Finite Group Databasebeta

← #9993

Subgroups — #9993

Order 256.

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

normal subgroups only

This group has 268 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
2C2×#481C211yesmaximal, characteristic
3C2×#483C211yesmaximal, characteristic, complemented
4#1821C211yesmaximal, characteristic, complemented
5#1472C211yesmaximal, characteristic, complemented
6#1471C211yesmaximal, characteristic
7#1298C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8C2211yescharacteristic
9C2×#121C2211yescharacteristic
10C22×C4⋊C4C2211yescharacteristic
11#484—21no—
12#483—22no—
13#483—22no—
14#481—22no—
15#481—22no—
16#480—21no—
17#468—22no—
18#450—21no—
19#448—21no—
20C2×#106—21no—
21C2×#106—21no—
22C2×#103—21no—
23C2×#102—21no—
24#407C2211yescharacteristic
25#407C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28#343—22no—
29#324—22no—
30#323—22no—
31C22×Q8D411yescharacteristic
32C22×D4—21no—
33C22×D4D411yescharacteristic
34C23×C4—21no—
35C23×C4C2311yescharacteristic
36C2×SD16—22no—
37C2×M16—21no—
38C2×M16D412yes—
39C22×C8—21no—
40C22×C8—21no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43C22⋊Q8—21no—
44C22⋊Q8—21no—
45#121—21no—
46#121—21no—
47#121—21no—
48#121—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—22no—
61#106—24no—
62#106—24no—
63#103—24no—
64#103—22no—
65#102—24no—
66#102—22no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8D412yes—
74C22⋊C8D412yes—
75C24—21no—
76C24#4111yescharacteristic
77C2×Q8—21no—
78C2×Q8—21no—
79C2×Q8—42no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—42no—
83C2×D4—41no—
84C2×D4—41no—
85C2×D4—82no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—22no—
90C22×C4—42no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—82no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100SD16—82no—
101M16—42no—
102M16—42no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—42no—
107C2×C8—42no—
108C2×C8—41no—
109C4⋊C4—22no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C23—22no—
124C23—22no—
125C23—41no—
126C23—41no—
127C23—42no—
128C23—81no—
129C23—81no—
130C23#12111yescharacteristic
131C23#12111yescharacteristic
132C23#12111yescharacteristic
133Q8—42no—
134Q8—42no—
135D4—42no—
136D4—42no—
137D4—82no—
138D4—82no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—22no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—42no—
150C2×C4—82no—
151C2×C4—82no—
152C2×C4—82no—
153C2×C4—81no—
154C2×C4—81no—
155C2×C4#12011yescharacteristic
156C2×C4#12011yescharacteristic
157C2×C4#12011yescharacteristic
158C2×C4#12711yescharacteristic
159C8—42no—
160C8—42no—
161C8—42no—
162C8—81no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—44no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22—162no—
175C22#39211yescharacteristic
176C22#47112yes—
177C22#47112yes—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—81no—
182C4—82no—
183C4—81no—
184C4—81no—
185C4#45812yes—
186C2—22no—
187C2—22no—
188C2—41no—
189C2—161no—
190C2#201911yescharacteristic
191C2#201911yescharacteristic
192C2#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
3. C2×#483C2semidirect
4. #1821C2semidirect
5. #1472C2semidirect
7. #1298C2semidirect

Maximal subgroups

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

subgroupindex
#31932
C2×#4812
C2×#4832
#18212
#14722
#14712
#12982