Finite Group Databasebeta

← #122539

Subgroups — #122539

Order 512.

358 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 72 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#23699C211yesmaximal, characteristic
2#11459C211yesmaximal, characteristic
3#7980C211yesmaximal, characteristic
4#7079C212yesmaximal
5#7078C212yesmaximal
6C2×C4×Q16C411yescharacteristic, complemented
7C2×#431C2211yescharacteristic
8C4×C22⋊Q8—21no—
9#2283—21no—
10#2162—21no—
11#2086C411yescharacteristic, complemented
12#2075C2211yescharacteristic
13#2066—21no—
14#2056C411yescharacteristic, complemented
15#2035—21no—
16#2001C2211yescharacteristic
17C4×#106C411yescharacteristic, complemented
18C4×#103—21no—
19#1322—21no—
20#1314C2212yes—
21#1289—21no—
22#1289C2212yes—
23C22×Q16C2×C411yescharacteristic
24C2×C22⋊Q8—21no—
25C2×C4×Q8—21no—
26C22×C42D411yescharacteristic
27C4×Q16—41no—
28C4×Q16—41no—
29#431—41no—
30#431—41no—
31C2×#107C2×C411yescharacteristic
32C2×#106—21no—
33C2×C4⋊C8C2×C411yescharacteristic
34C2×C4⋊C8D411yescharacteristic
35C2×#103—21no—
36C2×#103—21no—
37C2×#103—21no—
38C2×C22⋊C8D411yescharacteristic
39C2×C22⋊C8C2311yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42#391—21no—
43#389—21no—
44#387—41no—
45#385—41no—
46#383—22no—
47#382—41no—
48#381—41no—
49#380—22no—
50#380—21no—
51C4×C4⋊C4—21no—
52C4×C4⋊C4—41no—
53C4×C4⋊C4—41no—
54C4×C22⋊C4—41no—
55C4×C22⋊C4—41no—
56#376—22no—
57C4⋊C16—41no—
58C22⋊C16—22no—
59C4×C16—41no—
60#340—21no—
61#340—21no—
62#340C2×C411yescharacteristic
63#340C2×C411yescharacteristic
64C22×Q8—21no—
65C23×C4D811yescharacteristic
66C23×C4D811yescharacteristic
67C23×C4C2×D411yescharacteristic
68C2×Q16—41no—
69C2×Q16—42no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22×C411yescharacteristic
73C22×C8C2×D411yescharacteristic
74C22⋊Q8—42no—
75C22⋊Q8—42no—
76C4×Q8—41no—
77C4×Q8—41no—
78C4×Q8—41no—
79C4×Q8—41no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—41no—
84C2×C4⋊C4—42no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—42no—
89C2×C22⋊C4—41no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42—42no—
95C2×C42—41no—
96C2×C42—41no—
97C2×C42C22⋊C411yescharacteristic
98C2×C42C22⋊C411yescharacteristic
99C2×C42SD1611yescharacteristic
100C2×C42SD1611yescharacteristic
101C2×C16—42no—
102#107—41no—
103#106—42no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106#103—42no—
107#103—42no—
108#103—42no—
109C22⋊C8—41no—
110C22⋊C8—41no—
111C4×C8—21no—
112C4×C8—21no—
113C4×C8—21no—
114C4×C8—21no—
115#95—24no—
116#95—42no—
117#95—42no—
118#95—42no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124C24C2×D811yescharacteristic
125C2×Q8—22no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—44no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4#10211yescharacteristic
148C22×C4#10211yescharacteristic
149C22×C4#10211yescharacteristic
150C22×C4#10211yescharacteristic
151C22×C4C2×C22⋊C411yescharacteristic
152C22×C4C2×SD1611yescharacteristic
153Q16—42no—
154Q16—42no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C4⋊C4—22no—
164C4⋊C4—41no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—44no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—82no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C4⋊C4—81no—
174C4⋊C4—81no—
175C22⋊C4—44no—
176C22⋊C4—82no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—21no—
185C42—82no—
186C42—84no—
187C42—81no—
188C42—81no—
189C42#9911yescharacteristic
190C42#9911yescharacteristic
191C42#10411yescharacteristic
192C42#10411yescharacteristic
193C16—82no—
194C23—21no—
195C23—21no—
196C23—41no—
197C23—41no—
198C23C2×#10211yescharacteristic
199C23#50812yes—
200Q8—42no—
201Q8—42no—
202Q8—81no—
203Q8—81no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—84no—
231C2×C4—81no—
232C2×C4—84no—
233C2×C4—84no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—81no—
238C2×C4#32711yescharacteristic
239C2×C4#32711yescharacteristic
240C2×C4#32711yescharacteristic
241C2×C4#32711yescharacteristic
242C2×C4C2×#9911yescharacteristic
243C2×C4C2×#10411yescharacteristic
244C8—41no—
245C8—41no—
246C8—41no—
247C8—41no—
248C22—21no—
249C22—21no—
250C22—22no—
251C22—42no—
252C22—41no—
253C22—41no—
254C22—81no—
255C22C2×#32711yescharacteristic
256C22#213811yescharacteristic
257C22#214511yescharacteristic
258C4—21no—
259C4—21no—
260C4—21no—
261C4—21no—
262C4—41no—
263C4—41no—
264C4—41no—
265C4—82no—
266C4—82no—
267C4—161no—
268C4—164no—
269C4—161no—
270C4#134012yes—
271C4#212011yescharacteristic
272C4#212611yescharacteristic
273C2—21no—
274C2—21no—
275C2—42no—
276C2C2×#134011yescharacteristic
277C2#1169111yescharacteristic
278C2#1170311yescharacteristic
subgroupG/H
centerC2×C4#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×C22⋊C8C23
socleC22C2×#327
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. C2×C4×Q16C4semidirect
11. #2086C4semidirect
14. #2056C4semidirect
17. C4×#106C4semidirect

Maximal subgroups

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

subgroupindex
#236992
#114592
#79802
#70792
#70782