Finite Group Databasebeta

← #101274

Subgroups — #101274

Order 512.

482 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 90 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#23734C211yesmaximal, characteristic
2#21909C211yesmaximal, characteristic
3#8725C212yesmaximal, complemented
4#7597C211yesmaximal, characteristic
5#7045C212yesmaximal, complemented
6C2×#428—21no—
7C2×#417—21no—
8#2297C2211yescharacteristic
9C2×#384C2211yescharacteristic
10#2096C412yes—
11#2095—21no—
12#2094—21no—
13#2093C412yes—
14#1924—22no—
15#1923—22no—
16#1797C2212yes—
17#1797C2212yes—
18#1780—21no—
19#1750—21no—
20#1728C2211yescharacteristic
21#1296—22no—
22#1296—22no—
23#1272—21no—
24#1270—21no—
25C22×#41D411yescharacteristic
26C22×M16—21no—
27C2×#117D411yescharacteristic
28C22×C4⋊C4—21no—
29C22×C4⋊C4—22no—
30C22×C42C2311yescharacteristic
31#428—42no—
32#428—42no—
33C2×#107—22no—
34C2×#106—22no—
35C2×C4⋊C8—22no—
36C2×C4⋊C8D412yes—
37C2×C4⋊C8D412yes—
38#417—42no—
39#417—42no—
40C2×#103—22no—
41C2×#102—22no—
42C2×C22⋊C8—22no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—22no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—22no—
47C2×#97—21no—
48C2×C4×C8—21no—
49#391C2×C411yescharacteristic
50#391D411yescharacteristic
51#390C2×C411yescharacteristic
52#390D411yescharacteristic
53#386—22no—
54#384—22no—
55#384—21no—
56#384C2×C412yes—
57C4×C22⋊C4—22no—
58#336—22no—
59#336—22no—
60#336—21no—
61#336—22no—
62#336C2×C412yes—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—42no—
66C2×#41—41no—
67C2×#41—41no—
68C22×Q8C22⋊C411yescharacteristic
69C22×D4—21no—
70C22×D4C22⋊C411yescharacteristic
71C23×C4—21no—
72C23×C4—41no—
73C23×C4C2×D411yescharacteristic
74C23×C4#4112yes—
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—42no—
78C2×M16—42no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—42no—
83#117—41no—
84#117—41no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—24no—
87C2×C4⋊C4—41no—
88C2×C4⋊C4—41no—
89C2×C4⋊C4—41no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—84no—
94C2×C4⋊C4C22⋊C411yescharacteristic
95C2×C4⋊C4C22⋊C411yescharacteristic
96C2×C22⋊C4—22no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—22no—
99C2×C42—21no—
100C2×C42—42no—
101C2×C42C22×C411yescharacteristic
102C2×C42C2×D412yes—
103C2×C42C2×D411yescharacteristic
104#107—44no—
105#106—44no—
106C4⋊C8—42no—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8—42no—
110#103—44no—
111#102—44no—
112C22⋊C8—42no—
113C22⋊C8—42no—
114C22⋊C8—81no—
115C22⋊C8—81no—
116C22⋊C8—82no—
117C22⋊C8—82no—
118#97—81no—
119C4×C8—81no—
120#95—22no—
121#95—22no—
122C24—21no—
123C24#12311yescharacteristic
124#41—44no—
125#41—44no—
126#41—82no—
127#41—82no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—42no—
137C2×D4—81no—
138C2×D4—81no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4—81no—
155C22×C4—81no—
156C22×C4—84no—
157C22×C4C2×C22⋊C411yescharacteristic
158C22×C4C4×D412yes—
159C22×C4#12011yescharacteristic
160C22×C4#12112yes—
161C22×C4#13512yes—
162C22×C4#13712yes—
163M16—44no—
164M16—44no—
165C2×C8—24no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—82no—
175C2×C8—82no—
176C2×C8—82no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—84no—
182C4⋊C4—84no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—82no—
188C22⋊C4—81no—
189C42—22no—
190C42—22no—
191C42—41no—
192C42—41no—
193C42—41no—
194C42—41no—
195C23—21no—
196C23—21no—
197C23—22no—
198C23—22no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—42no—
203C23—42no—
204C23#38611yescharacteristic
205C23#48512yes—
206Q8—42no—
207Q8—81no—
208D4—42no—
209D4—81no—
210D4—84no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—44no—
223C2×C4—44no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—84no—
235C2×C4—162no—
236C2×C4—162no—
237C2×C4#41611yescharacteristic
238C2×C4#41911yescharacteristic
239C2×C4#45212yes—
240C2×C4#45311yescharacteristic
241C2×C4#45311yescharacteristic
242C2×C4#45812yes—
243C2×C4#47112yes—
244C2×C4#47312yes—
245C8—44no—
246C8—82no—
247C8—82no—
248C8—162no—
249C22—22no—
250C22—24no—
251C22—42no—
252C22—41no—
253C22—41no—
254C22—82no—
255C22—81no—
256C22—81no—
257C22—82no—
258C22—81no—
259C22#187311yescharacteristic
260C22#188711yescharacteristic
261C22#189011yescharacteristic
262C22#190312yes—
263C22#190812yes—
264C4—22no—
265C4—22no—
266C4—21no—
267C4—21no—
268C4—42no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4—162no—
273C4#161812yes—
274C4#161812yes—
275C2—42no—
276C2—161no—
277C2#902411yescharacteristic
278C2#917012yes—
279C2#942311yescharacteristic
280C2#943511yescharacteristic
281C2#952112yes—
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8725C2semidirect
5. #7045C2semidirect

Maximal subgroups

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

subgroupindex
#237342
#219092
#87252
#75972
#70452