Finite Group Databasebeta

← #136576

Subgroups — #136576

Order 512.

402 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 116 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#33397C211yesmaximal, characteristic
2#21601C211yesmaximal, characteristic
3#12602C211yesmaximal, characteristic
4#11607C212yesmaximal, complemented
5#7169C212yesmaximal, complemented
6C22×M32—21no—
7C2×C8×Q8C2211yescharacteristic
8C2×#405C2211yescharacteristic
9#2396C411yescharacteristic
10#2151—24no—
11#2150—21no—
12#2150C2212yes—
13C2×#349—21no—
14C4×M32C2212yes—
15#1985C411yescharacteristic
16#1940C411yescharacteristic
17#1917—21no—
18#1851—21no—
19#1798C411yescharacteristic
20C8×C4⋊C4—21no—
21C8×C4⋊C4C2211yescharacteristic
22C8×C22⋊C4—21no—
23#1377—21no—
24Q8⋊C16—24no—
25C22×M16D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4×Q8C2×C411yescharacteristic
28C2×#117C2×C411yescharacteristic
29C2×M32—22no—
30C2×M32—22no—
31C2×M32—22no—
32C2×M32—42no—
33C2×M32—42no—
34C22×C16—22no—
35C8×Q8—22no—
36C8×Q8—22no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C22⋊C8—21no—
42#405—22no—
43#405—22no—
44C2×#97C2×C411yescharacteristic
45C2×#97D411yescharacteristic
46C2×C4×C8—21no—
47C2×C4×C8—21no—
48C2×C4×C8D411yescharacteristic
49C2×C4×C8C2311yescharacteristic
50#400C811yescharacteristic
51#399C811yescharacteristic
52#398—21no—
53#396—21no—
54#395C811yescharacteristic
55#393C811yescharacteristic
56C4×C4⋊C4—21no—
57C4×C4⋊C4C2×C411yescharacteristic
58C4×C22⋊C4—21no—
59C4⋊C16—24no—
60C4⋊C16—22no—
61C4⋊C16D414yes—
62#349—24no—
63#346—22no—
64C4×C16—22no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#336—21no—
70C22×Q8C2×C811yescharacteristic
71C23×C4C22⋊C411yescharacteristic
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—41no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8SD1612yes—
84C22×C8Q1612yes—
85C22×C8C2×D411yescharacteristic
86C22×C8C2×D411yescharacteristic
87C4×Q8—22no—
88C4×Q8—22no—
89#117—22no—
90#117—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4C2×C811yescharacteristic
96C2×C4⋊C4C2×C811yescharacteristic
97C2×C4⋊C4M1611yescharacteristic
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4C2×C811yescharacteristic
101C2×C22⋊C4M1611yescharacteristic
102C2×C42—21no—
103C2×C42—21no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42C22×C411yescharacteristic
106M32—24no—
107M32—42no—
108M32—44no—
109M32—42no—
110M32—42no—
111C2×C16—24no—
112C2×C16—44no—
113C2×C16—42no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116C4⋊C8—42no—
117C4⋊C8—42no—
118C22⋊C8—42no—
119#97—22no—
120#97—22no—
121C4×C8—21no—
122C4×C8—21no—
123C4×C8—41no—
124C4×C8—41no—
125C4×C8—41no—
126C4×C8—41no—
127C4×C8C2×D412yes—
128C4×C8#4112yes—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95C2×C811yescharacteristic
134#95C2×C811yescharacteristic
135C24C22⋊C811yescharacteristic
136C2×Q8—22no—
137C2×Q8—42no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4C22⋊C811yescharacteristic
147C22×C4C22⋊C811yescharacteristic
148C22×C4C22⋊C811yescharacteristic
149C22×C4#10312yes—
150C22×C4#10412yes—
151C22×C4C2×C22⋊C411yescharacteristic
152C22×C4C22×C811yescharacteristic
153C22×C4C2×M1611yescharacteristic
154M16—44no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—22no—
160C2×C8—22no—
161C2×C8—22no—
162C2×C8—22no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8#10312yes—
173C2×C8#12011yescharacteristic
174C2×C8#12112yes—
175C2×C8#12311yescharacteristic
176C2×C8C2×SD1611yescharacteristic
177C2×C8C2×Q1611yescharacteristic
178C2×C8AGL(1,ℤ/8)11yescharacteristic
179C2×C8#13711yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C42—21no—
192C42—21no—
193C42—41no—
194C42—41no—
195C42—41no—
196C42—41no—
197C42C4×D412yes—
198C42#13112yes—
199C16—44no—
200C16—44no—
201C23—21no—
202C23—21no—
203C23—41no—
204C23—41no—
205C23Q8⋊C812yes—
206C23C2×C22⋊C811yescharacteristic
207Q8—24no—
208Q8—42no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4Q8⋊C812yes—
234C2×C4#38611yescharacteristic
235C2×C4#40811yescharacteristic
236C2×C4C2×#10311yescharacteristic
237C2×C4#41711yescharacteristic
238C2×C4C2×#10411yescharacteristic
239C2×C4#42111yescharacteristic
240C2×C4C8×D412yes—
241C2×C4#43512yes—
242C8—22no—
243C8—22no—
244C8—41no—
245C8—81no—
246C8—81no—
247C8#45011yescharacteristic
248C8#45111yescharacteristic
249C8#48311yescharacteristic
250C8#48411yescharacteristic
251C22—21no—
252C22—21no—
253C22—22no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22—81no—
258C22C2×Q8⋊C811yescharacteristic
259C22#147711yescharacteristic
260C22#186911yescharacteristic
261C4—22no—
262C4—22no—
263C4—22no—
264C4—42no—
265C4—41no—
266C4—42no—
267C4—41no—
268C4—81no—
269C4—81no—
270C4#148111yescharacteristic
271C4#148311yescharacteristic
272C4#187111yescharacteristic
273C4#187211yescharacteristic
274C2—21no—
275C2—21no—
276C2—42no—
277C2#893711yescharacteristic
278C2#1298111yescharacteristic
279C2#1298211yescharacteristic
subgroupG/H
centerC2×C8#120
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC22#1869
prime radicalC24C22⋊C8
LCSG ▷ C22×C4 ▷ C22 ▷ 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
4. #11607C2semidirect
5. #7169C2semidirect

Maximal subgroups

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

subgroupindex
#333972
#216012
#126022
#116072
#71692