Finite Group Databasebeta

← #215192

Subgroups — #215192

Order 512.

990 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 206 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#36639C211yesmaximal, characteristic
2#20778C212yesmaximal, complemented
3#9688C218yesmaximal, complemented
4#8338C212yesmaximal, complemented
5#8337C212yesmaximal
6C2×#533C2211yescharacteristic
7C2×#532C2211yescharacteristic
8#2862C412yes—
9#2861C2214yes—
10#2858C412yes—
11#2850—22no—
12#2850—22no—
13C4×#124—22no—
14C4×#124—22no—
15#2297C412yes—
16#2297C412yes—
17C2×C43C2211yescharacteristic
18#1839—22no—
19#1831—22no—
20#1817—22no—
21#1797—22no—
22#1767C2214yes—
23#1749C2214yes—
24#1745C2214yes—
25C2×#330C2218yes—
26C2×#329C2218yes—
27C22×#41C2×C412yes—
28#533—44no—
29#532—44no—
30C2×C4⋊Q8C2×C411yescharacteristic
31C2×C4⋊Q8C2×C412yes—
32C2×#127C2×C411yescharacteristic
33C2×#125C2312yes—
34C2×#124—22no—
35C2×#124—22no—
36C2×#124C2312yes—
37C2×C22⋊Q8—22no—
38C2×#121—22no—
39C2×C4×Q8—22no—
40C2×C4×D4—22no—
41C2×#117C2×C412yes—
42C22×C42C2×C411yescharacteristic
43C22×C42C2×C411yescharacteristic
44C22×C42D412yes—
45C22×C42D412yes—
46C22×C42C2311yescharacteristic
47C2×C4⋊C8—24no—
48C2×C22⋊C8—24no—
49C2×#97C2318yes—
50#391—22no—
51#391—22no—
52#391C2×C414yes—
53#391C2×C414yes—
54#390—22no—
55#390—22no—
56#390C2×C414yes—
57#390C2×C414yes—
58#388—24no—
59#388—24no—
60#388—24no—
61#388—24no—
62#386—22no—
63#386—24no—
64#386—22no—
65#382—22no—
66#382—22no—
67C4×C22⋊C4—24no—
68C4×C22⋊C4—22no—
69C4×C22⋊C4—22no—
70C43C2×C412yes—
71C43D412yes—
72C43D412yes—
73C43C2312yes—
74#336—24no—
75#336—24no—
76#336—24no—
77#336—24no—
78#330—48no—
79#329—48no—
80C2×#41—44no—
81C2×#41—44no—
82C2×#41—44no—
83C22×Q8—22no—
84C22×Q8C22×C412yes—
85C22×D4—22no—
86C22×D4—22no—
87C22×D4C22×C412yes—
88C23×C4—22no—
89C23×C4C22⋊C411yescharacteristic
90C23×C4C22⋊C412yes—
91C23×C4C22⋊C411yescharacteristic
92C23×C4C22×C411yescharacteristic
93C23×C4C2×D412yes—
94C22×C8—28no—
95C4⋊Q8—44no—
96C4⋊Q8—42no—
97#127—42no—
98#125—28no—
99#124—28no—
100#124—48no—
101#124—48no—
102C22⋊Q8—48no—
103#121—48no—
104C4×Q8—48no—
105C4×D4—48no—
106#117—44no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4C22×C414yes—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—24no—
113C2×C22⋊C4—24no—
114C2×C22⋊C4—24no—
115C2×C22⋊C4—24no—
116C2×C22⋊C4—24no—
117C2×C42—24no—
118C2×C42—22no—
119C2×C42—42no—
120C2×C42—44no—
121C2×C42—44no—
122C2×C42—42no—
123C2×C42C22⋊C412yes—
124C2×C42C22⋊C412yes—
125C2×C42C22⋊C412yes—
126C2×C42C22⋊C412yes—
127C2×C42C22⋊C412yes—
128C2×C42C22⋊C411yescharacteristic
129C2×C42C22⋊C411yescharacteristic
130C2×C42C22×C411yescharacteristic
131C2×C42C22×C412yes—
132C2×C42C22×C412yes—
133C2×C42C2×D412yes—
134C2×C42C2×D412yes—
135C2×C42C2×D412yes—
136C2×C42C2×D412yes—
137C2×C42C2×D412yes—
138C2×C42C2411yescharacteristic
139C4⋊C8—48no—
140C22⋊C8—48no—
141#97—216no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—24no—
146C24—22no—
147C24C2×C22⋊C411yescharacteristic
148#41—816no—
149C2×Q8—24no—
150C2×Q8—44no—
151C2×Q8—44no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—44no—
155C2×D4—24no—
156C2×D4—42no—
157C2×D4—42no—
158C2×D4—44no—
159C2×D4—44no—
160C2×D4—44no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4—44no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—44no—
181C22×C4—44no—
182C22×C4C2×C22⋊C412yes—
183C22×C4C2×C22⋊C412yes—
184C22×C4C2×C22⋊C412yes—
185C22×C4C2×C22⋊C411yescharacteristic
186C22×C4C2×C22⋊C412yes—
187C22×C4C2×C22⋊C412yes—
188C22×C4C23×C411yescharacteristic
189C22×C4C22×D412yes—
190C2×C8—48no—
191C2×C8—48no—
192C2×C8—48no—
193C4⋊C4—28no—
194C4⋊C4—44no—
195C4⋊C4—44no—
196C4⋊C4—48no—
197C22⋊C4—48no—
198C22⋊C4—48no—
199C22⋊C4—48no—
200C22⋊C4—44no—
201C22⋊C4—48no—
202C22⋊C4—48no—
203C42—22no—
204C42—24no—
205C42—24no—
206C42—24no—
207C42—24no—
208C42—24no—
209C42—24no—
210C42—41no—
211C42—41no—
212C42—42no—
213C42—42no—
214C42—42no—
215C42—42no—
216C42—42no—
217C42—42no—
218C42—44no—
219C42H(5,𝔽2)14yes—
220C23—21no—
221C23—21no—
222C23—22no—
223C23—22no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23—44no—
228C23—44no—
229C23#41312yes—
230C23C22×C22⋊C411yescharacteristic
231Q8—84no—
232Q8—88no—
233D4—84no—
234D4—88no—
235D4—88no—
236C2×C4—28no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—44no—
257C2×C4—48no—
258C2×C4—48no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—88no—
267C2×C4#41312yes—
268C2×C4#42114yes—
269C2×C4#42114yes—
270C2×C4#51812yes—
271C2×C4#53514yes—
272C8—88no—
273C22—24no—
274C22—22no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—82no—
279C22—84no—
280C22—82no—
281C22—82no—
282C22—84no—
283C22#230911yescharacteristic
284C22C2×#41311yescharacteristic
285C22C2×#41311yescharacteristic
286C22C2×#42112yes—
287C22C2×#42112yes—
288C4—24no—
289C4—22no—
290C4—22no—
291C4—24no—
292C4—24no—
293C4—42no—
294C4—42no—
295C4—42no—
296C4—82no—
297C4—84no—
298C2—42no—
299C2—162no—
300C2#1766511yescharacteristic
301C2#1767412yes—
302C2#1767811yescharacteristic
303C2#2179811yescharacteristic
304C2#2181612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20778C2semidirect
3. #9688C2semidirect
4. #8338C2semidirect

Maximal subgroups

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

subgroupindex
#366392
#207782
#96882
#83382
#83372