Finite Group Databasebeta

← #145403

Subgroups — #145403

Order 512.

285 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 78 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#19204C211yesmaximal, characteristic
2#13265C211yesmaximal, characteristic
3#13212C211yesmaximal, characteristic
4#12009C211yesmaximal, characteristic
5C2×#1362C211yesmaximal, characteristic
6#8685C211yesmaximal, characteristic
7#7277C211yesmaximal, characteristic
8#3229—21no—
9C2×#494C2211yescharacteristic
10#2960C2211yescharacteristic
11C2×C4⋊C16C2211yescharacteristic
12C2×#362—21no—
13C2×#362—21no—
14C4×M32C2211yescharacteristic
15#1848C2211yescharacteristic
16#1690—22no—
17#1670—21no—
18#1670—21no—
19#1564C2211yescharacteristic
20C2×#335C2211yescharacteristic
21#1550—21no—
22#1546—21no—
23#1379—21no—
24#1379—21no—
25#1372—21no—
26#1371—21no—
27#1362—22no—
28#1292—21no—
29C22×Q16—21no—
30C22×Q16—21no—
31#533D411yescharacteristic
32C2×C4⋊Q8—21no—
33C2×M32D411yescharacteristic
34C2×M32D411yescharacteristic
35C22×C16—21no—
36#498—41no—
37#494—21no—
38#494—21no—
39#494—21no—
40#494D412yes—
41#487—41no—
42#483—42no—
43#475—22no—
44#475—22no—
45#446—21no—
46#446—21no—
47C8×Q8—21no—
48C8×Q8—21no—
49#431—21no—
50C2×#108—21no—
51C2×#108—21no—
52#424—21no—
53#423—21no—
54#423D411yescharacteristic
55C2×C4⋊C8—21no—
56#405—21no—
57C2×C4×C8C2311yescharacteristic
58C4⋊C16—21no—
59C4⋊C16—21no—
60#362—42no—
61#362—42no—
62#346—21no—
63C4×C16—21no—
64#335—21no—
65#335—21no—
66#335D411yescharacteristic
67#335D411yescharacteristic
68#334—21no—
69#333—41no—
70#332—22no—
71#332—22no—
72#332—41no—
73C22×Q8—41no—
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—41no—
77C2×Q16—41no—
78C2×Q16—41no—
79C2×Q16—41no—
80C2×Q16—41no—
81C2×Q16—41no—
82C2×Q16—81no—
83C2×Q16—81no—
84C2×M16—21no—
85C2×M16—41no—
86C22×C8—41no—
87C22×C8C2×D411yescharacteristic
88C22×C8#4111yescharacteristic
89C4⋊Q8—22no—
90C4⋊Q8—41no—
91C4⋊Q8—41no—
92C4⋊Q8—41no—
93C4⋊Q8D811yescharacteristic
94C4⋊Q8D811yescharacteristic
95#125—21no—
96C22⋊Q8—21no—
97C22⋊Q8—21no—
98C4×Q8—21no—
99C4×Q8—21no—
100#117—21no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C42C2×D411yescharacteristic
104M32—21no—
105M32—21no—
106M32—21no—
107M32SD1612yes—
108C2×C16—21no—
109C2×C16—21no—
110C2×C16—41no—
111C2×C16—41no—
112C2×C16—41no—
113#108—41no—
114#108—41no—
115#108—81no—
116#106—42no—
117#106—42no—
118C4⋊C8—21no—
119C4⋊C8—21no—
120C4⋊C8—21no—
121C4⋊C8—21no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128#103—42no—
129#103—42no—
130C22⋊C8—41no—
131#97—41no—
132C4×C8—41no—
133C4×C8—41no—
134C4×C8C2×D411yescharacteristic
135C4×C8C2×D411yescharacteristic
136C4×C8#4111yescharacteristic
137C4×C8#4111yescharacteristic
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—42no—
141C2×Q8—81no—
142C2×Q8—81no—
143C2×Q8—81no—
144C2×Q8—81no—
145C2×Q8—81no—
146C22×C4—81no—
147C22×C4#12111yescharacteristic
148C22×C4AGL(1,ℤ/8)11yescharacteristic
149C22×C4#13711yescharacteristic
150Q16—42no—
151Q16—42no—
152Q16—82no—
153Q16—82no—
154Q16—81no—
155Q16—81no—
156M16—41no—
157M16—41no—
158M16—81no—
159M16—81no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—81no—
166C2×C8—81no—
167C2×C8—161no—
168C2×C8#12111yescharacteristic
169C2×C8#12111yescharacteristic
170C2×C8#12311yescharacteristic
171C2×C8#12411yescharacteristic
172C2×C8C2×SD1611yescharacteristic
173C2×C8#13511yescharacteristic
174C2×C8AGL(1,ℤ/8)11yescharacteristic
175C2×C8#13711yescharacteristic
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C22⋊C4—41no—
187C42—41no—
188C42#12011yescharacteristic
189C42#12311yescharacteristic
190C42C2×D811yescharacteristic
191C42#13511yescharacteristic
192C16—22no—
193C16—41no—
194C16—81no—
195C23#46811yescharacteristic
196Q8—41no—
197Q8—41no—
198Q8—81no—
199Q8—82no—
200Q8—82no—
201Q8—161no—
202Q8—161no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—82no—
209C2×C4—161no—
210C2×C4—161no—
211C2×C4#39411yescharacteristic
212C2×C4#44711yescharacteristic
213C2×C4#45211yescharacteristic
214C2×C4#46011yescharacteristic
215C2×C4#46411yescharacteristic
216C2×C4#46611yescharacteristic
217C2×C4#47111yescharacteristic
218C2×C4#47211yescharacteristic
219C2×C4#48011yescharacteristic
220C2×C4#48211yescharacteristic
221C8—21no—
222C8—21no—
223C8—81no—
224C8—81no—
225C8—81no—
226C8—161no—
227C8—161no—
228C8#46111yescharacteristic
229C8#46311yescharacteristic
230C8#46511yescharacteristic
231C8#46811yescharacteristic
232C22—21no—
233C22—21no—
234C22#168611yescharacteristic
235C22#169811yescharacteristic
236C22#203311yescharacteristic
237C4—21no—
238C4—21no—
239C4—21no—
240C4—81no—
241C4—81no—
242C4—162no—
243C4#166811yescharacteristic
244C4#168211yescharacteristic
245C4#205111yescharacteristic
246C4#205511yescharacteristic
247C4#219311yescharacteristic
248C4#219511yescharacteristic
249C2—21no—
250C2—21no—
251C2#1047411yescharacteristic
252C2#1338811yescharacteristic
253C2#1339011yescharacteristic
subgroupG/H
center, socleC22#2033
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC23#468
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ 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
#192042
#132652
#132122
#120092
C2×#13622
#86852
#72772