Finite Group Databasebeta

← #241166

Subgroups — #241166

Order 512.

1086 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 218 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#35167C211yesmaximal, characteristic
2#21365C212yesmaximal
3#13856C211yesmaximal, characteristic
4#10570C214yesmaximal
5#10567C214yesmaximal
6#8022C212yesmaximal
7#7912C211yesmaximal, characteristic
8C2×#544—21no—
9C2×#432C2212yes—
10#2857—22no—
11#2659C2214yes—
12#2552—22no—
13#2551—22no—
14#2550—24no—
15#2316—21no—
16#2285C2212yes—
17C2×C4×C4⋊C4—22no—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×#376C2211yescharacteristic
21#1948—22no—
22#1947—22no—
23#1853C2214yes—
24#1852C2214yes—
25#1776—24no—
26#1444—21no—
27#1444—21no—
28#1444C2212yes—
29#1444C2212yes—
30#1294C2218yes—
31#1280C2214yes—
32#544—48no—
33C2×C4⋊Q8—24no—
34C2×C4⋊Q8—22no—
35C2×#125—24no—
36C2×C22⋊Q8—24no—
37C2×#117—22no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4C2311yescharacteristic
43C22×C4⋊C4C2311yescharacteristic
44C22×C42—21no—
45C22×C42—21no—
46C22×C42D412yes—
47C22×C42C2311yescharacteristic
48#432—24no—
49#432Q818yescomplemented
50C2×#107—24no—
51C2×#106—24no—
52C2×C4⋊C8—22no—
53C2×C22⋊C8C2314yes—
54C2×#97—22no—
55#398—28no—
56#395—28no—
57#391—24no—
58#389—24no—
59#389—24no—
60#389—24no—
61#389—24no—
62#384—22no—
63#384D412yes—
64#384D412yes—
65#382—22no—
66#380—24no—
67#380—22no—
68#380C2314yes—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—48no—
75C4×C4⋊C4D414yes—
76#376—22no—
77#376Q814yescomplemented
78C2×#95—21no—
79C2×#95—21no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95C2312yes—
83C2×#95C2312yes—
84#340—24no—
85#340—24no—
86C22×Q8—24no—
87C23×C4—22no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2411yescharacteristic
95C22×C8—24no—
96C4⋊Q8—416no—
97C4⋊Q8—84no—
98#125—416no—
99C22⋊Q8—48no—
100C22⋊Q8—48no—
101#117—48no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4C2×D414yes—
125C2×C4⋊C4C2×D414yes—
126C2×C22⋊C4—24no—
127C2×C42—24no—
128C2×C42—24no—
129C2×C42—24no—
130C2×C42—24no—
131C2×C42—22no—
132C2×C42—22no—
133C2×C42—44no—
134C2×C42—44no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42C2×D412yes—
138C2×C42C2×D412yes—
139C2×C42C2×Q814yes—
140C2×C42C2×Q812yes—
141C2×C42#4112yes—
142#107—48no—
143#106—48no—
144C4⋊C8—28no—
145C22⋊C8—24no—
146C22⋊C8C2×Q818yes—
147#97—28no—
148#95—28no—
149#95—24no—
150#95—24no—
151#95—24no—
152#95—24no—
153#95—22no—
154#95—44no—
155#95—44no—
156#95—42no—
157#95C2×Q814yes—
158C24C22×D411yescharacteristic
159C2×Q8—48no—
160C2×Q8—84no—
161C2×Q8—84no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—41no—
178C22×C4—41no—
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—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4#12011yescharacteristic
196C22×C4#12012yes—
197C22×C4#12011yescharacteristic
198C22×C4C22⋊Q812yes—
199C22×C4C22⋊Q812yes—
200C22×C4C4⋊Q812yes—
201C22×C4C4⋊Q812yes—
202C22×C4AGL(1,ℤ/8)14yes—
203C22×C4#13714yes—
204C22×C4C22×D412yes—
205C22×C4C22×Q812yes—
206C22×C4C22×Q811yescharacteristic
207C22×C4C2×#4111yescharacteristic
208C2×C8—28no—
209C2×C8—44no—
210C2×C8—44no—
211C4⋊C4—28no—
212C4⋊C4—28no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—48no—
222C4⋊C4—416no—
223C4⋊C4—416no—
224C4⋊C4—416no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C22⋊C4—48no—
230C42—216no—
231C42—48no—
232C42—44no—
233C42—82no—
234C42—82no—
235C42—82no—
236C42C22⋊Q814yes—
237C42C22⋊Q814yes—
238C42C4⋊Q814yes—
239C42C4⋊Q814yes—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23C2×#12011yescharacteristic
245C23C2×C22⋊Q811yescharacteristic
246C23C2×C4⋊Q811yescharacteristic
247C23C2×AGL(1,ℤ/8)12yes—
248C23C2×#13712yes—
249Q8—168no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—28no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4#39514yes—
297C2×C4#39512yes—
298C2×C4#39512yes—
299C2×C4#45312yes—
300C2×C4#45612yes—
301C2×C4C2×C22⋊Q812yes—
302C2×C4C2×C4⋊Q812yes—
303C8—48no—
304C22—22no—
305C22—22no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22C2×#39511yescharacteristic
313C22#300311yescharacteristic
314C22#300411yescharacteristic
315C22C2×#45311yescharacteristic
316C22C2×#45611yescharacteristic
317C22#302511yescharacteristic
318C22#302511yescharacteristic
319C22#307912yes—
320C22#316512yes—
321C4—24no—
322C4—22no—
323C4—22no—
324C4—22no—
325C4—48no—
326C4—44no—
327C4—44no—
328C4—82no—
329C4—84no—
330C4—82no—
331C4—84no—
332C4#206014yes—
333C2—22no—
334C2—22no—
335C2#1817811yescharacteristic
336C2#1818011yescharacteristic
337C2#1818711yescharacteristic
338C2#1818911yescharacteristic
339C2#2355211yescharacteristic
340C2C2×#206011yescharacteristic
341C2#2356411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
49. #432Q8semidirect
77. #376Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#213652
#138562
#105702
#105672
#80222
#79122