Finite Group Databasebeta

← #98997

Subgroups — #98997

Order 512.

328 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 102 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#21673C211yesmaximal, characteristic
2#8580C212yesmaximal
3#8274C211yesmaximal, characteristic
4#8273C211yesmaximal, characteristic
5#7049C212yesmaximal
6C4×#125C2211yescharacteristic
7#1939C411yescharacteristic
8#1939C411yescharacteristic
9#1916—21no—
10#1916—21no—
11#1831C2211yescharacteristic
12#1771—21no—
13#1771—21no—
14#1770C411yescharacteristic
15#1770C411yescharacteristic
16#1767C2212yes—
17#1767C2212yes—
18C42×C8C2211yescharacteristic
19C2×#335—21no—
20C2×#334—21no—
21C2×#330—22no—
22#1295—22no—
23#1277—21no—
24#1277—21no—
25C2×#125D411yescharacteristic
26C2×C4⋊C8—22no—
27C2×C4⋊C8—22no—
28C2×C4⋊C8—21no—
29C2×C4⋊C8—21no—
30C2×C4⋊C8C2×C411yescharacteristic
31C2×C4⋊C8C2×C411yescharacteristic
32C2×#97—21no—
33C2×#97—21no—
34C2×#97—22no—
35C2×#97—22no—
36C2×C4×C8C2×C411yescharacteristic
37C2×C4×C8C2×C411yescharacteristic
38C2×C4×C8D411yescharacteristic
39C2×C4×C8D411yescharacteristic
40C2×C4×C8D411yescharacteristic
41C2×C4×C8D411yescharacteristic
42#389Q811yescharacteristic
43#389Q811yescharacteristic
44#387—21no—
45#387—21no—
46#382D411yescharacteristic
47C4×C4⋊C4—21no—
48C4×C4⋊C4—21no—
49C4×C4⋊C4C2×C411yescharacteristic
50C4×C4⋊C4C2×C411yescharacteristic
51C43C2311yescharacteristic
52#336—21no—
53#336—21no—
54#336—21no—
55#336—21no—
56#336—21no—
57#336—21no—
58#336—22no—
59#336—22no—
60#335—24no—
61#334—24no—
62#330—44no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—42no—
66C22×C8—42no—
67C22×C8C22⋊C411yescharacteristic
68C22×C8C22⋊C411yescharacteristic
69C22×C8C22⋊C411yescharacteristic
70C22×C8C22⋊C411yescharacteristic
71#125—22no—
72#125—22no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4C4⋊C411yescharacteristic
78C2×C4⋊C4C4⋊C411yescharacteristic
79C2×C4⋊C4C4⋊C411yescharacteristic
80C2×C4⋊C4C4⋊C411yescharacteristic
81C2×C42—21no—
82C2×C42—21no—
83C2×C42C22×C411yescharacteristic
84C2×C42C2×D411yescharacteristic
85C2×C42C2×D411yescharacteristic
86C2×C42C2×D411yescharacteristic
87C2×C42C2×Q811yescharacteristic
88C2×C42#4111yescharacteristic
89C2×C42#4111yescharacteristic
90C4⋊C8—24no—
91C4⋊C8—24no—
92C4⋊C8—41no—
93C4⋊C8—41no—
94C4⋊C8—42no—
95C4⋊C8—42no—
96C4⋊C8—42no—
97C4⋊C8—42no—
98#97—42no—
99#97—42no—
100#97—42no—
101#97—42no—
102#97—42no—
103#97—42no—
104C4×C8—22no—
105C4×C8—22no—
106C4×C8—22no—
107C4×C8—41no—
108C4×C8—41no—
109C4×C8—41no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114#95—21no—
115#95—21no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4C2×C22⋊C411yescharacteristic
121C22×C4C2×C4⋊C411yescharacteristic
122C22×C4#11711yescharacteristic
123C22×C4#12111yescharacteristic
124C22×C4#12111yescharacteristic
125C22×C4C22⋊Q811yescharacteristic
126C22×C4C22⋊Q811yescharacteristic
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—44no—
137C2×C8—44no—
138C2×C8—41no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—41no—
142C2×C8—41no—
143C2×C8—41no—
144C2×C8—82no—
145C2×C8—82no—
146C2×C8—82no—
147C2×C8—82no—
148C2×C8#10112yes—
149C2×C8#10412yes—
150C2×C8#10412yes—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—21no—
165C42—21no—
166C42—41no—
167C42—41no—
168C42—41no—
169C42—41no—
170C42—41no—
171C42—41no—
172C42—41no—
173C42AGL(1,ℤ/8)12yes—
174C42#13712yes—
175C23#38011yescharacteristic
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4C2×#10111yescharacteristic
204C2×C4#41311yescharacteristic
205C2×C4C2×#10411yescharacteristic
206C2×C4C2×#10411yescharacteristic
207C2×C4#42111yescharacteristic
208C2×C4#42111yescharacteristic
209C2×C4#42812yes—
210C2×C4#46812yes—
211C2×C4#46911yescharacteristic
212C2×C4#47011yescharacteristic
213C8—41no—
214C8—41no—
215C8—41no—
216C8—84no—
217C8—84no—
218C8—81no—
219C8—81no—
220C8#33014yescomplemented
221C22#180111yescharacteristic
222C22#180611yescharacteristic
223C22#180611yescharacteristic
224C22#180911yescharacteristic
225C22#181311yescharacteristic
226C22#181411yescharacteristic
227C22#181411yescharacteristic
228C4—22no—
229C4—21no—
230C4—21no—
231C4—22no—
232C4—41no—
233C4—41no—
234C4—41no—
235C4—81no—
236C4—81no—
237C4—81no—
238C4—81no—
239C4C2×#33011yescharacteristic
240C4#152911yescharacteristic
241C4#152911yescharacteristic
242C4#153711yescharacteristic
243C2#858011yescharacteristic
244C2#861911yescharacteristic
245C2#863511yescharacteristic
246C2#866811yescharacteristic
247C2#866911yescharacteristic
248C2#869211yescharacteristic
249C2#869311yescharacteristic
subgroupG/H
center, socle, prime radicalC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
220. C8#330semidirect

Maximal subgroups

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

subgroupindex
#216732
#85802
#82742
#82732
#70492