Finite Group Databasebeta

← #114645

Subgroups — #114645

Order 512.

1116 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1116 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×#3152C211yesmaximal, characteristic
2#21799C211yesmaximal, characteristic, complemented
3#17746C211yesmaximal, characteristic, complemented
4#17490C212yesmaximal
5#7992C212yesmaximal, complemented
6C22×#137—22no—
7C2×#532C2211yescharacteristic
8C22×C4⋊Q8C2211yescharacteristic
9#3152—24no—
10#3152—24no—
11C2×#497—22no—
12C2×#494C412yes—
13C2×#492C412yes—
14#2920C2212yes—
15#2920C2212yes—
16C2×C4×M16C2211yescharacteristic
17#1966—24no—
18#1784—22no—
19#1558—22no—
20#1548—24no—
21#1496—24no—
22#1486—24no—
23#1484—24no—
24#1291—24no—
25C22×#41—21no—
26C23×Q8D412yes—
27C2×#137—216no—
28C2×#137—416no—
29C22×Q16—24no—
30C22×SD16—24no—
31C22×M16—21no—
32#532—22no—
33#532—22no—
34#532—22no—
35#532D414yes—
36C2×C4⋊Q8—22no—
37C2×C4⋊Q8—22no—
38C2×C4⋊Q8—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8C2×C412yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8D412yes—
44C2×#127C2×C411yescharacteristic
45C2×#124—21no—
46C2×#121—22no—
47C2×C4×D4—22no—
48C22×C4⋊C4—22no—
49C22×C42C2311yescharacteristic
50#497—216no—
51#494—24no—
52#494—24no—
53#492—24no—
54#492—24no—
55#432—22no—
56#432—22no—
57#432—22no—
58#432—22no—
59#431—22no—
60#431—22no—
61#431—22no—
62#431D414yes—
63#423—22no—
64#423D414yes—
65C2×#101—28no—
66#407—22no—
67#407—22no—
68C4×M16—22no—
69C4×M16—22no—
70C4×M16—22no—
71C4×M16D414yes—
72C2×#97—21no—
73C2×C4×C8C2×C412yes—
74#333—28no—
75D4⋊C8—216no—
76C2×#41—24no—
77C2×#41—48no—
78C2×#41—42no—
79C22×Q8—22no—
80C22×Q8—21no—
81C22×Q8—24no—
82C22×Q8—24no—
83C22×Q8—48no—
84C22×Q8C22⋊C414yes—
85C22×D4—21no—
86C22×D4—22no—
87C22×D4—41no—
88C23×C4—22no—
89C23×C4—41no—
90C23×C4C2×D411yescharacteristic
91C23×C4#4112yes—
92#137—464no—
93C2×Q16—216no—
94C2×Q16—416no—
95C2×SD16—216no—
96C2×SD16—416no—
97C2×M16—28no—
98C2×M16—24no—
99C2×M16—24no—
100C2×M16—42no—
101C22×C8—22no—
102C4⋊Q8—22no—
103C4⋊Q8—22no—
104C4⋊Q8—21no—
105C4⋊Q8—22no—
106C4⋊Q8—24no—
107C4⋊Q8—22no—
108C4⋊Q8—24no—
109C4⋊Q8—22no—
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112C4⋊Q8—42no—
113C4⋊Q8—42no—
114C4⋊Q8C22⋊C412yes—
115#127—21no—
116#127—22no—
117#127C22⋊C412yes—
118#124—24no—
119#124—41no—
120#124—41no—
121#121—24no—
122#121—24no—
123#121—42no—
124#121—42no—
125C4×D4—24no—
126C4×D4—24no—
127C4×D4—42no—
128C4×D4—42no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—24no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C42C22×C411yescharacteristic
138C2×C42C2×D412yes—
139C2×C42C2×D412yes—
140C2×C42C2×D412yes—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D412yes—
143C2×C42#4112yes—
144C4⋊C8—24no—
145C4⋊C8—24no—
146C4⋊C8—24no—
147C4⋊C8—42no—
148#101—88no—
149C22⋊C8—24no—
150C22⋊C8—42no—
151C22⋊C8—42no—
152C22⋊C8—42no—
153#97—24no—
154#97—42no—
155#97—42no—
156C4×C8—22no—
157C4×C8—24no—
158C4×C8—42no—
159C4×C8C22⋊C414yes—
160C24—41no—
161C24#12311yescharacteristic
162#41—416no—
163#41—88no—
164C2×Q8—22no—
165C2×Q8—24no—
166C2×Q8—24no—
167C2×Q8—48no—
168C2×Q8—44no—
169C2×Q8—416no—
170C2×Q8—41no—
171C2×Q8—44no—
172C2×Q8—48no—
173C2×Q8—88no—
174C2×D4—22no—
175C2×D4—24no—
176C2×D4—44no—
177C2×D4—44no—
178C2×D4—41no—
179C2×D4—42no—
180C2×D4—84no—
181C2×D4—84no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—44no—
185C22×C4—41no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—84no—
189C22×C4—84no—
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4C4×D412yes—
192C22×C4#12012yes—
193C22×C4#12011yescharacteristic
194C22×C4#12112yes—
195C22×C4#12112yes—
196C22×C4#12112yes—
197C22×C4#12112yes—
198C22×C4#12312yes—
199C22×C4#12412yes—
200Q16—432no—
201SD16—432no—
202M16—216no—
203M16—48no—
204M16—84no—
205C2×C8—28no—
206C2×C8—24no—
207C2×C8—44no—
208C2×C8—44no—
209C4⋊C4—24no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—44no—
214C4⋊C4—44no—
215C22⋊C4—44no—
216C22⋊C4—44no—
217C42C2×C22⋊C411yescharacteristic
218C42C2×C22⋊C411yescharacteristic
219C42C4×D412yes—
220C42#12012yes—
221C42#12012yes—
222C42#12011yescharacteristic
223C42#12112yes—
224C42#12112yes—
225C42#12311yescharacteristic
226C42#12712yes—
227C23—22no—
228C23—42no—
229C23—42no—
230C23—84no—
231C23—82no—
232C23#38611yescharacteristic
233C23#39412yes—
234Q8—48no—
235Q8—44no—
236Q8—84no—
237Q8—82no—
238Q8—88no—
239D4—44no—
240D4—48no—
241D4—82no—
242D4—84no—
243D4—88no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—48no—
254C2×C4—42no—
255C2×C4—44no—
256C2×C4—88no—
257C2×C4—82no—
258C2×C4—88no—
259C2×C4#37911yescharacteristic
260C2×C4#38611yescharacteristic
261C2×C4#38612yes—
262C2×C4#39012yes—
263C2×C4#39212yes—
264C2×C4#39212yes—
265C2×C4#39412yes—
266C2×C4#45314yes—
267C8—28no—
268C8—84no—
269C8—84no—
270C22—22no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—44no—
275C22—84no—
276C22—82no—
277C22—164no—
278C22#143811yescharacteristic
279C22#188712yes—
280C4—22no—
281C4—22no—
282C4—22no—
283C4—21no—
284C4—21no—
285C4—84no—
286C4—82no—
287C4#190214yes—
288C4#200214yes—
289C2—22no—
290C2—42no—
291C2—162no—
292C2#1097511yescharacteristic
293C2#1101512yes—
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. #21799C2semidirect
3. #17746C2semidirect
5. #7992C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31522
#217992
#177462
#174902
#79922