Finite Group Databasebeta

← #9950

Subgroups — #9950

Order 256.

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

normal subgroups only

This group has 318 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#3186C211yesmaximal, characteristic
2C2×#481C211yesmaximal, characteristic, complemented
3#2504C211yesmaximal, characteristic, complemented
4#1964C211yesmaximal, characteristic, complemented
5#1824C211yesmaximal, characteristic, complemented
6#1519C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8C2×#121C2211yescharacteristic
9C2×#121C2211yescharacteristic
10C22×C4⋊C4C2211yescharacteristic
11#481—22no—
12#481—22no—
13#480—21no—
14#480—21no—
15#466—22no—
16#447—21no—
17#447—21no—
18C2×#106—21no—
19C2×#102—21no—
20C2×#99—22no—
21C2×C22⋊C8C2211yescharacteristic
22C2×C22⋊C8C2211yescharacteristic
23#397—22no—
24#390—21no—
25#386—22no—
26#382—21no—
27#380C2211yescharacteristic
28C4×C22⋊C4C2211yescharacteristic
29#340—22no—
30#327—22no—
31C22×D4—21no—
32C22×D4—21no—
33C22×D4D411yescharacteristic
34C22×D4D411yescharacteristic
35C23×C4—21no—
36C23×C4C2311yescharacteristic
37C2×D8—22no—
38C22×C8—21no—
39C22×C8D412yes—
40#121—21no—
41#121—21no—
42#121—21no—
43#121—21no—
44#121—21no—
45#121—21no—
46#121—21no—
47#121—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⋊C4D412yes—
55C2×C4⋊C4D411yescharacteristic
56C2×C4⋊C4D411yescharacteristic
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—24no—
61C2×C22⋊C4D412yes—
62C2×C42—21no—
63#107—22no—
64#106—24no—
65#102—24no—
66#102—22no—
67#102—22no—
68#99—44no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8D412yes—
73#95—24no—
74#95—21no—
75#95—21no—
76C24—21no—
77C24—21no—
78C24C2×D411yescharacteristic
79C2×D4—21no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—42no—
84C2×D4—42no—
85C2×D4—41no—
86C2×D4—41no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—42no—
90C2×D4—42no—
91C2×D4—42no—
92C2×D4—42no—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—22no—
96C22×C4—21no—
97C22×C4—42no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—82no—
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4#4111yescharacteristic
108D8—82no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—42no—
112C2×C8—42no—
113C4⋊C4—22no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—42no—
120C4⋊C4—41no—
121C4⋊C4—41no—
122C22⋊C4—22no—
123C22⋊C4—22no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—41no—
128C22⋊C4—42no—
129C22⋊C4—84no—
130C42—42no—
131C23—41no—
132C23—41no—
133C23—42no—
134C23—42no—
135C23—41no—
136C23—41no—
137C23—42no—
138C23—42no—
139C23—81no—
140C23—81no—
141C23#12011yescharacteristic
142C23#12111yescharacteristic
143C23#12711yescharacteristic
144C23#13512yes—
145C23AGL(1,ℤ/8)12yes—
146D4—42no—
147D4—42no—
148D4—42no—
149D4—81no—
150D4—82no—
151D4—82no—
152D4—81no—
153D4—81no—
154D4—81no—
155D4—81no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—44no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—82no—
172C2×C4—82no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4#12011yescharacteristic
177C2×C4#12011yescharacteristic
178C2×C4#12111yescharacteristic
179C2×C4#12111yescharacteristic
180C8—42no—
181C8—42no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—42no—
189C22—42no—
190C22—84no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—162no—
198C22#39211yescharacteristic
199C22#44912yes—
200C22#46312yes—
201C22#49511yescharacteristic
202C22#49611yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—82no—
207C4—82no—
208C4—81no—
209C4—81no—
210C4—161no—
211C4#45812yes—
212C2—22no—
213C2—21no—
214C2—22no—
215C2—21no—
216C2—161no—
217C2—161no—
218C2#200811yescharacteristic
219C2#201211yescharacteristic
220C2#202211yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radical#3186C2
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. C2×#481C2semidirect
3. #2504C2semidirect
4. #1964C2semidirect
5. #1824C2semidirect
6. #1519C2semidirect
7. #1282C2semidirect

Maximal subgroups

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

subgroupindex
#31862
C2×#4812
#25042
#19642
#18242
#15192
#12822