Finite Group Databasebeta

← #134833

Subgroups — #134833

Order 512.

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

normal subgroups only

This group has 315 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#22210C211yesmaximal, characteristic
2#11530C212yesmaximal
3#8623C211yesmaximal, characteristic, complemented
4#8621C211yesmaximal, characteristic, complemented
5#7248C212yesmaximal, complemented
6C2×C8×Q8C2211yescharacteristic
7C2×#436C411yescharacteristic
8C4×#124C2211yescharacteristic
9#1984C2211yescharacteristic
10#1954C411yescharacteristic
11#1923—21no—
12#1923—21no—
13#1851—21no—
14#1851—21no—
15#1842C411yescharacteristic
16#1744C411yescharacteristic
17#1568—21no—
18#1568—21no—
19#1467C2212yes—
20#1467C2212yes—
21#1324—22no—
22#1318—22no—
23#1290—21no—
24#1290—21no—
25C22×M16—21no—
26C2×#124Q811yescharacteristic
27C2×C4×Q8C2311yescharacteristic
28C2×C4×D4C2×C411yescharacteristic
29C8×Q8—41no—
30C8×Q8—41no—
31#436—41no—
32#436—41no—
33#432—22no—
34#432—22no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C4⋊C8D411yescharacteristic
40C2×C4⋊C8D411yescharacteristic
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#391Q811yescharacteristic
49#390D411yescharacteristic
50#388—21no—
51#388—21no—
52#382D411yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4—21no—
55C43C2×C411yescharacteristic
56#336—21no—
57#336—21no—
58#336—21no—
59#336—21no—
60#335—41no—
61#335—41no—
62#334—41no—
63#334—41no—
64#324—22no—
65#324—22no—
66#324—22no—
67#324—22no—
68C22×Q8C2×Q811yescharacteristic
69C22×D4C4⋊C411yescharacteristic
70C23×C4—21no—
71C2×M16—42no—
72C2×M16—81no—
73C2×M16—81no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8C22⋊C411yescharacteristic
77C22×C8C22⋊C411yescharacteristic
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80#124—42no—
81C4×Q8—41no—
82C4×Q8—41no—
83C4×D4—41no—
84C4×D4—41no—
85C2×C4⋊C4C4⋊C411yescharacteristic
86C2×C4⋊C4C2×D411yescharacteristic
87C2×C4⋊C4#4111yescharacteristic
88C2×C4⋊C4#4111yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C42—21no—
95C2×C42—21no—
96C2×C42C4⋊C411yescharacteristic
97C2×C42C4⋊C411yescharacteristic
98C2×C42C22×C411yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C42C2×D411yescharacteristic
101C4⋊C8—41no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107C4⋊C8—81no—
108C4⋊C8—81no—
109C22⋊C8—41no—
110C22⋊C8—41no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113C22⋊C8—81no—
114C22⋊C8—81no—
115#97—42no—
116#97—42no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—21no—
120C4×C8—21no—
121C4×C8—21no—
122C4×C8—21no—
123C4×C8—41no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95—21no—
128C24—21no—
129C2×Q8—22no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×D4—22no—
133C2×D4—41no—
134C2×D4—41no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4C2×C22⋊C411yescharacteristic
145C22×C4C2×C4⋊C411yescharacteristic
146C22×C4#11711yescharacteristic
147C22×C4#12111yescharacteristic
148C22×C4#12111yescharacteristic
149C22×C4C22⋊Q811yescharacteristic
150C22×C4C22⋊Q811yescharacteristic
151M16—84no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—41no—
156C2×C8—41no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—82no—
166C2×C8—82no—
167C2×C8#9911yescharacteristic
168C2×C8#9911yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—22no—
185C42—41no—
186C42—41no—
187C42—41no—
188C42—42no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23#38011yescharacteristic
197Q8—44no—
198D4—82no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4C2×#9911yescharacteristic
229C2×C4#41011yescharacteristic
230C2×C4#43012yes—
231C8—21no—
232C8—21no—
233C8—41no—
234C8—41no—
235C8—41no—
236C8—81no—
237C8—81no—
238C8—162no—
239C8—162no—
240C22—21no—
241C22—21no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22#179911yescharacteristic
250C22#181111yescharacteristic
251C22#181311yescharacteristic
252C4—21no—
253C4—22no—
254C4—42no—
255C4—41no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—41no—
260C4—81no—
261C4—81no—
262C4#213311yescharacteristic
263C4#213411yescharacteristic
264C2—21no—
265C2—21no—
266C2—161no—
267C2#861311yescharacteristic
268C2#1280311yescharacteristic
269C2#1280711yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×Q8C23
prime radicalC22×D4C4⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8623C2semidirect
4. #8621C2semidirect
5. #7248C2semidirect

Maximal subgroups

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

subgroupindex
#222102
#115302
#86232
#86212
#72482