Finite Group Databasebeta

← #18399

Subgroups — #18399

Order 256.

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

normal subgroups only

This group has 447 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#3507C211yesmaximal, characteristic
2#3230C211yesmaximal, characteristic, complemented
3#3227C211yesmaximal, characteristic
4#2568C211yesmaximal, characteristic, complemented
5#1615C218yesmaximal, complemented
6#1503C211yesmaximal, characteristic, complemented
7#1502C211yesmaximal, characteristic
8#1461C211yesmaximal, characteristic, complemented
9#559—21no—
10#557—21no—
11#556—21no—
12#556—22no—
13#554—21no—
14#552—22no—
15#539C2214yes—
16#536—21no—
17#533C2211yescharacteristic
18#532C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20#529—22no—
21#520—21no—
22C2×C4×Q8C2211yescharacteristic
23#500—21no—
24#498—21no—
25#487—21no—
26#486—21no—
27#484C2214yes—
28#481C2214yes—
29#477—22no—
30#476—22no—
31#431C2211yescharacteristic
32#431C2211yescharacteristic
33#400C2214yes—
34#391—21no—
35#391—22no—
36#382—21no—
37C4×C4⋊C4C2211yescharacteristic
38#328C2214yes—
39#327C2214yes—
40#324C2214yes—
41C2×#41—21no—
42C22×Q8—21no—
43C22×Q8D411yescharacteristic
44C4⋊Q8—22no—
45C4⋊Q8—21no—
46C4⋊Q8—21no—
47C4⋊Q8—21no—
48C4⋊Q8—21no—
49C4⋊Q8D411yescharacteristic
50C4⋊Q8D411yescharacteristic
51C4⋊Q8D411yescharacteristic
52#127D411yescharacteristic
53#126—24no—
54#126—24no—
55#125—22no—
56#125—24no—
57#125—21no—
58#124—22no—
59#124—22no—
60#124—21no—
61#123—24no—
62#123—24no—
63C22⋊Q8—22no—
64C22⋊Q8—22no—
65C22⋊Q8—24no—
66C22⋊Q8—21no—
67C22⋊Q8C2312yes—
68#121—21no—
69#121C2312yes—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×Q8—21no—
73C4×Q8—21no—
74C4×Q8—21no—
75C4×Q8—21no—
76C4×Q8—21no—
77C4×Q8D412yes—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—24no—
81C4×D4—21no—
82C4×D4D412yes—
83#117—22no—
84#117—24no—
85#117—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2312yes—
91C2×C4⋊C4C2312yes—
92C2×C42—21no—
93C2×C42D412yes—
94C2×C42C2311yescharacteristic
95#107—24no—
96#106—24no—
97C4⋊C8—21no—
98C4⋊C8—21no—
99#103—24no—
100#102—24no—
101C22⋊C8C2312yes—
102C22⋊C8C2312yes—
103C4×C8—21no—
104C4×C8—21no—
105#95—24no—
106#95—24no—
107#95C2312yes—
108#41—42no—
109#41—42no—
110C2×Q8—21no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—44no—
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8C2×D412yes—
120C2×D4—22no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4C2×D412yes—
124C22×C4—24no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4C2×D412yes—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D412yes—
131C22×C4C2×D411yescharacteristic
132C22×C4C2411yescharacteristic
133C2×C8—22no—
134C2×C8—22no—
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4—24no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—42no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4C2×D412yes—
154C4⋊C4C2×D412yes—
155C22⋊C4—24no—
156C22⋊C4—24no—
157C22⋊C4—24no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C42—24no—
162C42—22no—
163C42—22no—
164C42—22no—
165C42—42no—
166C42C2×D411yescharacteristic
167C42C2×D411yescharacteristic
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C23—22no—
171C23C22×D411yescharacteristic
172Q8—42no—
173Q8—42no—
174Q8—42no—
175Q8—42no—
176Q8—42no—
177Q8—82no—
178Q8—82no—
179D4—42no—
180D4—42no—
181D4—42no—
182D4—42no—
183D4—42no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—44no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—84no—
201C2×C4#12011yescharacteristic
202C2×C4#12011yescharacteristic
203C2×C4#12011yescharacteristic
204C2×C4#12011yescharacteristic
205C2×C4#13512yes—
206C2×C4#13712yes—
207C2×C4C22×D411yescharacteristic
208C2×C4C22×D411yescharacteristic
209C8—42no—
210C8—42no—
211C22—21no—
212C22—21no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22C2×#12011yescharacteristic
217C22C2×#13511yescharacteristic
218C22C2×#13711yescharacteristic
219C4—21no—
220C4—21no—
221C4—21no—
222C4—42no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—42no—
227C4—82no—
228C4—82no—
229C4#45212yes—
230C4#45212yes—
231C4#45412yes—
232C2—21no—
233C2—21no—
234C2—82no—
235C2C2×#45211yescharacteristic
236C2C2×#45411yescharacteristic
237C2#302811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#532C22
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
4. #2568C2semidirect
5. #1615C2semidirect
6. #1503C2semidirect
8. #1461C2semidirect

Maximal subgroups

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

subgroupindex
#35072
#32302
#32272
#25682
#16152
#15032
#15022
#14612