Finite Group Databasebeta

← #98256

Subgroups — #98256

Order 512.

709 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 709 conjugacy classes of subgroups — too many to draw. Only its 152 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#36593C211yesmaximal, characteristic
2#9896C212yesmaximal
3#8587C211yesmaximal, characteristic
4#8564C212yesmaximal
5#7571C211yesmaximal, characteristic
6C2×#557C412yes—
7#2868—21no—
8#2868C2211yescharacteristic
9#2552—24no—
10#2552—22no—
11#2546—22no—
12#2546C412yes—
13Q8×C4⋊C4—22no—
14C4×C4⋊Q8—21no—
15C4×C4⋊Q8C2211yescharacteristic
16C42×Q8C2211yescharacteristic
17#1842C2212yes—
18C4×C4⋊C8C2212yes—
19C2×#333—21no—
20C2×Q8⋊C8—22no—
21#1296—21no—
22#1296—24no—
23#1279—22no—
24#1272—22no—
25#557—44no—
26#557—44no—
27C2×C4⋊Q8—22no—
28C2×C4⋊Q8—22no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8C2×C412yes—
32C2×C4⋊Q8D411yescharacteristic
33C2×#125—24no—
34C2×#125—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8C2×C411yescharacteristic
38C2×C4×Q8C2×C412yes—
39C2×C4×Q8D412yes—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4⋊C8—22no—
44C2×C4×C8—22no—
45#398—28no—
46#398—24no—
47#391—22no—
48#391D412yes—
49#389—24no—
50#389—28no—
51#389—24no—
52#389—22no—
53#384—24no—
54#384—24no—
55#384—22no—
56#384D411yescharacteristic
57#384D414yes—
58C4×C4⋊C4—24no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—21no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4C2×C411yescharacteristic
63C4×C4⋊C4D412yes—
64C43—21no—
65C43C2311yescharacteristic
66#336—22no—
67#333—42no—
68Q8⋊C8—44no—
69C22×Q8—22no—
70C22×Q8C22⋊C411yescharacteristic
71C22×Q8C22⋊C412yes—
72C22×C8—42no—
73C22×C8—42no—
74C4⋊Q8—22no—
75C4⋊Q8—21no—
76C4⋊Q8—21no—
77C4⋊Q8—48no—
78C4⋊Q8—42no—
79C4⋊Q8—42no—
80C4⋊Q8—42no—
81C4⋊Q8—41no—
82C4⋊Q8—41no—
83C4⋊Q8—82no—
84C4⋊Q8—84no—
85#125—416no—
86#125—44no—
87#125—44no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—48no—
93C4×Q8—42no—
94C4×Q8—42no—
95C4×Q8—44no—
96C4×Q8—44no—
97C4×Q8SD1614yes—
98C4×Q8Q1614yes—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4C22⋊C411yescharacteristic
112C2×C4⋊C4C22⋊C412yes—
113C2×C4⋊C4C22⋊C412yes—
114C2×C4⋊C4C22⋊C414yes—
115C2×C42—21no—
116C2×C42—22no—
117C2×C42—24no—
118C2×C42—22no—
119C2×C42—21no—
120C2×C42—22no—
121C2×C42C22×C411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D412yes—
125C2×C42C2×D411yescharacteristic
126C4⋊C8—24no—
127C4⋊C8—42no—
128C4⋊C8—44no—
129C4⋊C8—42no—
130C4⋊C8—42no—
131C4⋊C8—42no—
132C4⋊C8—42no—
133C4×C8—42no—
134C4×C8—42no—
135#95—28no—
136#95—21no—
137#95—22no—
138#95—24no—
139C2×Q8—21no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×Q8—44no—
144C2×Q8—44no—
145C2×Q8—42no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8#10012yes—
150C2×Q8#10314yes—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4C2×C22⋊C411yescharacteristic
158C22×C4C2×C22⋊C411yescharacteristic
159C22×C4C2×C22⋊C411yescharacteristic
160C22×C4#12012yes—
161C22×C4#12012yes—
162C2×C8—44no—
163C2×C8—82no—
164C2×C8—82no—
165C2×C8—82no—
166C2×C8—82no—
167C2×C8—82no—
168C4⋊C4—24no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—44no—
176C4⋊C4—48no—
177C4⋊C4—48no—
178C4⋊C4—48no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—44no—
183C4⋊C4—42no—
184C4⋊C4—44no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—84no—
188C4⋊C4—88no—
189C4⋊C4#10012yes—
190C4⋊C4#10314yes—
191C42—24no—
192C42—21no—
193C42—21no—
194C42—24no—
195C42—24no—
196C42—22no—
197C42—22no—
198C42—41no—
199C42—48no—
200C42—42no—
201C42—42no—
202C42—42no—
203C42—42no—
204C42—42no—
205C42C2×SD1612yes—
206C42C2×Q1612yes—
207C42#13512yes—
208C42AGL(1,ℤ/8)11yescharacteristic
209C42#13711yescharacteristic
210C42#13712yes—
211C42#13712yes—
212C23#37911yescharacteristic
213Q8—24no—
214Q8—22no—
215Q8—82no—
216Q8—82no—
217Q8—84no—
218Q8—84no—
219Q8#33314yescomplemented
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—42no—
240C2×C4—44no—
241C2×C4C2×#10011yescharacteristic
242C2×C4C2×#10011yescharacteristic
243C2×C4C2×#10312yes—
244C2×C4#41611yescharacteristic
245C2×C4#41711yescharacteristic
246C2×C4#41912yes—
247C2×C4#44812yes—
248C2×C4#44812yes—
249C2×C4#44912yes—
250C2×C4#45112yes—
251C2×C4#45112yes—
252C2×C4#45212yes—
253C2×C4#45412yes—
254C2×C4#45612yes—
255C8—84no—
256C8—162no—
257C22#178311yescharacteristic
258C22#178712yes—
259C22#178811yescharacteristic
260C22#179312yes—
261C22#179511yescharacteristic
262C4—24no—
263C4—21no—
264C4—21no—
265C4—22no—
266C4—22no—
267C4—41no—
268C4—42no—
269C4—44no—
270C4—82no—
271C4—82no—
272C4—84no—
273C4C2×#33311yescharacteristic
274C4#154812yes—
275C4#155811yescharacteristic
276C4#162612yes—
277C4#163612yes—
278C4#164812yes—
279C4#165312yes—
280C2#834211yescharacteristic
281C2#838511yescharacteristic
282C2#842412yes—
283C2#844812yes—
284C2#847111yescharacteristic
subgroupG/H
center, socle, prime radicalC23#379
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
219. Q8#333semidirect

Maximal subgroups

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

subgroupindex
#365932
#98962
#85872
#85642
#75712