Finite Group Databasebeta

← #98694

Subgroups — #98694

Order 512.

908 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 128 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#59817C211yesmaximal, characteristic
2#9644C211yesmaximal, characteristic
3#8589C212yesmaximal, complemented
4#7651C211yesmaximal, characteristic
5#7588C212yesmaximal, complemented
6C2×D4×Q8—22no—
7#2683—22no—
8#2551C412yes—
9#2538—22no—
10#2452—24no—
11#2409—22no—
12#2366C412yes—
13C4×C4⋊Q8C2212yes—
14C4×#127C2211yescharacteristic
15#1842C2211yescharacteristic
16#1831C2212yes—
17#1744C2211yescharacteristic
18C2×#332—22no—
19C2×D4⋊C8—22no—
20#1296—22no—
21#1271—24no—
22#1270—22no—
23C23×Q8—22no—
24D4×Q8—44no—
25D4×Q8—44no—
26D4×Q8—44no—
27D4×Q8—44no—
28C2×C4⋊Q8—24no—
29C2×C4⋊Q8D412yes—
30C2×#127C2×C411yescharacteristic
31C2×C22⋊Q8—24no—
32C2×C22⋊Q8—28no—
33C2×C4×Q8—22no—
34C2×C4×D4—24no—
35C2×C4×D4—22no—
36C2×C4×D4D412yes—
37C2×C4⋊C8—22no—
38C2×C4⋊C8—22no—
39C2×C4⋊C8—21no—
40C2×C4×C8—21no—
41#396—28no—
42#395—24no—
43#393—24no—
44#391—22no—
45#390—24no—
46#390—22no—
47#390D412yes—
48#389—24no—
49#389—22no—
50#389D414yes—
51#386—24no—
52#385—24no—
53#384C2×C411yescharacteristic
54#384D412yes—
55#380—22no—
56C4×C4⋊C4—22no—
57C4×C4⋊C4C2×C414yes—
58C43C2311yescharacteristic
59#336—22no—
60#336—22no—
61#336—22no—
62#332—44no—
63D4⋊C8—44no—
64C22×Q8—24no—
65C22×Q8—22no—
66C22×Q8—48no—
67C22×Q8—48no—
68C22×Q8—82no—
69C22×Q8—82no—
70C22×D4—22no—
71C22×D4C22⋊C412yes—
72C23×C4—24no—
73C23×C4—22no—
74C22×C8—41no—
75C22×C8—42no—
76C22×C8—41no—
77C4⋊Q8—22no—
78C4⋊Q8—22no—
79C4⋊Q8—48no—
80C4⋊Q8—48no—
81C4⋊Q8D814yes—
82C4⋊Q8SD1614yes—
83#127—24no—
84C22⋊Q8—416no—
85C22⋊Q8—416no—
86C22⋊Q8—48no—
87C22⋊Q8—48no—
88C4×Q8—42no—
89C4×Q8—42no—
90C4×Q8—42no—
91C4×Q8—42no—
92C4×D4—42no—
93C4×D4—42no—
94C4×D4—48no—
95C4×D4—48no—
96C4×D4—42no—
97C4×D4—42no—
98C4×D4—42no—
99C4×D4—42no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—28no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4C22⋊C414yes—
107C2×C4⋊C4C22⋊C412yes—
108C2×C4⋊C4C22⋊C414yes—
109C2×C22⋊C4—28no—
110C2×C22⋊C4—24no—
111C2×C22⋊C4—24no—
112C2×C22⋊C4—22no—
113C2×C42—24no—
114C2×C42C22×C411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D412yes—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D412yes—
119C4⋊C8—42no—
120C4⋊C8—42no—
121C4⋊C8—42no—
122C4⋊C8—42no—
123C4⋊C8—42no—
124C4×C8—22no—
125C4×C8—41no—
126#95—24no—
127#95—24no—
128#95—24no—
129C24—22no—
130C2×Q8—24no—
131C2×Q8—24no—
132C2×Q8—416no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8—42no—
136C2×Q8—42no—
137C2×Q8—84no—
138C2×Q8—88no—
139C2×Q8—88no—
140C2×Q8—816no—
141C2×D4—28no—
142C2×D4—22no—
143C2×D4—44no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4#10114yes—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—416no—
153C22×C4—44no—
154C22×C4—44no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—84no—
158C22×C4C2×C22⋊C411yescharacteristic
159C22×C4C2×C22⋊C412yes—
160C22×C4#12011yescharacteristic
161C22×C4#12012yes—
162C22×C4#12011yescharacteristic
163C2×C8—42no—
164C2×C8—82no—
165C2×C8—82no—
166C2×C8—82no—
167C2×C8—82no—
168C2×C8—82no—
169C2×C8—81no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—44no—
174C4⋊C4—42no—
175C4⋊C4—416no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—44no—
179C4⋊C4—42no—
180C4⋊C4—44no—
181C4⋊C4—42no—
182C4⋊C4—82no—
183C4⋊C4#10218yes—
184C22⋊C4—416no—
185C22⋊C4—44no—
186C22⋊C4—44no—
187C22⋊C4—82no—
188C22⋊C4—84no—
189C42—22no—
190C42—21no—
191C42—21no—
192C42—41no—
193C42—42no—
194C42—44no—
195C42—44no—
196C42C2×D812yes—
197C42C2×SD1612yes—
198C42AGL(1,ℤ/8)12yes—
199C42AGL(1,ℤ/8)12yes—
200C23—24no—
201C23—42no—
202C23—42no—
203C23—44no—
204C23—44no—
205C23#37911yescharacteristic
206Q8—44no—
207Q8—44no—
208Q8—44no—
209Q8—44no—
210Q8—816no—
211D4—44no—
212D4—48no—
213D4—48no—
214D4—82no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—28no—
224C2×C4—44no—
225C2×C4—44no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—44no—
229C2×C4—44no—
230C2×C4—44no—
231C2×C4—816no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—84no—
235C2×C4C2×#10112yes—
236C2×C4C2×#10212yes—
237C2×C4#41812yes—
238C2×C4#44712yes—
239C2×C4#44712yes—
240C2×C4#45012yes—
241C2×C4#45012yes—
242C2×C4#45311yescharacteristic
243C2×C4#45511yescharacteristic
244C8—82no—
245C8—162no—
246C8—161no—
247C22—44no—
248C22—48no—
249C22—82no—
250C22—84no—
251C22—82no—
252C22#178411yescharacteristic
253C22#178612yes—
254C22#179211yescharacteristic
255C22#179411yescharacteristic
256C22#179612yes—
257C4—22no—
258C4—22no—
259C4—22no—
260C4—22no—
261C4—22no—
262C4—41no—
263C4—48no—
264C4—84no—
265C4—82no—
266C4#161911yescharacteristic
267C4#162911yescharacteristic
268C4#163912yes—
269C2—84no—
270C2#838812yes—
271C2#842912yes—
272C2#844311yescharacteristic
273C2#845711yescharacteristic
274C2#849211yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
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
3. #8589C2semidirect
5. #7588C2semidirect

Maximal subgroups

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

subgroupindex
#598172
#96442
#85892
#76512
#75882