Finite Group Databasebeta

← #9953

Subgroups — #9953

Order 256.

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

normal subgroups only

This group has 297 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#3187C211yesmaximal, characteristic
2C2×#480C211yesmaximal, characteristic
3#2503C211yesmaximal, characteristic, complemented
4#1964C211yesmaximal, characteristic
5#1823C211yesmaximal, characteristic, complemented
6#1520C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8C2211yescharacteristic
9C2×#121C2211yescharacteristic
10C22×C4⋊C4C2211yescharacteristic
11#483—21no—
12#481—21no—
13#480—22no—
14#480—22no—
15#465—22no—
16#450—21no—
17#448—21no—
18C2×#107—21no—
19C2×#102—21no—
20C2×#99—22no—
21C2×C22⋊C8C2211yescharacteristic
22C2×C22⋊C8C2211yescharacteristic
23#400—22no—
24#391—21no—
25#382—21no—
26#381—22no—
27#380C2211yescharacteristic
28C4×C22⋊C4C2211yescharacteristic
29#340—22no—
30#328—22no—
31C22×Q8D411yescharacteristic
32C22×D4—21no—
33C22×D4D411yescharacteristic
34C23×C4—21no—
35C23×C4C2311yescharacteristic
36C2×SD16—22no—
37C22×C8—21no—
38C22×C8D412yes—
39C22⋊Q8—21no—
40C22⋊Q8—21no—
41C22⋊Q8—21no—
42C22⋊Q8—21no—
43#121—21no—
44#121—21no—
45#121—21no—
46#121—21no—
47C2×C4⋊C4—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⋊C4D412yes—
61C2×C42—21no—
62#107—24no—
63#106—22no—
64#103—22no—
65#102—24no—
66#102—22no—
67#99—44no—
68C22⋊C8—21no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8D412yes—
72#95—24no—
73#95—24no—
74#95—21no—
75#95—21no—
76C24—21no—
77C24C2×D411yescharacteristic
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—42no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—42no—
84C2×D4—41no—
85C2×D4—41no—
86C2×D4—42no—
87C2×D4—42no—
88C22×C4—22no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—22no—
92C22×C4—21no—
93C22×C4—42no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4—41no—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4#4111yescharacteristic
105SD16—82no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—42no—
109C2×C8—42no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—21no—
116C4⋊C4—42no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C22⋊C4—41no—
127C22⋊C4—42no—
128C42—42no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—42no—
133C23—81no—
134C23—81no—
135C23#12011yescharacteristic
136C23#12111yescharacteristic
137C23#12711yescharacteristic
138C23#13512yes—
139C23AGL(1,ℤ/8)12yes—
140Q8—42no—
141Q8—81no—
142D4—42no—
143D4—42no—
144D4—82no—
145D4—81no—
146D4—81no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—44no—
163C2×C4—42no—
164C2×C4—82no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4#12011yescharacteristic
171C2×C4#12011yescharacteristic
172C2×C4#12111yescharacteristic
173C2×C4#12111yescharacteristic
174C8—42no—
175C8—42no—
176C22—21no—
177C22—21no—
178C22—21no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—42no—
183C22—42no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22—162no—
188C22#39211yescharacteristic
189C22#44912yes—
190C22#46312yes—
191C22#49511yescharacteristic
192C22#49611yescharacteristic
193C4—21no—
194C4—21no—
195C4—21no—
196C4—82no—
197C4—81no—
198C4—82no—
199C4—81no—
200C4—81no—
201C4—161no—
202C4#45812yes—
203C2—22no—
204C2—21no—
205C2—22no—
206C2—21no—
207C2—161no—
208C2#200911yescharacteristic
209C2#201211yescharacteristic
210C2#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. #2503C2semidirect
5. #1823C2semidirect
6. #1520C2semidirect
7. #1282C2semidirect

Maximal subgroups

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

subgroupindex
#31872
C2×#4802
#25032
#19642
#18232
#15202
#12822