Finite Group Databasebeta

← #9995

Subgroups — #9995

Order 256.

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

normal subgroups only

This group has 255 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#3195C211yesmaximal, characteristic
2C2×#484C211yesmaximal, characteristic
3C2×#483C211yesmaximal, characteristic
4#1821C211yesmaximal, characteristic
5#1472C211yesmaximal, characteristic
6#1472C211yesmaximal, characteristic
7#1298C211yesmaximal, characteristic
8C2×C22⋊Q8C2211yescharacteristic
9C2×C22⋊Q8C2211yescharacteristic
10C22×C4⋊C4C2211yescharacteristic
11#484—22no—
12#484—22no—
13#483—22no—
14#483—21no—
15#483—21no—
16#483—22no—
17#470—22no—
18#451—21no—
19#451—21no—
20C2×#107—21no—
21C2×#106—21no—
22C2×#103—21no—
23C2×#103—21no—
24#407C2211yescharacteristic
25#407C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28#343—22no—
29#324—22no—
30#324—22no—
31C22×Q8D411yescharacteristic
32C22×Q8D411yescharacteristic
33C23×C4—21no—
34C23×C4C2311yescharacteristic
35C2×Q16—22no—
36C2×M16—21no—
37C2×M16D412yes—
38C22×C8—21no—
39C22×C8—21no—
40C22⋊Q8—21no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43C22⋊Q8—21no—
44C22⋊Q8—21no—
45C22⋊Q8—21no—
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48C2×C4⋊C4—21no—
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⋊C4D412yes—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60#107—24no—
61#106—24no—
62#106—22no—
63#103—24no—
64#103—24no—
65#103—22no—
66#103—22no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8D412yes—
74C22⋊C8D412yes—
75C24#4111yescharacteristic
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—42no—
81C2×Q8—42no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—22no—
87C22×C4—42no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4—82no—
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97Q16—82no—
98M16—42no—
99M16—42no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—22no—
103C2×C8—42no—
104C2×C8—42no—
105C2×C8—41no—
106C4⋊C4—22no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—22no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C23—22no—
123C23—22no—
124C23—41no—
125C23#12111yescharacteristic
126C23#12111yescharacteristic
127C23#12111yescharacteristic
128Q8—42no—
129Q8—42no—
130Q8—42no—
131Q8—42no—
132C2×C4—22no—
133C2×C4—22no—
134C2×C4—22no—
135C2×C4—41no—
136C2×C4—41no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—42no—
143C2×C4—42no—
144C2×C4—42no—
145C2×C4—82no—
146C2×C4—82no—
147C2×C4—82no—
148C2×C4—81no—
149C2×C4—81no—
150C2×C4#12011yescharacteristic
151C2×C4#12011yescharacteristic
152C2×C4#12011yescharacteristic
153C2×C4#12711yescharacteristic
154C8—42no—
155C8—42no—
156C8—42no—
157C8—81no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—21no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—44no—
166C22#39211yescharacteristic
167C22#47112yes—
168C22#47312yes—
169C4—21no—
170C4—21no—
171C4—21no—
172C4—81no—
173C4—82no—
174C4—81no—
175C4—81no—
176C4—81no—
177C4#45812yes—
178C2—22no—
179C2—22no—
180C2—41no—
181C2#201911yescharacteristic
182C2#202011yescharacteristic
183C2#202211yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radicalC24#41
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#31952
C2×#4842
C2×#4832
#18212
#14722
#12982