Finite Group Databasebeta

← #21285

Subgroups — #21285

Order 256.

407 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 171 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×C4×SD16C211yesmaximal, characteristic
2#2320C211yesmaximal, characteristic, complemented
3#2319C211yesmaximal, characteristic
4#1973C212yesmaximal
5#1944C212yesmaximal, complemented
6#1934C212yesmaximal
7#1933C212yesmaximal, complemented
8C4×#107C211yesmaximal, characteristic, complemented
9#1763C211yesmaximal, characteristic, complemented
10#1762C211yesmaximal, characteristic
11#1726C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15C4×SD16C418yescomplemented
16C2×#107C2212yes—
17C2×#106C2211yescharacteristic
18C2×#103C2212yes—
19C2×#103C2211yescharacteristic
20C2×#102C2212yes—
21C2×#102C2211yescharacteristic
22C2×#97C2212yes—
23C2×#97C2211yescharacteristic
24C2×C4×C8C2212yes—
25C2×C4×C8C2211yescharacteristic
26#391C2212yes—
27#390C2212yes—
28#389C2212yes—
29#389C2212yes—
30#388—22no—
31#387—22no—
32#383—21no—
33#382—21no—
34#381—21no—
35#380—21no—
36C4×C4⋊C4C2211yescharacteristic
37C4×C4⋊C4C2211yescharacteristic
38#376C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42C22×Q8C2311yescharacteristic
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C2×SD16—24no—
46C2×SD16C2×C414yes—
47C22×C8C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2311yescharacteristic
50C4×Q8—22no—
51C4×Q8C2×C414yes—
52C4×D4—22no—
53C4×D4C2×C414yes—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C22⋊C4—21no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C42—21no—
65C2×C42—21no—
66C2×C42—21no—
67C2×C42C2312yes—
68C2×C42C2311yescharacteristic
69#107—24no—
70#106C2×C414yes—
71#103—24no—
72#103C2×C414yes—
73#102—24no—
74#102C2×C414yes—
75#97—24no—
76#97—22no—
77C4×C8—24no—
78C4×C8C2×C414yes—
79#95—21no—
80#95—21no—
81#95—22no—
82#95—22no—
83#95—22no—
84#95—22no—
85#95D411yescharacteristic
86#95D411yescharacteristic
87#95D412yes—
88C24—21no—
89C2×Q8—21no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8C22×C412yes—
93C2×D4—22no—
94C2×D4—22no—
95C2×D4—21no—
96C2×D4C22×C412yes—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—24no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—82no—
104C22×C4C2×D412yes—
105C22×C4C2×D412yes—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2411yescharacteristic
109SD16—28no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8C22×C412yes—
114C2×C8C22×C412yes—
115C2×C8#4114yes—
116C2×C8#4114yes—
117C4⋊C4—24no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—82no—
125C4⋊C4—82no—
126C4⋊C4C22×C412yes—
127C4⋊C4C22×C412yes—
128C22⋊C4—24no—
129C22⋊C4—82no—
130C22⋊C4—82no—
131C42—24no—
132C42—22no—
133C42—22no—
134C42—81no—
135C42—81no—
136C42C22×C412yes—
137C42#4112yes—
138C23—22no—
139C23—42no—
140C23—41no—
141C23—42no—
142C23—41no—
143C23C22×D411yescharacteristic
144Q8—22no—
145Q8—22no—
146Q8—42no—
147D4—22no—
148D4—22no—
149D4—42no—
150C2×C4—21no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—84no—
173C2×C4C4×D412yes—
174C2×C4C4×D412yes—
175C2×C4C23×C411yescharacteristic
176C2×C4C2×#4112yes—
177C2×C4C2×#4112yes—
178C2×C4C2×#4111yescharacteristic
179C8—22no—
180C8—24no—
181C8#11714yes—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—82no—
186C22—82no—
187C22C2×C4×D411yescharacteristic
188C22#53212yes—
189C22#57611yescharacteristic
190C22#57712yes—
191C22#57811yescharacteristic
192C4—22no—
193C4—22no—
194C4—42no—
195C4—42no—
196C4—42no—
197C4—42no—
198C4—42no—
199C4—82no—
200C4—82no—
201C4C2×#11711yescharacteristic
202C4C4×#4111yescharacteristic
203C4#52912yes—
204C2—82no—
205C2#229311yescharacteristic
206C2#294911yescharacteristic
207C2#295111yescharacteristic
208C2#317812yes—
209C2#317912yes—
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2320C2semidirect
5. #1944C2semidirect
7. #1933C2semidirect
8. C4×#107C2semidirect
9. #1763C2semidirect
11. #1726C2semidirect
15. C4×SD16C4semidirect

Maximal subgroups

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

subgroupindex
C2×C4×SD162
#23202
#23192
#19732
#19442
#19342
#19332
C4×#1072
#17632
#17622
#17262