Finite Group Databasebeta

← #122675

Subgroups — #122675

Order 512.

366 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 80 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#23666C211yesmaximal, characteristic
2#11440C211yesmaximal, characteristic
3C2×#1292C211yesmaximal, characteristic
4#7091C212yesmaximal, complemented
5#7089C212yesmaximal, complemented
6C2×#427C411yescharacteristic
7C2×#423—21no—
8C22×C4⋊C8C2211yescharacteristic
9#2296—21no—
10#2112—21no—
11#2088—21no—
12#2088—21no—
13#2087C411yescharacteristic
14#2087C2211yescharacteristic
15#2086C411yescharacteristic
16#2086C2211yescharacteristic
17#1735—21no—
18#1735C411yescharacteristic
19#1322—21no—
20#1322C2212yes—
21#1292—21no—
22#1292C2212yes—
23C22×M16—41no—
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#117—21no—
27C2×#117—21no—
28C22×C42D411yescharacteristic
29C2×M32—41no—
30#427—41no—
31#427—41no—
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C411yescharacteristic
34C2×#106—21no—
35#423—22no—
36#423—22no—
37#423—41no—
38#423—41no—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—41no—
41C2×C4⋊C8—41no—
42C2×C4⋊C8—41no—
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C4⋊C8D411yescharacteristic
45C2×C4⋊C8D411yescharacteristic
46C2×C4⋊C8C2311yescharacteristic
47#389—21no—
48#389—21no—
49#389—21no—
50#389—21no—
51#385—41no—
52#385—41no—
53C4×C4⋊C4—41no—
54C4×C4⋊C4—41no—
55C22⋊C16—22no—
56C22⋊C16—41no—
57#340—21no—
58#340—21no—
59#340—21no—
60#340—21no—
61#340—21no—
62#340C2×C411yescharacteristic
63#340C2×C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4D811yescharacteristic
66C23×C4SD1611yescharacteristic
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—82no—
70C2×M16—82no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—42no—
76C22×C8—42no—
77C22×C8—42no—
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22×C411yescharacteristic
81C22×C8C2×D411yescharacteristic
82#117—24no—
83#117—42no—
84#117—41no—
85#117—41no—
86#117—41no—
87#117—41no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—41no—
103C2×C42—41no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42D811yescharacteristic
106C2×C42SD1611yescharacteristic
107C2×C42C2×D411yescharacteristic
108M32—82no—
109C2×C16—42no—
110#107—41no—
111#107—41no—
112#106—42no—
113C4⋊C8—24no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—42no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121#95—41no—
122#95—41no—
123#95—41no—
124#95—41no—
125C24#10211yescharacteristic
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—81no—
140C22×C4#9911yescharacteristic
141C22×C4#9911yescharacteristic
142C22×C4#10211yescharacteristic
143C22×C4#10211yescharacteristic
144C22×C4#10211yescharacteristic
145C22×C4#10411yescharacteristic
146C22×C4#10411yescharacteristic
147C22×C4C2×C22⋊C411yescharacteristic
148C22×C4C2×D811yescharacteristic
149C22×C4C2×SD1611yescharacteristic
150M16—84no—
151M16—82no—
152M16—82no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—44no—
165C2×C8—81no—
166C2×C8—84no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—82no—
176C4⋊C4—82no—
177C4⋊C4—82no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C22⋊C4—44no—
181C22⋊C4—81no—
182C22⋊C4—81no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—44no—
192C42—81no—
193C42—81no—
194C16—82no—
195C23—22no—
196C23—21no—
197C23—21no—
198C23—41no—
199C23#32711yescharacteristic
200C23#32711yescharacteristic
201C23C2×#10211yescharacteristic
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4#32711yescharacteristic
236C2×C4#32711yescharacteristic
237C2×C4#35212yes—
238C2×C4#35612yes—
239C2×C4#36012yes—
240C2×C4C2×#9911yescharacteristic
241C2×C4C2×#10411yescharacteristic
242C2×C4#50911yescharacteristic
243C2×C4#51011yescharacteristic
244C8—41no—
245C8—41no—
246C8—41no—
247C8—41no—
248C8—84no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—42no—
257C22—42no—
258C22#134612yes—
259C22C2×#32711yescharacteristic
260C22#213911yescharacteristic
261C22C2×#36011yescharacteristic
262C4—21no—
263C4—21no—
264C4—21no—
265C4—22no—
266C4—21no—
267C4—41no—
268C4—41no—
269C4—82no—
270C4—82no—
271C4—161no—
272C4—161no—
273C4#134612yes—
274C4C2×#35211yescharacteristic
275C4C2×#35611yescharacteristic
276C2—21no—
277C2—21no—
278C2—22no—
279C2—41no—
280C2#1168811yescharacteristic
281C2C2×#134611yescharacteristic
282C2#1170411yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7091C2semidirect
5. #7089C2semidirect

Maximal subgroups

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

subgroupindex
#236662
#114402
C2×#12922
#70912
#70892