Finite Group Databasebeta

← #9344

Subgroups — #9344

Order 256.

260 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 59 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#3193C211yesmaximal, characteristic
2#2378C211yesmaximal, characteristic, complemented
3#1964C211yesmaximal, characteristic
4#1510C211yesmaximal, characteristic
5#1508C211yesmaximal, characteristic, complemented
6#1305C211yesmaximal, characteristic, complemented
7#1284C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8C2211yescharacteristic
9C2×#121C2211yescharacteristic
10#484C412yescomplemented
11#480—21no—
12#468—22no—
13#450C412yescomplemented
14#448—21no—
15C2×#99—21no—
16C2×#99—21no—
17#407C2211yescharacteristic
18#407C2211yescharacteristic
19#391—21no—
20#389—21no—
21#388—21no—
22#388—21no—
23#387—21no—
24#387—21no—
25#381C2211yescharacteristic
26#380C2211yescharacteristic
27C4×C22⋊C4C2211yescharacteristic
28#328—22no—
29#327—21no—
30#327—21no—
31C22×Q8D411yescharacteristic
32C22×D4—21no—
33C22×D4C2×C411yescharacteristic
34C23×C4C2311yescharacteristic
35C2×SD16—22no—
36C2×M16—21no—
37C2×M16—21no—
38C22⋊Q8—21no—
39C22⋊Q8—21no—
40C22⋊Q8C2×C412yes—
41C22⋊Q8D412yes—
42#121—21no—
43#121—21no—
44#121—21no—
45#121—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4C2×C411yescharacteristic
51C2×C4⋊C4D411yescharacteristic
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4D411yescharacteristic
57C2×C22⋊C4D411yescharacteristic
58C2×C42—21no—
59C2×C42—21no—
60C2×C42—21no—
61#107—22no—
62#103—22no—
63#102—22no—
64#99—42no—
65#99—42no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8C2×C412yes—
70#95—21no—
71#95—21no—
72#95—21no—
73#95—21no—
74#95—21no—
75#95—21no—
76#95—21no—
77#95D411yescharacteristic
78#95D411yescharacteristic
79C24—21no—
80C24C2×D411yescharacteristic
81C2×Q8—21no—
82C2×Q8—81no—
83C2×Q8C22⋊C412yes—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—81no—
93C22×C4—21no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—41no—
101C22×C4—81no—
102C22×C4C22×C411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4#4111yescharacteristic
107C22×C4#4111yescharacteristic
108SD16—82no—
109M16—81no—
110M16—81no—
111C2×C8—22no—
112C2×C8—41no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—81no—
119C4⋊C4—81no—
120C4⋊C4—81no—
121C4⋊C4C22⋊C412yes—
122C22⋊C4—22no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—81no—
131C42—81no—
132C42—81no—
133C42—81no—
134C23—21no—
135C23—21no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—81no—
142C23—81no—
143C23C4×D411yescharacteristic
144C23#12011yescharacteristic
145C23#12111yescharacteristic
146Q8—42no—
147Q8—81no—
148D4—42no—
149D4—81no—
150D4—81no—
151D4—81no—
152D4—81no—
153D4—81no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—81no—
183C2×C4—81no—
184C2×C4C2×C22⋊C411yescharacteristic
185C2×C4C4×D411yescharacteristic
186C2×C4#12111yescharacteristic
187C2×C4#12311yescharacteristic
188C8—42no—
189C8—81no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22—81no—
202C22—81no—
203C22—161no—
204C22—161no—
205C22#38611yescharacteristic
206C22#45311yescharacteristic
207C22#45411yescharacteristic
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—42no—
213C4—42no—
214C4—81no—
215C4—81no—
216C4—81no—
217C4—81no—
218C4—82no—
219C4—81no—
220C2—21no—
221C2—21no—
222C2—41no—
223C2—41no—
224C2—161no—
225C2#188311yescharacteristic
226C2#190511yescharacteristic
227C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2378C2semidirect
5. #1508C2semidirect
6. #1305C2semidirect
7. #1284C2semidirect
10. #484C4semidirect
13. #450C4semidirect

Maximal subgroups

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

subgroupindex
#31932
#23782
#19642
#15102
#15082
#13052
#12842