Finite Group Databasebeta

← #329660

Subgroups — #329660

Order 512.

603 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 138 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#29878C211yesmaximal, characteristic
2#29691C212yesmaximal
3#21821C211yesmaximal, characteristic
4#10624C212yesmaximal, complemented
5#10482C214yesmaximal, complemented
6#10279C212yesmaximal, complemented
7#9136C212yesmaximal
8#8598C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#500C2211yescharacteristic
11C2×#477C2214yes—
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2857C2212yes—
15#2087—22no—
16#2076—22no—
17#2057—22no—
18#2043—22no—
19#2036—24no—
20#1990C2212yes—
21#1985—22no—
22#1984C2212yes—
23#1984C2212yes—
24#1834C2211yescharacteristic
25#1818—22no—
26#1803C2214yes—
27#1797C2212yes—
28C2×#334C2212yes—
29C2×#333C2214yes—
30#1296C2214yes—
31#1277C2212yes—
32#533—42no—
33#533—42no—
34C2×C4⋊Q8D411yescharacteristic
35C2×C4⋊Q8D411yescharacteristic
36C2×C4⋊Q8C2312yes—
37C2×#125—21no—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8D412yes—
41C2×C4×Q8—21no—
42C2×C4×Q8C2312yes—
43C2×#117—21no—
44C22×C42C2311yescharacteristic
45#500—42no—
46#477—44no—
47#477—44no—
48#432—42no—
49#431—42no—
50C2×#107—22no—
51C2×#107—24no—
52C2×C4⋊C8D412yes—
53C2×C4⋊C8C2314yes—
54C2×C4⋊C8C2312yes—
55C2×#103—24no—
56C2×#103—24no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8D412yes—
60C2×#97—21no—
61C2×C4×C8C2311yescharacteristic
62C2×C4×C8C2311yescharacteristic
63#391—22no—
64#389—22no—
65#389—22no—
66#384C2312yes—
67#380—22no—
68#380—22no—
69#340—22no—
70#340—24no—
71#340—24no—
72#336—24no—
73#336—24no—
74#336—22no—
75#334—42no—
76#333—44no—
77C22×Q8—21no—
78C22×Q8—22no—
79C22×Q8C2×D412yes—
80C23×C4C2×D411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4#4111yescharacteristic
83C22×C8—22no—
84C22×C8—24no—
85C22×C8C2×D412yes—
86C22×C8C2×D412yes—
87C4⋊Q8—24no—
88C4⋊Q8—24no—
89C4⋊Q8—42no—
90C4⋊Q8—42no—
91#125—44no—
92C22⋊Q8—44no—
93C22⋊Q8—44no—
94C22⋊Q8—84no—
95C4×Q8—24no—
96C4×Q8—24no—
97C4×Q8—42no—
98C4×Q8—42no—
99#117—42no—
100#117—42no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4C2×D412yes—
113C2×C22⋊C4—22no—
114C2×C22⋊C4—42no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42—22no—
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42#4111yescharacteristic
121C2×C42C2411yescharacteristic
122#107—48no—
123#107—44no—
124C4⋊C8—24no—
125C4⋊C8—24no—
126C4⋊C8—24no—
127C4⋊C8—42no—
128#103—48no—
129#103—84no—
130C22⋊C8—42no—
131C22⋊C8—42no—
132C22⋊C8—82no—
133#97—42no—
134C4×C8—22no—
135C4×C8—22no—
136#95—42no—
137#95—42no—
138C24#12111yescharacteristic
139C2×Q8—24no—
140C2×Q8—24no—
141C2×Q8—24no—
142C2×Q8—42no—
143C2×Q8—44no—
144C2×Q8—82no—
145C2×Q8—82no—
146C2×Q8—82no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4#12111yescharacteristic
160C22×C4#12111yescharacteristic
161C22×C4#12111yescharacteristic
162C22×C4C22×D411yescharacteristic
163C22×C4C22×D411yescharacteristic
164C22×C4C2×#4111yescharacteristic
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—24no—
168C2×C8—24no—
169C2×C8—44no—
170C2×C8—42no—
171C2×C8—44no—
172C2×C8—44no—
173C2×C8—44no—
174C4⋊C4—24no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—42no—
179C4⋊C4—44no—
180C4⋊C4—44no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—44no—
186C4⋊C4—82no—
187C4⋊C4—84no—
188C22⋊C4—44no—
189C22⋊C4—84no—
190C42—21no—
191C42—22no—
192C42—21no—
193C42—22no—
194C42—42no—
195C42—42no—
196C42H(5,𝔽2)12yes—
197C42H(5,𝔽2)12yes—
198C23—21no—
199C23—22no—
200C23—21no—
201C23—21no—
202C23—22no—
203C23C2×#12111yescharacteristic
204Q8—48no—
205Q8—84no—
206Q8—164no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—44no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4#53412yes—
231C2×C4#53411yescharacteristic
232C2×C4#53511yescharacteristic
233C2×C4#53812yes—
234C2×C4#57712yes—
235C2×C4#57812yes—
236C2×C4#57814yes—
237C8—42no—
238C8—42no—
239C8—84no—
240C8—82no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—42no—
245C22—42no—
246C22#261711yescharacteristic
247C22#304412yes—
248C22#305911yescharacteristic
249C22#306011yescharacteristic
250C22#306812yes—
251C4—22no—
252C4—41no—
253C4—41no—
254C4—41no—
255C4—82no—
256C4—82no—
257C4—162no—
258C4—162no—
259C4#320114yes—
260C4#320212yes—
261C4#322312yes—
262C2—81no—
263C2#1953112yes—
264C2#1953212yes—
265C2#2972411yescharacteristic
266C2#2973711yescharacteristic
267C2#2975111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10624C2semidirect
5. #10482C2semidirect
6. #10279C2semidirect

Maximal subgroups

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

subgroupindex
#298782
#296912
#218212
#106242
#104822
#102792
#91362
#85982