Finite Group Databasebeta

← #23720

Subgroups — #23720

Order 256.

372 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C4×#137C212yesmaximal
2C2×#441C211yesmaximal, characteristic, complemented
3#2923C211yesmaximal, characteristic
4#2090C211yesmaximal, characteristic, complemented
5#2079C211yesmaximal, characteristic
6#2064C212yesmaximal, complemented
7#2052C212yesmaximal
8#2047C211yesmaximal, characteristic, complemented
9#2020C211yesmaximal, characteristic
10#1778C211yesmaximal, characteristic, complemented
11C4×#101C212yesmaximal, complemented
12C2×#137C2212yes—
13C22×Q16C2211yescharacteristic
14C4×#41C2211yescharacteristic
15C2×C4×Q8C2212yes—
16#441—22no—
17#441—22no—
18#441—22no—
19#440—22no—
20C4×Q16—22no—
21C4×SD16—22no—
22#433—22no—
23#432—22no—
24C2×#108C2212yes—
25#428C2212yes—
26C2×#106C2211yescharacteristic
27#424C2211yescharacteristic
28#424C2211yescharacteristic
29#419C2212yes—
30#417C2212yes—
31C2×#103C2212yes—
32C2×#101C2212yes—
33C2×#101C2212yes—
34#408C2211yescharacteristic
35#408C2211yescharacteristic
36C4×M16C2211yescharacteristic
37C4×M16C2212yes—
38C2×#97C2211yescharacteristic
39#343C2212yes—
40#343C2212yes—
41#341C2211yescharacteristic
42#341C2211yescharacteristic
43C2×#41C2311yescharacteristic
44C22×Q8C2312yes—
45#137—24no—
46#137—24no—
47C2×Q16—22no—
48C2×Q16—24no—
49C2×Q16—22no—
50C2×SD16—22no—
51C2×M16C2312yes—
52C2×M16C2311yescharacteristic
53C2×M16C2311yescharacteristic
54C2×M16C2312yes—
55C22×C8C2311yescharacteristic
56C22×C8C2311yescharacteristic
57C4×Q8—22no—
58C4×Q8—22no—
59C4×Q8—22no—
60C4×Q8—22no—
61C4×Q8D412yes—
62C4×D4—22no—
63C4×D4D412yes—
64#117—21no—
65#117C2311yescharacteristic
66C2×C4⋊C4—22no—
67C2×C4⋊C4C2312yes—
68C2×C42—21no—
69C2×C42—22no—
70C2×C42C2311yescharacteristic
71#108—24no—
72#107—22no—
73#106—22no—
74#106—22no—
75C4⋊C8—22no—
76C4⋊C8—22no—
77#103—22no—
78#103—24no—
79#103—22no—
80#103—22no—
81#102—22no—
82#101—24no—
83#101—24no—
84C22⋊C8—22no—
85C22⋊C8—22no—
86#97—21no—
87#97—22no—
88#97D412yes—
89#97D412yes—
90C4×C8—21no—
91C4×C8—22no—
92#41—22no—
93#41—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8C2×D411yescharacteristic
100C2×Q8#4114yes—
101C2×D4—21no—
102C2×D4C2×D411yescharacteristic
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4#4112yes—
108C22×C4C2411yescharacteristic
109Q16—28no—
110Q16—28no—
111SD16—28no—
112M16—22no—
113M16—22no—
114M16—22no—
115M16—22no—
116M16—22no—
117M16#4114yes—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8C2×D412yes—
125C2×C8C2×D412yes—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4C2×D412yes—
132C22⋊C4—21no—
133C22⋊C4—22no—
134C42—22no—
135C42—21no—
136C42—42no—
137C42—42no—
138C42C2×D412yes—
139C42C2×D411yescharacteristic
140C42C2×D411yescharacteristic
141C23—21no—
142C23C2×#4111yescharacteristic
143Q8—24no—
144Q8—24no—
145Q8—22no—
146Q8—42no—
147Q8—42no—
148D4—22no—
149D4—42no—
150C2×C4—22no—
151C2×C4—21no—
152C2×C4—24no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—82no—
160C2×C4—82no—
161C2×C4#12112yes—
162C2×C4#12112yes—
163C2×C4C22×D411yescharacteristic
164C2×C4C22×D411yescharacteristic
165C2×C4C2×#4112yes—
166C2×C4C2×#4112yes—
167C8—22no—
168C8—24no—
169C8—22no—
170C8—41no—
171C8—42no—
172C8—41no—
173C22—22no—
174C22—41no—
175C22—42no—
176C22C2×#12111yescharacteristic
177C22#52912yes—
178C4—22no—
179C4—22no—
180C4—41no—
181C4—44no—
182C4—42no—
183C4—82no—
184C4#52511yescharacteristic
185C4#52512yes—
186C4#53211yescharacteristic
187C2—22no—
188C2—81no—
189C2#239211yescharacteristic
190C2#306912yes—
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#441C2semidirect
4. #2090C2semidirect
6. #2064C2semidirect
8. #2047C2semidirect
10. #1778C2semidirect
11. C4×#101C2semidirect

Maximal subgroups

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

subgroupindex
C4×#1372
C2×#4412
#29232
#20902
#20792
#20642
#20522
#20472
#20202
#17782
C4×#1012