Finite Group Databasebeta

← #137638

Subgroups — #137638

Order 512.

355 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 102 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#33419C211yesmaximal, characteristic
2#21604C211yesmaximal, characteristic
3#12786C211yesmaximal, characteristic
4#11611C212yesmaximal, complemented
5#7169C212yesmaximal, complemented
6C2×#446C2211yescharacteristic
7C2×#405C2211yescharacteristic
8#2396C411yescharacteristic
9#2164—24no—
10#2150—21no—
11#2150C2212yes—
12C2×#349—21no—
13#2112—21no—
14C4×M32C2212yes—
15#1985C411yescharacteristic
16#1940C411yescharacteristic
17#1916—21no—
18#1850—21no—
19#1798C411yescharacteristic
20#1770—21no—
21C8×C4⋊C4C2211yescharacteristic
22#1753—21no—
23#1377—21no—
24#1338—24no—
25C22×M16D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4×Q8C2×C411yescharacteristic
28C2×#117C2×C411yescharacteristic
29C2×M32—22no—
30C2×M32—22no—
31#446—22no—
32#446—22no—
33C2×C4⋊C8—21no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C22⋊C8—21no—
38#405—22no—
39#405—22no—
40C2×#97—21no—
41C2×#97C2×C411yescharacteristic
42C2×#97D411yescharacteristic
43C2×C4×C8—21no—
44C2×C4×C8D411yescharacteristic
45C2×C4×C8C2311yescharacteristic
46#400C811yescharacteristic
47#399C811yescharacteristic
48#398—21no—
49#396—21no—
50#395C811yescharacteristic
51#393C811yescharacteristic
52C4×C4⋊C4—21no—
53C4×C4⋊C4C2×C411yescharacteristic
54C4×C22⋊C4—21no—
55C4⋊C16—24no—
56C4⋊C16—22no—
57C4⋊C16—22no—
58#349—42no—
59C22⋊C16—24no—
60#346D414yes—
61C4×C16—22no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67C22×Q8C2×C811yescharacteristic
68C23×C4C22⋊C411yescharacteristic
69C2×M16—22no—
70C2×M16—21no—
71C2×M16—21no—
72C2×M16—22no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C4×Q8—22no—
84C4×Q8—22no—
85#117—22no—
86#117—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4C2×C811yescharacteristic
92C2×C4⋊C4C2×C811yescharacteristic
93C2×C4⋊C4M1611yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4C2×C811yescharacteristic
97C2×C22⋊C4M1611yescharacteristic
98C2×C42—21no—
99C2×C42—21no—
100C2×C42C22⋊C411yescharacteristic
101C2×C42C22×C411yescharacteristic
102M32—42no—
103M32—42no—
104M32—42no—
105M32—42no—
106C2×C16—24no—
107C2×C16—42no—
108C4⋊C8—22no—
109C4⋊C8—22no—
110C4⋊C8—42no—
111C4⋊C8—41no—
112C4⋊C8—41no—
113C22⋊C8—42no—
114#97—22no—
115#97—22no—
116#97—42no—
117C4×C8—21no—
118C4×C8—21no—
119C4×C8—41no—
120C4×C8—41no—
121C4×C8C2×D412yes—
122C4×C8#4112yes—
123#95—21no—
124#95—21no—
125#95—21no—
126#95—21no—
127#95C2×C811yescharacteristic
128#95C2×C811yescharacteristic
129C24C22⋊C811yescharacteristic
130C2×Q8—22no—
131C2×Q8—81no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4C22⋊C811yescharacteristic
143C22×C4C22⋊C811yescharacteristic
144C22×C4C22⋊C811yescharacteristic
145C22×C4C2×C22⋊C411yescharacteristic
146C22×C4C22×C811yescharacteristic
147C22×C4C2×M1611yescharacteristic
148M16—42no—
149M16—42no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8#12011yescharacteristic
167C2×C8#12112yes—
168C2×C8#12311yescharacteristic
169C2×C8AGL(1,ℤ/8)11yescharacteristic
170C2×C8#13712yes—
171C2×C8#13711yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—81no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—42no—
182C22⋊C4—81no—
183C42—21no—
184C42—21no—
185C42—41no—
186C42—41no—
187C42—41no—
188C42—41no—
189C42C4×D412yes—
190C42#13112yes—
191C16—44no—
192C16—82no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—41no—
197C23—41no—
198C23C2×C22⋊C811yescharacteristic
199Q8—82no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
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—41no—
225C2×C4#38611yescharacteristic
226C2×C4#40811yescharacteristic
227C2×C4#41711yescharacteristic
228C2×C4#41911yescharacteristic
229C2×C4#42111yescharacteristic
230C2×C4#42111yescharacteristic
231C2×C4C8×D412yes—
232C2×C4#43512yes—
233C8—21no—
234C8—21no—
235C8—41no—
236C8—41no—
237C8—81no—
238C8—81no—
239C8#46812yes—
240C8#47012yes—
241C22—21no—
242C22—21no—
243C22—42no—
244C22—41no—
245C22—41no—
246C22—41no—
247C22—81no—
248C22#147711yescharacteristic
249C22#147911yescharacteristic
250C22#186911yescharacteristic
251C4—22no—
252C4—22no—
253C4—42no—
254C4—41no—
255C4—41no—
256C4—41no—
257C4—81no—
258C4—81no—
259C4—81no—
260C4#149511yescharacteristic
261C4#149711yescharacteristic
262C4#189411yescharacteristic
263C4#189711yescharacteristic
264C2—21no—
265C2—21no—
266C2—81no—
267C2#907311yescharacteristic
268C2#1298211yescharacteristic
269C2#1298911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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
4. #11611C2semidirect
5. #7169C2semidirect

Maximal subgroups

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

subgroupindex
#334192
#216042
#127862
#116112
#71692