Finite Group Databasebeta

← #98563

Subgroups — #98563

Order 512.

846 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×#2567C211yesmaximal, characteristic
2#8725C212yesmaximal
3#8560C212yesmaximal
4#7643C212yesmaximal
5C22×C4⋊Q8C411yescharacteristic
6C22×C4×Q8C411yescharacteristic
7#2567—24no—
8#2567—24no—
9C2×#395—22no—
10C2×#391C411yescharacteristic
11C2×#391C2211yescharacteristic
12C2×#389—22no—
13C2×#384C411yescharacteristic
14C2×#384C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1797C2214yes—
17C2×#324—22no—
18#1296—24no—
19#1280—22no—
20#1272—24no—
21C23×Q8—21no—
22C23×Q8C2×C411yescharacteristic
23C2×C4⋊Q8—22no—
24C2×C4⋊Q8—22no—
25C2×C4⋊Q8—42no—
26C2×C4⋊Q8—42no—
27C2×C4×Q8—22no—
28C2×C4×Q8—22no—
29C2×C4×Q8—42no—
30C2×C4×Q8—42no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4—21no—
33C22×C4⋊C4—22no—
34C22×C4⋊C4—22no—
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4D412yes—
39C22×C42—21no—
40C22×C42C2×C411yescharacteristic
41C22×C42C2×C411yescharacteristic
42C22×C42C2311yescharacteristic
43C2×C4⋊C8—24no—
44C2×C22⋊C8—24no—
45#395—216no—
46#391—22no—
47#391—22no—
48#391—22no—
49#391D414yes—
50#389—216no—
51#384—22no—
52#384—22no—
53#384—22no—
54#384D414yes—
55C4×C4⋊C4—22no—
56C4×C4⋊C4—22no—
57C2×#95—22no—
58C2×#95—22no—
59C2×#95D412yes—
60#336—24no—
61#324—44no—
62C22×Q8—24no—
63C22×Q8—24no—
64C22×Q8—21no—
65C22×Q8—42no—
66C22×Q8—42no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8—42no—
70C22×Q8—84no—
71C22×Q8C22⋊C412yes—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—21no—
76C23×C4C22⋊C411yescharacteristic
77C23×C4C22⋊C411yescharacteristic
78C23×C4C22⋊C412yes—
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D412yes—
81C22×C8—44no—
82C4⋊Q8—48no—
83C4⋊Q8—48no—
84C4×Q8—48no—
85C4×Q8—48no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—28no—
94C2×C4⋊C4—28no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4—42no—
105C2×C4⋊C4—84no—
106C2×C4⋊C4—84no—
107C2×C4⋊C4C22⋊C412yes—
108C2×C4⋊C4C22⋊C412yes—
109C2×C4⋊C4C22⋊C412yes—
110C2×C4⋊C4SD1614yes—
111C2×C4⋊C4Q1614yes—
112C2×C42—22no—
113C2×C42—22no—
114C2×C42—21no—
115C2×C42—21no—
116C2×C42—24no—
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—42no—
120C2×C42—42no—
121C2×C42—42no—
122C2×C42—42no—
123C2×C42C2×D412yes—
124C2×C42C2×D412yes—
125C4⋊C8—48no—
126C22⋊C8—44no—
127C22⋊C8—44no—
128#95—28no—
129#95—28no—
130#95—24no—
131C24C2×C22⋊C411yescharacteristic
132C2×Q8—24no—
133C2×Q8—22no—
134C2×Q8—42no—
135C2×Q8—48no—
136C2×Q8—48no—
137C2×Q8—48no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—88no—
142C2×Q8—88no—
143C2×Q8—88no—
144C2×Q8#10114yes—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—44no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4#10012yes—
179C22×C4#10012yes—
180C22×C4#10314yes—
181C22×C4#10314yes—
182C22×C4C2×C22⋊C411yescharacteristic
183C22×C4C2×C22⋊C411yescharacteristic
184C22×C4#12014yes—
185C22×C4C2×SD1612yes—
186C22×C4C2×Q1612yes—
187C22×C4#13714yes—
188C2×C8—84no—
189C2×C8—84no—
190C2×C8—84no—
191C4⋊C4—28no—
192C4⋊C4—28no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—48no—
198C4⋊C4—48no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—88no—
202C4⋊C4—88no—
203C42—24no—
204C42—24no—
205C42—24no—
206C42—42no—
207C42—42no—
208C42—48no—
209C42—42no—
210C23—22no—
211C23—21no—
212C23—21no—
213C23#37911yescharacteristic
214C23C2×#10011yescharacteristic
215C23C2×#10011yescharacteristic
216C23C2×#10312yes—
217C23#41912yes—
218Q8—44no—
219Q8—48no—
220Q8—44no—
221Q8—816no—
222C2×C4—24no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—24no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—44no—
253C2×C4—44no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4#33314yes—
264C2×C4#41012yes—
265C2×C4C2×#10112yes—
266C2×C4#44814yes—
267C2×C4#45114yes—
268C8—164no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22C2×#33311yescharacteristic
278C22#154812yes—
279C22#155811yescharacteristic
280C22#178211yescharacteristic
281C22#178311yescharacteristic
282C22#178411yescharacteristic
283C22#178712yes—
284C22#179512yes—
285C4—22no—
286C4—22no—
287C4—24no—
288C4—24no—
289C4—24no—
290C4—42no—
291C4—44no—
292C4—44no—
293C4—82no—
294C4—82no—
295C4—82no—
296C2—21no—
297C2—22no—
298C2—21no—
299C2#834211yescharacteristic
300C2#838012yes—
301C2#839612yes—
302C2#852211yescharacteristic
303C2#855511yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C23×C4 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25672
#87252
#85602
#76432