Finite Group Databasebeta

← #214054

Subgroups — #214054

Order 512.

935 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 935 conjugacy classes of subgroups — too many to draw. Only its 186 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#37298C211yesmaximal, characteristic
2#17408C212yesmaximal, complemented
3#9457C214yesmaximal, complemented
4#9449C214yesmaximal, complemented
5#8743C211yesmaximal, characteristic, complemented
6#8742C211yesmaximal, characteristic, complemented
7#8526C212yesmaximal, complemented
8C2×#540C2211yescharacteristic
9C2×#540C2211yescharacteristic
10#2973C2214yes—
11#2856C412yes—
12#2682—24no—
13#2675—24no—
14#2619C412yes—
15#2566C412yes—
16#2566C2212yes—
17#2541C412yes—
18#2541C2212yes—
19#2296C2211yescharacteristic
20#1555C2214yes—
21#1534C2214yes—
22#1471—24no—
23#1468—24no—
24#1466C2214yes—
25#1464C2212yes—
26#1464C2212yes—
27#1285C2214yes—
28#1285C2214yes—
29#540—44no—
30#540—44no—
31C2×#127C2×C411yescharacteristic
32C2×#127C2×C412yes—
33C2×#127C2×C412yes—
34C2×#127C2311yescharacteristic
35C2×#125C2×C411yescharacteristic
36C2×#125C2311yescharacteristic
37C2×#123—22no—
38C2×#123D414yes—
39C2×#121—24no—
40C2×#121—24no—
41C2×#121—24no—
42C2×#121—22no—
43C2×#121C2×C412yes—
44C2×#121C2×C412yes—
45C2×#121D414yes—
46C2×C4×D4C2×C412yes—
47C2×C4×D4C2312yes—
48C2×#117C2×C411yescharacteristic
49C2×#117C2311yescharacteristic
50C22×C42C2×C411yescharacteristic
51#435—44no—
52#435—44no—
53#432C2314yes—
54#432C2314yes—
55#407—24no—
56C2×C22⋊C8—24no—
57#396—24no—
58#396—24no—
59#394—24no—
60#392—24no—
61#392—24no—
62#392—24no—
63#390—24no—
64#390C2×C412yes—
65#390C2×C412yes—
66#390C2×C412yes—
67#390C2×C412yes—
68#389—22no—
69#385—24no—
70#385—24no—
71#385—22no—
72#384C2×C412yes—
73#382C2×C412yes—
74C4×C4⋊C4C2×C412yes—
75C4×C4⋊C4C2312yes—
76#331—44no—
77#330—44no—
78#324—22no—
79#324—22no—
80#323—22no—
81#323—22no—
82#323—28no—
83C22×D4—22no—
84C22×D4—24no—
85C22×D4—24no—
86C22×D4—22no—
87C22×D4C22⋊C414yes—
88C22×D4C22⋊C414yes—
89C22×D4C22×C412yes—
90C22×D4C22×C412yes—
91C23×C4—21no—
92C23×C4—22no—
93C23×C4C22×C411yescharacteristic
94C23×C4C2×D414yes—
95C2×M16—44no—
96C22×C8—44no—
97#127—24no—
98#127—42no—
99#127—82no—
100#127—82no—
101#125—24no—
102#125—42no—
103#123—44no—
104#123—44no—
105#123—48no—
106#121—44no—
107#121—44no—
108#121—48no—
109#121—42no—
110#121—42no—
111#121—44no—
112#121—88no—
113#121—88no—
114#121—88no—
115C4×D4—24no—
116C4×D4—24no—
117C4×D4—44no—
118#117—24no—
119#117—81no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4C22⋊C414yes—
126C2×C4⋊C4C22×C412yes—
127C2×C4⋊C4C22×C412yes—
128C2×C4⋊C4C2×D412yes—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4C22⋊C414yes—
137C2×C22⋊C4C2×D412yes—
138C2×C22⋊C4C2×D414yes—
139C2×C42—21no—
140C2×C42—81no—
141C2×C42C22×C411yescharacteristic
142C2×C42C22×C412yes—
143C2×C42C22×C412yes—
144C2×C42C2411yescharacteristic
145C4⋊C8—24no—
146C4⋊C8—24no—
147C22⋊C8—24no—
148C22⋊C8—24no—
149C22⋊C8—44no—
150C22⋊C8—44no—
151#97—24no—
152#97—24no—
153#95—22no—
154#95—24no—
155#95—22no—
156C24—22no—
157C24C2×C22⋊C414yes—
158C24C2×C22⋊C411yescharacteristic
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—24no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—44no—
167C2×D4—44no—
168C2×D4—48no—
169C2×D4—44no—
170C2×D4—44no—
171C2×D4—84no—
172C2×D4—84no—
173C2×D4—84no—
174C2×D4—84no—
175C2×D4—84no—
176C2×D4—84no—
177C2×D4—84no—
178C2×D4—84no—
179C2×D4—84no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—81no—
191C22×C4—84no—
192C22×C4—84no—
193C22×C4C2×C22⋊C411yescharacteristic
194C22×C4C2×C22⋊C412yes—
195C22×C4C2×C22⋊C414yes—
196C22×C4C23×C411yescharacteristic
197C22×C4C22×D412yes—
198M16—164no—
199C2×C8—44no—
200C2×C8—44no—
201C2×C8—164no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C4⋊C4—44no—
207C4⋊C4—44no—
208C4⋊C4—44no—
209C4⋊C4—44no—
210C4⋊C4—81no—
211C4⋊C4—84no—
212C22⋊C4—24no—
213C22⋊C4—24no—
214C22⋊C4—24no—
215C22⋊C4—44no—
216C22⋊C4—44no—
217C22⋊C4—44no—
218C22⋊C4—48no—
219C22⋊C4—48no—
220C22⋊C4—81no—
221C22⋊C4—84no—
222C22⋊C4—84no—
223C42—22no—
224C42—42no—
225C42—42no—
226C42—81no—
227C42H(5,𝔽2)14yes—
228C23—21no—
229C23—24no—
230C23—21no—
231C23—24no—
232C23—21no—
233C23—24no—
234C23—44no—
235C23—44no—
236C23—42no—
237C23—44no—
238C23—44no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23C22×C22⋊C411yescharacteristic
243D4—82no—
244D4—82no—
245D4—84no—
246D4—164no—
247D4—164no—
248D4—164no—
249D4—168no—
250D4—164no—
251D4—168no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—84no—
272C2×C4—84no—
273C2×C4—84no—
274C2×C4—164no—
275C2×C4#51812yes—
276C2×C4#53414yes—
277C8—84no—
278C8—84no—
279C22—22no—
280C22—44no—
281C22—42no—
282C22—44no—
283C22—41no—
284C22—44no—
285C22—81no—
286C22—84no—
287C22—81no—
288C22—84no—
289C22—84no—
290C22—84no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22#230811yescharacteristic
295C22#288811yescharacteristic
296C22#288911yescharacteristic
297C4—24no—
298C4—22no—
299C4—44no—
300C4—44no—
301C4—42no—
302C4—41no—
303C4—82no—
304C4—82no—
305C2—21no—
306C2—21no—
307C2—84no—
308C2—81no—
309C2—162no—
310C2#2175711yescharacteristic
311C2#2178212yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#37298C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17408C2semidirect
3. #9457C2semidirect
4. #9449C2semidirect
5. #8743C2semidirect
6. #8742C2semidirect
7. #8526C2semidirect

Maximal subgroups

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

subgroupindex
#372982
#174082
#94572
#94492
#87432
#87422
#85262