44no—
154#95—41no—
155#95—41no—
156#95—82no—
157#95—82no—
158#95C2×Q814yes—
159C24—22no—
160C24—22no—
161C24—41no—
162C24—42no—
163C24—41no—
164C24#12012yes—
165C24C22⋊Q812yes—
166C24C4⋊Q812yes—
167C24C22×D411yescharacteristic
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—84no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4—82no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4#12011yescharacteristic
210C22×C4#12011yescharacteristic
211C22×C4C22⋊Q814yes—
212C22×C4C22⋊Q814yes—
213C22×C4C22⋊Q812yes—
214C22×C4C4⋊Q814yes—
215C22×C4C4⋊Q814yes—
216C22×C4C4⋊Q812yes—
217C22×C4C22×D412yes—
218C22×C4C22×Q812yes—
219C22×C4C22×Q811yescharacteristic
220C22×C4C2×#4111yescharacteristic
221M16—88no—
222C2×C8—28no—
223C2×C8—44no—
224C2×C8—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C4⋊C4—88no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C22⋊C4—48no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C22⋊C4—84no—
242C22⋊C4—84no—
243C42—44no—
244C42—82no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—24no—
255C23—22no—
256C23—22no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—82no—
262C23—82no—
263C23—82no—
264C23—82no—
265C23#39514yes—
266C23C2×#12011yescharacteristic
267C23C2×C22⋊Q811yescharacteristic
268C23C2×C4⋊Q811yescharacteristic
269C23#57512yes—
270C23#57512yes—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—164no—
315C2×C4—164no—
316C2×C4#39512yes—
317C2×C4#39512yes—
318C2×C4C2×C22⋊Q812yes—
319C2×C4C2×C4⋊Q812yes—
320C8—48no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—24no—
330C22—24no—
331C22—24no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—84no—
337C22—82no—
338C22—82no—
339C22C2×#39511yescharacteristic
340C22#300512yes—
341C22#302111yescharacteristic
342C22#302211yescharacteristic
343C22#302612yes—
344C22#308012yes—
345C22#316611yescharacteristic
346C22#316611yescharacteristic
347C4—22no—
348C4—22no—
349C4—22no—
350C4—22no—
351C4—84no—
352C4—82no—
353C4—84no—
354C4—82no—
355C4—84no—
356C4—162no—
357C2—22no—
358C2—21no—
359C2—21no—
360C2—44no—
361C2#1835412yes—
362C2#1835512yes—
363C2#2355311yescharacteristic
364C2#2356211yescharacteristic
365C2#2356511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10572C2semidirect
5. #10571C2semidirect

Maximal subgroups

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

subgroupindex
#350202
#219422
#142952
#105722
#105712
#80152
#79112