Finite Group Databasebeta

← #144191

Subgroups — #144191

Order 512.

392 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 96 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#33731C211yesmaximal, characteristic
2#33705C211yesmaximal, characteristic
3C2×#1707C211yesmaximal, characteristic
4#13290C211yesmaximal, characteristic
5#13212C211yesmaximal, characteristic
6#9855C211yesmaximal, characteristic
7#7138C211yesmaximal, characteristic
8#3160C2211yescharacteristic
9C2×#500C2211yescharacteristic
10C2×#498—21no—
11C2×#494C2211yescharacteristic
12C2×#487—21no—
13#2255—22no—
14#2254—21no—
15#2252—21no—
16#2245—22no—
17#2241—21no—
18#2241—21no—
19#2159—22no—
20C2×#362—22no—
21#2141—22no—
22C4×M32C2211yescharacteristic
23C4×M32C2211yescharacteristic
24#1848C2211yescharacteristic
25#1707—22no—
26#1707—22no—
27#1493—21no—
28#1493—21no—
29C2×C82C2211yescharacteristic
30#1387—22no—
31C22×Q16—21no—
32C22×Q16—21no—
33#533—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×M32D412yes—
37C2×M32D412yes—
38#500—21no—
39#500—21no—
40#500—21no—
41#500D412yes—
42#500D411yescharacteristic
43#500D411yescharacteristic
44#499—21no—
45#499—21no—
46#498—24no—
47#498—21no—
48#498—42no—
49#494—21no—
50#494—21no—
51#494—21no—
52#494D412yes—
53#487—24no—
54#487—42no—
55C2×#108—21no—
56C2×#108—21no—
57#427—21no—
58#427—21no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#106—42no—
62#423—21no—
63C2×#103—42no—
64C2×C4×C8D411yescharacteristic
65C2×C4×C8D411yescharacteristic
66C2×C4×C8C2311yescharacteristic
67#367—24no—
68#366—24no—
69#362—44no—
70#358—24no—
71#358—24no—
72#346—21no—
73#346—21no—
74C4×C16—21no—
75C4×C16—21no—
76Q8⋊C8—42no—
77Q8⋊C8—42no—
78C82—21no—
79C82—21no—
80C22×Q8—41no—
81C22×Q8—41no—
82C2×Q16—22no—
83C2×Q16—22no—
84C2×Q16—41no—
85C2×Q16—41no—
86C2×Q16—42no—
87C2×Q16—42no—
88C2×Q16—42no—
89C2×Q16—42no—
90C2×M16—41no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8C2×D411yescharacteristic
94C22×C8C2×D411yescharacteristic
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97C4⋊Q8—21no—
98C4⋊Q8—21no—
99C4⋊Q8—41no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C4⋊Q8—41no—
103C4⋊Q8—41no—
104C4⋊Q8—41no—
105#125—41no—
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108C4×Q8—41no—
109C4×Q8—41no—
110#117—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C42C2×D411yescharacteristic
116M32—22no—
117M32—22no—
118M32—22no—
119M32SD1614yes—
120C2×C16—22no—
121C2×C16—22no—
122#108—42no—
123#108—42no—
124#107—21no—
125#107—21no—
126#107—21no—
127#107—22no—
128#107—21no—
129#107—22no—
130#107—41no—
131#107—41no—
132#106—48no—
133#106—41no—
134#106—41no—
135C4⋊C8—41no—
136C4⋊C8—41no—
137#103—48no—
138C4×C8—21no—
139C4×C8—21no—
140C4×C8D812yes—
141C4×C8D812yes—
142C4×C8C2×D411yescharacteristic
143C4×C8C2×D411yescharacteristic
144C4×C8C2×D411yescharacteristic
145C4×C8#4111yescharacteristic
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—84no—
151C2×Q8—81no—
152C2×Q8—81no—
153C2×Q8—81no—
154C2×Q8—81no—
155C2×Q8—81no—
156C2×Q8—81no—
157C22×C4—81no—
158C22×C4—81no—
159C22×C4#12711yescharacteristic
160C22×C4AGL(1,ℤ/8)11yescharacteristic
161C22×C4AGL(1,ℤ/8)11yescharacteristic
162Q16—44no—
163Q16—44no—
164Q16—82no—
165Q16—82no—
166M16—81no—
167M16—81no—
168C2×C8—24no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—42no—
172C2×C8—81no—
173C2×C8#12011yescharacteristic
174C2×C8#12011yescharacteristic
175C2×C8#12111yescharacteristic
176C2×C8#12111yescharacteristic
177C2×C8C2×SD1612yes—
178C2×C8#13512yes—
179C2×C8#13712yes—
180C2×C8#13712yes—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—81no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C22⋊C4—81no—
196C42—81no—
197C42#12011yescharacteristic
198C42#12111yescharacteristic
199C42C2×D811yescharacteristic
200C42C2×D811yescharacteristic
201C16—24no—
202C16—42no—
203C23#49611yescharacteristic
204Q8—84no—
205Q8—82no—
206Q8—82no—
207Q8—82no—
208Q8—82no—
209Q8—162no—
210Q8—161no—
211Q8—161no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—81no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4—82no—
218C2×C4—162no—
219C2×C4—161no—
220C2×C4—161no—
221C2×C4#39211yescharacteristic
222C2×C4#44711yescharacteristic
223C2×C4#44711yescharacteristic
224C2×C4#45911yescharacteristic
225C2×C4#45911yescharacteristic
226C2×C4#49211yescharacteristic
227C2×C4#49311yescharacteristic
228C2×C4#49511yescharacteristic
229C2×C4#49711yescharacteristic
230C2×C4#49711yescharacteristic
231C8—24no—
232C8—42no—
233C8—161no—
234C8#44812yes—
235C8#45212yes—
236C8#46112yes—
237C8#46412yes—
238C22—21no—
239C22—21no—
240C22#171411yescharacteristic
241C22#171911yescharacteristic
242C22#201111yescharacteristic
243C4—21no—
244C4—21no—
245C4—162no—
246C4—161no—
247C4—161no—
248C4—162no—
249C4#170611yescharacteristic
250C4#172211yescharacteristic
251C4#201311yescharacteristic
252C4#201511yescharacteristic
253C4#221212yes—
254C4#221412yes—
255C2—21no—
256C2—21no—
257C2#1030811yescharacteristic
258C2#1333211yescharacteristic
259C2#1333311yescharacteristic
subgroupG/H
center, socleC22#2011
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#496
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#337312
#337052
C2×#17072
#132902
#132122
#98552
#71382