Finite Group Databasebeta

← #318576

Subgroups — #318576

Order 512.

579 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 156 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#28295C212yesmaximal
2#19256C212yesmaximal, complemented
3#17977C211yesmaximal, characteristic, complemented
4C2×#1580C211yesmaximal, characteristic
5#10461C214yesmaximal, complemented
6#8826C211yesmaximal, characteristic, complemented
7#8683C212yesmaximal, complemented
8#8616C212yesmaximal, complemented
9#3170C2211yescharacteristic
10#3077C2214yes—
11#2960C2212yes—
12C2×C4×D8C2211yescharacteristic
13C2×C8×D4C2212yes—
14#2063—24no—
15#1978—22no—
16#1952—22no—
17#1848C2212yes—
18#1800C2214yes—
19#1683—24no—
20#1675—22no—
21#1669—22no—
22#1584—24no—
23#1580—24no—
24#1567C2212yes—
25#1567C2211yescharacteristic
26#1567C2212yes—
27C2×#334C2211yescharacteristic
28#1550C2214yes—
29#1493C2212yes—
30C2×D4⋊C8C2212yes—
31C2×#322C2211yescharacteristic
32#1292C2214yes—
33C22×D8D411yescharacteristic
34C23×C8—22no—
35#533C2312yes—
36C2×C4×D4C2312yes—
37#500D411yescharacteristic
38#494D411yescharacteristic
39#487D412yes—
40#478—44no—
41#476—24no—
42#474—24no—
43#467—22no—
44#446—22no—
45#446—22no—
46C8×Q8—22no—
47C8×Q8—22no—
48C4×Q16—22no—
49C4×D8—22no—
50C4×D8—22no—
51C8×D4—22no—
52C8×D4—22no—
53C8×D4—22no—
54C8×D4—22no—
55C2×#108—22no—
56C2×#108—22no—
57#428—24no—
58C2×#107D411yescharacteristic
59#424—22no—
60#423C2314yes—
61#423C2312yes—
62C2×C4⋊C8C2312yes—
63#418—24no—
64C2×#102D412yes—
65C2×#100—24no—
66C2×C22⋊C8—22no—
67#405—22no—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70C2×C4×C8C2311yescharacteristic
71#343—24no—
72#335—22no—
73#335—22no—
74#335—22no—
75#335—42no—
76#334—22no—
77#333—24no—
78#332—24no—
79Q8⋊C8—24no—
80D4⋊C8—24no—
81#322D414yes—
82C22×D4C2×D412yes—
83C23×C4—22no—
84C2×Q16—22no—
85C2×D8—22no—
86C2×D8—22no—
87C2×D8Q1614yescomplemented
88C2×M16—22no—
89C2×M16—24no—
90C22×C8—21no—
91C22×C8—22no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8—84no—
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8#4112yes—
98C4⋊Q8C2×D412yes—
99C4⋊Q8C2×D412yes—
100#125—22no—
101C22⋊Q8—22no—
102C22⋊Q8—22no—
103C4×Q8—22no—
104C4×Q8—22no—
105C4×D4—22no—
106C4×D4—22no—
107C4×D4—22no—
108C4×D4—22no—
109#117—22no—
110C2×C4⋊C4C2×D412yes—
111C2×C22⋊C4—22no—
112C2×C42C2411yescharacteristic
113#108—44no—
114#108—82no—
115#107—22no—
116#107—22no—
117#107—44no—
118#106—44no—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—24no—
123C4⋊C8—22no—
124C4⋊C8—24no—
125C4⋊C8—42no—
126C4⋊C8—42no—
127#103—24no—
128#102—24no—
129#102—44no—
130#102—44no—
131#100—84no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134#97—42no—
135C4×C8—21no—
136C4×C8—42no—
137C4×C8C2×D412yes—
138C4×C8C2×D412yes—
139C4×C8C2×D412yes—
140C4×C8C2×D411yescharacteristic
141C4×C8C2×D411yescharacteristic
142C4×C8#4112yes—
143C24—22no—
144C2×Q8—22no—
145C2×Q8—22no—
146C2×D4—22no—
147C2×D4—22no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4C2×Q1614yes—
152C22×C4—22no—
153C22×C4—42no—
154C22×C4—42no—
155C22×C4—82no—
156C22×C4—82no—
157C22×C4C22×D411yescharacteristic
158C22×C4C2×#4111yescharacteristic
159C22×C42−1+411yescharacteristic
160Q16—82no—
161D8—28no—
162M16—44no—
163M16—42no—
164M16—44no—
165M16—42no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—22no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—44no—
173C2×C8—84no—
174C2×C8—84no—
175C2×C8—81no—
176C2×C8—82no—
177C2×C8#12112yes—
178C2×C8#12112yes—
179C2×C8#12112yes—
180C2×C8#12112yes—
181C2×C8C2×Q1612yes—
182C2×C8#13512yes—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C22⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—42no—
194C42—42no—
195C42C22×D411yescharacteristic
196C42C22×D411yescharacteristic
197C42C2×#4111yescharacteristic
198C42H(5,𝔽2)11yescharacteristic
199C23—24no—
200C23—42no—
201C23—42no—
202C23—82no—
203C23—82no—
204C23#54711yescharacteristic
205Q8—44no—
206Q8—82no—
207D4—24no—
208D4—24no—
209D4—42no—
210D4—42no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—84no—
217C2×C4—82no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4C2×#12111yescharacteristic
221C2×C4C2×#12111yescharacteristic
222C2×C4#53711yescharacteristic
223C2×C4D4211yescharacteristic
224C2×C4#54711yescharacteristic
225C2×C4#55511yescharacteristic
226C2×C4C22×Q1611yescharacteristic
227C2×C4C2×#13511yescharacteristic
228C2×C4#57611yescharacteristic
229C2×C4#57811yescharacteristic
230C8—22no—
231C8—22no—
232C8—22no—
233C8—42no—
234C8—84no—
235C8—82no—
236C8—82no—
237C8#46714yes—
238C22—21no—
239C22—21no—
240C22—44no—
241C22—84no—
242C22—84no—
243C22—82no—
244C22—82no—
245C22#253711yescharacteristic
246C22#333811yescharacteristic
247C22#334211yescharacteristic
248C4—21no—
249C4—21no—
250C4—21no—
251C4—82no—
252C4—82no—
253C4—82no—
254C4C2×#46711yescharacteristic
255C4#306111yescharacteristic
256C4#306812yes—
257C4D4×Q1611yescharacteristic
258C4#328711yescharacteristic
259C2—21no—
260C2—21no—
261C2—84no—
262C2#2822811yescharacteristic
263C2#2824311yescharacteristic
264C2#2825011yescharacteristic
subgroupG/H
center, socleC22#2537
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
LCSG ▷ C2×C42 ▷ 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. #19256C2semidirect
3. #17977C2semidirect
5. #10461C2semidirect
6. #8826C2semidirect
7. #8683C2semidirect
8. #8616C2semidirect
87. C2×D8Q16semidirect

Maximal subgroups

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

subgroupindex
#282952
#192562
#179772
C2×#15802
#104612
#88262
#86832
#86162