Finite Group Databasebeta

← #112804

Subgroups — #112804

Order 512.

862 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×#3131C211yesmaximal, characteristic
2#21826C211yesmaximal, characteristic, complemented
3#17676C211yesmaximal, characteristic, complemented
4#17595C212yesmaximal
5#7898C212yesmaximal, complemented
6C2×#533C2211yescharacteristic
7C2×#532C2211yescharacteristic
8#3131—24no—
9#3131—24no—
10C2×#488—22no—
11C2×#487C412yescomplemented
12C2×#486C412yescomplemented
13C2×#432C2212yes—
14C2×#416—22no—
15C22×C4×C8C2211yescharacteristic
16#2283C2212yes—
17#1996—24no—
18#1820—22no—
19#1534—22no—
20#1531—24no—
21#1519—24no—
22#1514—24no—
23#1505—24no—
24#1279—24no—
25C22×#41—21no—
26C23×C8—21no—
27#533—22no—
28#533—22no—
29#533—22no—
30#533—22no—
31#532—22no—
32#532—22no—
33#532—22no—
34#532D414yes—
35C2×C4⋊Q8C2×C411yescharacteristic
36C2×C4⋊Q8C2×C412yes—
37C2×#127C2×C411yescharacteristic
38C2×#125D412yes—
39C2×#124—21no—
40C2×C22⋊Q8—22no—
41C2×#121—22no—
42C2×C4×Q8—22no—
43C2×C4×D4—22no—
44C2×#117D412yes—
45C22×C42C2311yescharacteristic
46#488—216no—
47#487—24no—
48#487—24no—
49#486—24no—
50#486—24no—
51#432—22no—
52#432—22no—
53#432—22no—
54#432D414yes—
55C2×C4⋊C8—22no—
56#416—216no—
57C2×#103—24no—
58C2×#102—24no—
59C2×C22⋊C8—22no—
60C2×#97—22no—
61C2×C4×C8—21no—
62C2×C4×C8—22no—
63C2×C4×C8C2×C412yes—
64C2×C4×C8D414yes—
65#383—22no—
66#382—22no—
67#381—22no—
68#380D414yes—
69C4×C4⋊C4—22no—
70C4×C22⋊C4—22no—
71#376—22no—
72#342—28no—
73#330—28no—
74#327—216no—
75C2×#41—24no—
76C2×#41—48no—
77C2×#41—42no—
78C22×Q8—22no—
79C22×Q8—21no—
80C22×D4—21no—
81C22×D4—22no—
82C22×D4—41no—
83C23×C4—41no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D412yes—
86C22×C8—22no—
87C22×C8—24no—
88C22×C8—44no—
89C22×C8—42no—
90C4⋊Q8—22no—
91C4⋊Q8—22no—
92C4⋊Q8—21no—
93C4⋊Q8—22no—
94C4⋊Q8—22no—
95C4⋊Q8—22no—
96C4⋊Q8C22⋊C412yes—
97#127—21no—
98#127—22no—
99#127C22⋊C412yes—
100#125—22no—
101#125—24no—
102#125—22no—
103#124—24no—
104#124—41no—
105#124—41no—
106C22⋊Q8—42no—
107C22⋊Q8—42no—
108C22⋊Q8—42no—
109C22⋊Q8—42no—
110#121—24no—
111#121—24no—
112#121—42no—
113#121—42no—
114C4×Q8—42no—
115C4×Q8—42no—
116C4×Q8—42no—
117C4×Q8—42no—
118C4×D4—24no—
119C4×D4—24no—
120C4×D4—42no—
121C4×D4—42no—
122#117—24no—
123#117—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4C22⋊C414yes—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—42no—
132C2×C42—22no—
133C2×C42—42no—
134C2×C42C22×C411yescharacteristic
135C2×C42C2×D412yes—
136C2×C42C2×D412yes—
137C2×C42C2×D412yes—
138C2×C42C2×D411yescharacteristic
139C2×C42#4112yes—
140C2×C42#4112yes—
141C4⋊C8—24no—
142C4⋊C8—42no—
143#103—216no—
144#102—216no—
145C22⋊C8—24no—
146C22⋊C8—42no—
147#97—24no—
148#97—42no—
149C4×C8—22no—
150C4×C8—24no—
151C4×C8C22⋊C414yes—
152#95—24no—
153#95—42no—
154#95—42no—
155#95—42no—
156C24—41no—
157C24#12011yescharacteristic
158#41—416no—
159#41—88no—
160C2×Q8—22no—
161C2×Q8—44no—
162C2×Q8—42no—
163C2×Q8—41no—
164C2×Q8—42no—
165C2×Q8—84no—
166C2×D4—22no—
167C2×D4—24no—
168C2×D4—44no—
169C2×D4—44no—
170C2×D4—41no—
171C2×D4—42no—
172C2×D4—84no—
173C2×D4—84no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—44no—
177C22×C4—41no—
178C22×C4—44no—
179C22×C4—44no—
180C22×C4—84no—
181C22×C4C2×C22⋊C411yescharacteristic
182C22×C4C4×D412yes—
183C22×C4#12012yes—
184C22×C4#12012yes—
185C22×C4#12112yes—
186C22×C4#12112yes—
187C22×C4#12112yes—
188C22×C4#12112yes—
189C22×C4#12311yescharacteristic
190C22×C4#12712yes—
191C2×C8—24no—
192C2×C8—44no—
193C2×C8—48no—
194C2×C8—44no—
195C2×C8—84no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—24no—
199C4⋊C4—48no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—44no—
204C4⋊C4—44no—
205C22⋊C4—44no—
206C22⋊C4—44no—
207C22⋊C4—44no—
208C22⋊C4—48no—
209C42—44no—
210C42—84no—
211C42C2×C22⋊C411yescharacteristic
212C42C2×C22⋊C411yescharacteristic
213C42C4×D412yes—
214C42#12012yes—
215C42#12011yescharacteristic
216C42#12112yes—
217C42#12112yes—
218C42#12312yes—
219C42#12311yescharacteristic
220C42#12412yes—
221C23—42no—
222C23—42no—
223C23—84no—
224C23—82no—
225C23#38611yescharacteristic
226C23#39212yes—
227C23#45414yes—
228Q8—44no—
229Q8—82no—
230Q8—84no—
231Q8—84no—
232D4—44no—
233D4—84no—
234D4—82no—
235D4—88no—
236D4—84no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—24no—
243C2×C4—24no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—88no—
250C2×C4—82no—
251C2×C4—84no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4#37911yescharacteristic
256C2×C4#38612yes—
257C2×C4#38611yescharacteristic
258C2×C4#39012yes—
259C2×C4#39212yes—
260C2×C4#39412yes—
261C2×C4#39412yes—
262C8—44no—
263C8—84no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—44no—
268C22—84no—
269C22—82no—
270C22—164no—
271C22#143811yescharacteristic
272C22#188812yes—
273C22#200314yes—
274C4—22no—
275C4—22no—
276C4—22no—
277C4—21no—
278C4—22no—
279C4—21no—
280C4—84no—
281C4—82no—
282C4—164no—
283C4#197614yes—
284C2—24no—
285C2—22no—
286C2—162no—
287C2#1097611yescharacteristic
288C2#1102612yes—
subgroupG/H
center, socleC22#1438
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21826C2semidirect
3. #17676C2semidirect
5. #7898C2semidirect
11. C2×#487C4semidirect
12. C2×#486C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31312
#218262
#176762
#175952
#78982