Finite Group Databasebeta

← #129666

Subgroups — #129666

Order 512.

554 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 122 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#60330C211yesmaximal, characteristic
2#13263C211yesmaximal, characteristic, complemented
3#11621C212yesmaximal, complemented
4#11613C212yesmaximal, complemented
5#11493C211yesmaximal, characteristic, complemented
6#3461C411yescharacteristic
7#2993—22no—
8#2992—22no—
9#2989C412yes—
10#2985—22no—
11#2983C411yescharacteristic
12#2966—22no—
13#2966—22no—
14C8×#41C2212yes—
15#2960—22no—
16C2×C8×Q8C2211yescharacteristic
17#2162C2212yes—
18C2×C4⋊C16C2211yescharacteristic
19C2×C4×C16C2211yescharacteristic
20#1374—22no—
21Q8⋊C16—22no—
22Q8⋊C16—22no—
23D4⋊C16—22no—
24D4⋊C16—22no—
25#1322—22no—
26#544C811yescharacteristic
27#542—22no—
28#541—22no—
29#539—22no—
30#537C811yescharacteristic
31#536C812yes—
32#533—22no—
33#529—22no—
34#529—22no—
35C4×#41C2×C412yes—
36C2×C4×Q8C2×C411yescharacteristic
37C22×C16—21no—
38C22×C16—21no—
39#446—24no—
40#446—22no—
41#446—22no—
42#446—22no—
43#446—22no—
44C8×Q8—22no—
45C8×Q8—22no—
46C8×Q8—22no—
47C8×Q8D412yes—
48C8×Q8D412yes—
49#436—22no—
50#436—22no—
51#435—24no—
52#435—24no—
53C8×D4—22no—
54C8×D4—22no—
55C8×D4—22no—
56C8×D4D412yes—
57C8×D4D412yes—
58#433—24no—
59#433—24no—
60#432—22no—
61#432—22no—
62#431—22no—
63#431C2×C412yes—
64#424—24no—
65#423—22no—
66C2×C4⋊C8C2×C411yescharacteristic
67C2×C4⋊C8D412yes—
68#405—24no—
69C2×C4×C8—22no—
70C2×C4×C8D412yes—
71C2×C4×C8C2311yescharacteristic
72C4⋊C16—22no—
73C4⋊C16—22no—
74C22⋊C16—22no—
75C4×C16—22no—
76C4×C16—22no—
77C2×#41C2×C812yes—
78C22×Q8C2×C811yescharacteristic
79C2×M16—24no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8C22⋊C412yes—
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D411yescharacteristic
85C4⋊Q8—22no—
86C4⋊Q8—22no—
87#126—24no—
88#126—24no—
89#125—22no—
90#125—24no—
91#125—22no—
92#124—22no—
93#124—22no—
94#123—24no—
95#123—24no—
96C22⋊Q8—24no—
97C22⋊Q8—22no—
98C22⋊Q8—22no—
99C22⋊Q8—22no—
100C22⋊Q8—22no—
101#121—22no—
102#121—22no—
103C4×Q8—22no—
104C4×Q8—22no—
105C4×Q8—22no—
106C4×Q8C22⋊C412yes—
107C4×Q8M1612yes—
108C4×D4—22no—
109C4×D4—22no—
110C4×D4—22no—
111C4×D4C22⋊C412yes—
112C4×D4M1612yes—
113#117—24no—
114#117—22no—
115#117C2×C812yes—
116C2×C4⋊C4C2×C811yescharacteristic
117C2×C4⋊C4M1612yes—
118C2×C42—22no—
119C2×C42C22⋊C412yes—
120C2×C42C22×C411yescharacteristic
121C2×C16—22no—
122C2×C16—41no—
123C2×C16—41no—
124C2×C16—42no—
125C2×C16—42no—
126C2×C16—41no—
127C4⋊C8—24no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—24no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—22no—
134C4⋊C8C22⋊C412yes—
135C4⋊C8C22⋊C412yes—
136C22⋊C8—24no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140#97—24no—
141#97—24no—
142C4×C8—22no—
143C4×C8—22no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8—22no—
147C4×C8C2×D412yes—
148C4×C8C2×D412yes—
149#41—28no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—24no—
153C2×Q8C22⋊C812yes—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4C22⋊C812yes—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4C22⋊C812yes—
161C22×C4C22⋊C812yes—
162C22×C4C2×C22⋊C411yescharacteristic
163C22×C4C22×C811yescharacteristic
164C22×C4C2×M1611yescharacteristic
165M16—84no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—41no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—82no—
176C2×C8#12011yescharacteristic
177C2×C8#12012yes—
178C2×C8#12011yescharacteristic
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—22no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4C22⋊C812yes—
189C4⋊C4C22⋊C812yes—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C42—22no—
197C42—22no—
198C42—22no—
199C42—22no—
200C42—22no—
201C42C2×C22⋊C412yes—
202C42C2×M1612yes—
203C16—42no—
204C16—82no—
205C16—81no—
206C23—22no—
207C23C2×C22⋊C811yescharacteristic
208Q8—24no—
209Q8—44no—
210Q8—82no—
211D4—24no—
212D4—44no—
213D4—82no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—24no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—82no—
228C2×C4#37911yescharacteristic
229C2×C4C2×C22⋊C812yes—
230C2×C4#40711yescharacteristic
231C2×C4#40712yes—
232C2×C4#50414yes—
233C8—21no—
234C8—22no—
235C8—42no—
236C8—42no—
237C8—42no—
238C8#45411yescharacteristic
239C8#45611yescharacteristic
240C22—21no—
241C22—21no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22#177911yescharacteristic
246C22#211312yes—
247C4—21no—
248C4—22no—
249C4—42no—
250C4—42no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4#179311yescharacteristic
255C4#179411yescharacteristic
256C4#211314yes—
257C2—22no—
258C2—82no—
259C2#831911yescharacteristic
260C2#1268011yescharacteristic
261C2#1268511yescharacteristic
subgroupG/H
centerC2×C8#120
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1779
prime radical#537C8
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13263C2semidirect
3. #11621C2semidirect
4. #11613C2semidirect
5. #11493C2semidirect

Maximal subgroups

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

subgroupindex
#603302
#132632
#116212
#116132
#114932