Finite Group Databasebeta

← #11342

Subgroups — #11342

Order 256.

411 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×#428C212yesmaximal
2#2316C211yesmaximal, characteristic
3C2×#343C211yesmaximal, characteristic
4C2×#339C212yesmaximal
5C4⋊#95C211yesmaximal, characteristic
6C22×M16C2212yes—
7C2×#117C2212yes—
8C22×C4⋊C4C2211yescharacteristic
9C22×C4⋊C4C2211yescharacteristic
10C22×C42C2211yescharacteristic
11#428—24no—
12#428—24no—
13C2×#107—22no—
14C2×#106—22no—
15#389—22no—
16#389—22no—
17#380—22no—
18C2×#95—22no—
19#343C414yescomplemented
20#339—24no—
21C23×C4—21no—
22C23×C4—21no—
23C23×C4D412yes—
24C23×C4C2311yescharacteristic
25C2×M16—22no—
26C2×M16—22no—
27C2×M16—22no—
28C2×M16C2×C414yes—
29C2×M16D414yes—
30C2×M16Q814yes—
31C22×C8—22no—
32#117—22no—
33#117—22no—
34#117—22no—
35#117—22no—
36C2×C4⋊C4—22no—
37C2×C4⋊C4—22no—
38C2×C4⋊C4—22no—
39C2×C4⋊C4—22no—
40C2×C4⋊C4—21no—
41C2×C4⋊C4—22no—
42C2×C4⋊C4—22no—
43C2×C4⋊C4C2×C412yes—
44C2×C4⋊C4D412yes—
45C2×C4⋊C4D412yes—
46C2×C4⋊C4D412yes—
47C2×C4⋊C4Q812yes—
48C2×C22⋊C4—22no—
49C2×C42—22no—
50C2×C42—22no—
51C2×C42—22no—
52C2×C42—21no—
53C2×C42Q812yes—
54#107—28no—
55#106—28no—
56#95—24no—
57#95—44no—
58C24C2×D411yescharacteristic
59C22×C4—22no—
60C22×C4—22no—
61C22×C4—22no—
62C22×C4—22no—
63C22×C4—22no—
64C22×C4—22no—
65C22×C4—22no—
66C22×C4—22no—
67C22×C4—41no—
68C22×C4—41no—
69C22×C4—41no—
70C22×C4—41no—
71C22×C4—42no—
72C22×C4—42no—
73C22×C4—42no—
74C22×C4—42no—
75C22×C4—41no—
76C22×C4—41no—
77C22×C4C4⋊C414yes—
78C22×C4C22×C411yescharacteristic
79C22×C4C2×D412yes—
80C22×C4C2×D411yescharacteristic
81C22×C4C2×D412yes—
82C22×C4C2×Q812yes—
83C22×C4C2×Q811yescharacteristic
84C22×C4C2×Q811yescharacteristic
85C22×C4#4112yes—
86C22×C4#4111yescharacteristic
87C22×C4#4111yescharacteristic
88M16—24no—
89M16—24no—
90M16—24no—
91M16C4⋊C418yescomplemented
92C2×C8—24no—
93C2×C8—24no—
94C2×C8—24no—
95C4⋊C4—24no—
96C4⋊C4—24no—
97C4⋊C4—22no—
98C4⋊C4—24no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4—42no—
103C4⋊C4—82no—
104C4⋊C4—82no—
105C22⋊C4—42no—
106C22⋊C4—42no—
107C42—24no—
108C42—42no—
109C42—42no—
110C42—42no—
111C23—21no—
112C23—21no—
113C23—21no—
114C23—21no—
115C23C2×C4⋊C411yescharacteristic
116C23#12011yescharacteristic
117C23#12111yescharacteristic
118C23C22⋊Q811yescharacteristic
119C23#12311yescharacteristic
120C23#12711yescharacteristic
121C23C4⋊Q811yescharacteristic
122C2×C4—24no—
123C2×C4—24no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4—42no—
127C2×C4—42no—
128C2×C4—42no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—44no—
133C2×C4—44no—
134C2×C4—42no—
135C2×C4—42no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—82no—
142C2×C4—82no—
143C2×C4—81no—
144C2×C4—81no—
145C2×C4C2×C4⋊C412yes—
146C2×C4C4×D411yescharacteristic
147C2×C4C4×D412yes—
148C2×C4C4×Q811yescharacteristic
149C2×C4#12012yes—
150C2×C4#12112yes—
151C2×C4C22⋊Q812yes—
152C2×C4C22⋊Q812yes—
153C2×C4C22⋊Q812yes—
154C2×C4C22⋊Q812yes—
155C2×C4#12312yes—
156C2×C4#12512yes—
157C2×C4#12511yescharacteristic
158C2×C4C4⋊Q812yes—
159C2×C4C4⋊Q811yescharacteristic
160C2×C4C4⋊Q812yes—
161C8—28no—
162C22—21no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—21no—
171C22—21no—
172C22—21no—
173C22—21no—
174C22#38411yescharacteristic
175C22#38511yescharacteristic
176C22#38911yescharacteristic
177C22#39211yescharacteristic
178C22#39511yescharacteristic
179C22#39811yescharacteristic
180C22#39911yescharacteristic
181C22#45311yescharacteristic
182C22#45411yescharacteristic
183C22#45511yescharacteristic
184C22#45611yescharacteristic
185C4—44no—
186C4—44no—
187C4—82no—
188C4—82no—
189C4—82no—
190C4#38512yes—
191C4#38912yes—
192C4#39512yes—
193C4#39812yes—
194C2—21no—
195C2—21no—
196C2—21no—
197C2—21no—
198C2#144411yescharacteristic
199C2#185611yescharacteristic
200C2#185711yescharacteristic
201C2#200611yescharacteristic
202C2#200711yescharacteristic
203C2#206011yescharacteristic
204C2#206111yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
19. #343C4semidirect
91. M16C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4282
#23162
C2×#3432
C2×#3392
C4⋊#952