Finite Group Databasebeta

← #131378

Subgroups — #131378

Order 512.

701 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 701 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C2×#2822C211yesmaximal, characteristic
2C2×#1346C212yesmaximal
3C2×#1324C211yesmaximal, characteristic
4#11211C211yesmaximal, characteristic
5#9027C212yesmaximal
6C22×#125—21no—
7#2822—24no—
8#2822—22no—
9#2822C414yescomplemented
10C2×#400—22no—
11C2×#398—22no—
12C2×#398—21no—
13C2×#398C2212yes—
14#2316C2211yescharacteristic
15C2×#382C2211yescharacteristic
16#1823—22no—
17C2×#343C2211yescharacteristic
18#1736—21no—
19C2×#339—21no—
20C2×#324C2212yes—
21#1346—24no—
22#1324—22no—
23#1294—22no—
24C22×M16—21no—
25C2×#125—24no—
26C2×#125—24no—
27C2×#125—42no—
28C2×#125—42no—
29C2×#125—42no—
30C2×#125—42no—
31C2×#117—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—21no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4D412yes—
38C22×C4⋊C4D411yescharacteristic
39C22×C4⋊C4C2311yescharacteristic
40C22×C42D411yescharacteristic
41C2×C22⋊C8—22no—
42#400—28no—
43#400—28no—
44#398—24no—
45#398—28no—
46#398—24no—
47#398—28no—
48#398—22no—
49#398—22no—
50#398C2×C414yes—
51#398D414yes—
52#389—42no—
53#382—22no—
54#382C2×C412yes—
55#382D412yes—
56#380—22no—
57C2×#95—22no—
58C2×#95—21no—
59C2×#95—22no—
60C2×#95—22no—
61C2×#95D412yes—
62#343—22no—
63#340—44no—
64#340—42no—
65#339—42no—
66#324—22no—
67#324—22no—
68C23×C4—21no—
69C23×C4—22no—
70C23×C4—22no—
71C23×C4—21no—
72C23×C4C2×D411yescharacteristic
73C23×C4C2×D412yes—
74C2×M16—22no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16—41no—
78C2×M16—41no—
79C2×M16—41no—
80C22×C8—41no—
81C22×C8—42no—
82#125—48no—
83#125—48no—
84#125—48no—
85#125—48no—
86#117—41no—
87#117—41no—
88#117—41no—
89#117—41no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—28no—
92C2×C4⋊C4—24no—
93C2×C4⋊C4—24no—
94C2×C4⋊C4—24no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—44no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4—44no—
107C2×C4⋊C4—42no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—81no—
117C2×C4⋊C4—81no—
118C2×C4⋊C4C22⋊C412yes—
119C2×C4⋊C4C22⋊C414yes—
120C2×C4⋊C4C22×C411yescharacteristic
121C2×C4⋊C4C2×D411yescharacteristic
122C2×C4⋊C4C2×D412yes—
123C2×C22⋊C4—41no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—41no—
128C2×C42C22⋊C412yes—
129C2×C42D812yes—
130C2×C42SD1612yes—
131C22⋊C8—42no—
132C22⋊C8—42no—
133#95—28no—
134#95—24no—
135#95—24no—
136#95—24no—
137#95—24no—
138#95—24no—
139#95—22no—
140#95—42no—
141#95C22⋊C414yes—
142C24#12011yescharacteristic
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C2×C22⋊C412yes—
174C22×C4#12011yescharacteristic
175C22×C4#12012yes—
176C22×C4C2×D811yescharacteristic
177C22×C4C2×SD1611yescharacteristic
178C22×C4AGL(1,ℤ/8)11yescharacteristic
179C22×C4AGL(1,ℤ/8)11yescharacteristic
180M16—42no—
181M16—42no—
182M16—81no—
183M16—81no—
184C2×C8—42no—
185C2×C8—81no—
186C2×C8—81no—
187C2×C8—82no—
188C2×C8—84no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—44no—
194C4⋊C4—48no—
195C4⋊C4—416no—
196C4⋊C4—48no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—44no—
200C4⋊C4—44no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—81no—
204C4⋊C4—82no—
205C4⋊C4—84no—
206C4⋊C4—84no—
207C22⋊C4—81no—
208C22⋊C4—81no—
209C42—22no—
210C42—22no—
211C42—22no—
212C42—81no—
213C42—81no—
214C42#10214yes—
215C23—22no—
216C23—22no—
217C23#37911yescharacteristic
218C23#44711yescharacteristic
219C23#45011yescharacteristic
220C23#45411yescharacteristic
221C23#45611yescharacteristic
222C23B(4,𝔽2)11yescharacteristic
223C23#45811yescharacteristic
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—44no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—42no—
254C2×C4—82no—
255C2×C4—84no—
256C2×C4—84no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—82no—
266C2×C4C2×#10211yescharacteristic
267C2×C4#41811yescharacteristic
268C2×C4#44711yescharacteristic
269C2×C4#45011yescharacteristic
270C8—82no—
271C8—162no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22#159811yescharacteristic
281C22#160411yescharacteristic
282C22#161011yescharacteristic
283C22#161611yescharacteristic
284C22#178611yescharacteristic
285C22#179011yescharacteristic
286C22#179311yescharacteristic
287C22#220112yes—
288C22#220312yes—
289C4—21no—
290C4—21no—
291C4—21no—
292C4—21no—
293C4—44no—
294C4—42no—
295C4—42no—
296C4—84no—
297C4—84no—
298C4—82no—
299C4—162no—
300C2—21no—
301C2—21no—
302C2—21no—
303C2—21no—
304C2#840411yescharacteristic
305C2#1215712yes—
306C2#1225112yes—
307C2#1276411yescharacteristic
308C2#1277511yescharacteristic
subgroupG/H
center, socleC23B(4,𝔽2)
commutatorC2×C4⋊C4C22×C4
Frattini, cosocleC22×C4⋊C4C23
prime radicalC24#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G
LECSG ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #2822C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28222
C2×#13462
C2×#13242
#112112
#90272