Finite Group Databasebeta

← #18413

Subgroups — #18413

Order 256.

515 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 121 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#3506C211yesmaximal, characteristic
2#3230C211yesmaximal, characteristic, complemented
3#3227C211yesmaximal, characteristic, complemented
4#2542C211yesmaximal, characteristic, complemented
5#1618C218yesmaximal, complemented
6#1503C211yesmaximal, characteristic, complemented
7#1502C211yesmaximal, characteristic, complemented
8#1460C211yesmaximal, characteristic, complemented
9#559—22no—
10#556—21no—
11#555—21no—
12#553—22no—
13#550—21no—
14#547—21no—
15#538C2214yes—
16#535—21no—
17#533C2211yescharacteristic
18#532C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20#525—22no—
21C2×C22⋊Q8—22no—
22#520—21no—
23C2×C4×D4C2211yescharacteristic
24#500—21no—
25#498—21no—
26#487—21no—
27#486—21no—
28#484C2214yes—
29#481C2214yes—
30#477—22no—
31#476—22no—
32#431C2211yescharacteristic
33#431C2211yescharacteristic
34#399C2214yes—
35#390—21no—
36#384—21no—
37C4×C4⋊C4C2211yescharacteristic
38#328C2214yes—
39#327C2214yes—
40#323C2214yes—
41C2×#41—22no—
42C2×#41—21no—
43C22×Q8—21no—
44C22×D4D411yescharacteristic
45C23×C4D412yes—
46C4⋊Q8—21no—
47C4⋊Q8—21no—
48C4⋊Q8—21no—
49C4⋊Q8—21no—
50C4⋊Q8D411yescharacteristic
51C4⋊Q8D411yescharacteristic
52C4⋊Q8D411yescharacteristic
53#127—22no—
54#127D411yescharacteristic
55#126—24no—
56#125—21no—
57#124—22no—
58#124—24no—
59#124—21no—
60#123—24no—
61#123—24no—
62C22⋊Q8—22no—
63C22⋊Q8—21no—
64C22⋊Q8—48no—
65C22⋊Q8C2312yes—
66#121—22no—
67#121—22no—
68#121—22no—
69#121—24no—
70#121—21no—
71#121C2312yes—
72#120—24no—
73C4×Q8—22no—
74C4×Q8—21no—
75C4×Q8D412yes—
76C4×D4—22no—
77C4×D4—24no—
78C4×D4—24no—
79C4×D4—21no—
80C4×D4—21no—
81C4×D4—21no—
82C4×D4—21no—
83C4×D4—21no—
84C4×D4D412yes—
85#117—22no—
86#117—22no—
87#117—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4C2312yes—
94C2×C22⋊C4—24no—
95C2×C22⋊C4C2312yes—
96C2×C42—21no—
97C2×C42C2311yescharacteristic
98#107—24no—
99#106—24no—
100C4⋊C8—21no—
101C4⋊C8—21no—
102#103—24no—
103#102—24no—
104C22⋊C8C2312yes—
105C22⋊C8C2312yes—
106C4×C8—21no—
107C4×C8—21no—
108#95C2312yes—
109C24C2×D412yes—
110#41—42no—
111#41—42no—
112#41—84no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—44no—
116C2×Q8—41no—
117C2×Q8—41no—
118C2×Q8C2×D412yes—
119C2×D4—22no—
120C2×D4—24no—
121C2×D4—24no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4C2×D412yes—
129C22×C4—22no—
130C22×C4—24no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—44no—
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D412yes—
138C22×C4C2×D411yescharacteristic
139C22×C4C2411yescharacteristic
140C2×C8—22no—
141C2×C8—22no—
142C4⋊C4—24no—
143C4⋊C4—24no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—48no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4C2×D412yes—
158C4⋊C4C2×D412yes—
159C22⋊C4—24no—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—48no—
168C42—24no—
169C42—22no—
170C42—42no—
171C42C2×D411yescharacteristic
172C42C2×D411yescharacteristic
173C42C2×D411yescharacteristic
174C42C2×D411yescharacteristic
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—42no—
180C23—42no—
181C23C22×D411yescharacteristic
182Q8—42no—
183Q8—42no—
184Q8—42no—
185Q8—82no—
186Q8—82no—
187D4—42no—
188D4—42no—
189D4—42no—
190D4—42no—
191D4—42no—
192D4—42no—
193D4—42no—
194D4—84no—
195D4—84no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—42no—
210C2×C4—82no—
211C2×C4—84no—
212C2×C4#12011yescharacteristic
213C2×C4#12011yescharacteristic
214C2×C4#12011yescharacteristic
215C2×C4#12011yescharacteristic
216C2×C4#13512yes—
217C2×C4#13712yes—
218C2×C4C22×D411yescharacteristic
219C2×C4C22×D411yescharacteristic
220C8—42no—
221C8—42no—
222C22—21no—
223C22—21no—
224C22—44no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—82no—
232C22C2×#12011yescharacteristic
233C22C2×#13511yescharacteristic
234C22C2×#13711yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—42no—
242C4—42no—
243C4—82no—
244C4#45212yes—
245C4#45212yes—
246C4#45312yes—
247C2—21no—
248C2—21no—
249C2—82no—
250C2—82no—
251C2C2×#45211yescharacteristic
252C2C2×#45311yescharacteristic
253C2#302811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3506C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3230C2semidirect
3. #3227C2semidirect
4. #2542C2semidirect
5. #1618C2semidirect
6. #1503C2semidirect
7. #1502C2semidirect
8. #1460C2semidirect

Maximal subgroups

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

subgroupindex
#35062
#32302
#32272
#25422
#16182
#15032
#15022
#14602