Finite Group Databasebeta

← #10331

Subgroups — #10331

Order 256.

336 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 53 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#2502C211yesmaximal, characteristic, complemented
3#2501C211yesmaximal, characteristic
4#1821C211yesmaximal, characteristic, complemented
5#1508C211yesmaximal, characteristic, complemented
6#1507C211yesmaximal, characteristic
7#1297C211yesmaximal, characteristic, complemented
8C2×#123—22no—
9C2×C22⋊Q8C2211yescharacteristic
10C2×#121C2211yescharacteristic
11C22×C4⋊C4C2211yescharacteristic
12#484—21no—
13#480—21no—
14#468—22no—
15#450—21no—
16#448—21no—
17#407C2211yescharacteristic
18#407C2211yescharacteristic
19#399—22no—
20#398—22no—
21#391—21no—
22#390—21no—
23#385—22no—
24#384—21no—
25#384—21no—
26C4×C22⋊C4C2211yescharacteristic
27C4×C22⋊C4C2211yescharacteristic
28#343—22no—
29#328—22no—
30#327—22no—
31C22×Q8D411yescharacteristic
32C22×D4—21no—
33C22×D4D411yescharacteristic
34C23×C4D412yes—
35C23×C4C2311yescharacteristic
36C2×SD16—22no—
37C2×M16—21no—
38C2×M16D412yes—
39#123—48no—
40C22⋊Q8—21no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43C22⋊Q8—21no—
44#121—21no—
45#121—21no—
46#121—21no—
47#121—21no—
48C2×C4⋊C4—24no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C42—21no—
65C2×C42—21no—
66#107—22no—
67#103—22no—
68#102—22no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73#95—24no—
74#95D412yes—
75#95D412yes—
76C24—21no—
77C24C2×D411yescharacteristic
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—42no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—42no—
84C2×D4—44no—
85C2×D4—41no—
86C2×D4—41no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—42no—
96C22×C4—44no—
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4#4111yescharacteristic
103SD16—82no—
104M16—42no—
105M16—42no—
106C2×C8—22no—
107C2×C8—41no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—48no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—84no—
120C22⋊C4—24no—
121C22⋊C4—24no—
122C22⋊C4—48no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C42—42no—
128C42—42no—
129C23—22no—
130C23—22no—
131C23—41no—
132C23—42no—
133C23—42no—
134C23—41no—
135C23—42no—
136C23#12011yescharacteristic
137C23#12011yescharacteristic
138C23#12111yescharacteristic
139Q8—42no—
140Q8—81no—
141D4—42no—
142D4—81no—
143D4—82no—
144D4—82no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—24no—
149C2×C4—24no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—44no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—44no—
166C2×C4—82no—
167C2×C4#12011yescharacteristic
168C2×C4#12111yescharacteristic
169C2×C4#12111yescharacteristic
170C2×C4#12711yescharacteristic
171C8—42no—
172C8—81no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—84no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22#39211yescharacteristic
186C22#45312yes—
187C22#45612yes—
188C4—21no—
189C4—21no—
190C4—21no—
191C4—44no—
192C4—44no—
193C4—81no—
194C4—82no—
195C4—81no—
196C4—81no—
197C4#45812yes—
198C2—22no—
199C2—22no—
200C2—41no—
201C2—161no—
202C2#200211yescharacteristic
203C2#200511yescharacteristic
204C2#202211yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C23×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. #2502C2semidirect
4. #1821C2semidirect
5. #1508C2semidirect
7. #1297C2semidirect

Maximal subgroups

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

subgroupindex
#31932
#25022
#25012
#18212
#15082
#15072
#12972