Finite Group Databasebeta

← #29874

Subgroups — #29874

Order 256.

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

normal subgroups only

This group has 354 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
1C2×#478C211yesmaximal, characteristic
2C2×#476C211yesmaximal, characteristic
3#2660C211yesmaximal, characteristic, complemented
4#2319C211yesmaximal, characteristic
5#2085C212yesmaximal, complemented
6#2084C212yesmaximal, complemented
7#2073C212yesmaximal
8#2071C212yesmaximal
9#1989C211yesmaximal, characteristic, complemented
10#1988C211yesmaximal, characteristic, complemented
11#1832C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×C4×D4C2211yescharacteristic
15#478—24no—
16#476—24no—
17C2×#107C2211yescharacteristic
18C2×#106C2211yescharacteristic
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×#102C2211yescharacteristic
23C2×#102C2211yescharacteristic
24C2×#102C2212yes—
25#398—21no—
26#398—21no—
27#395—22no—
28#390C2212yes—
29#389C2212yes—
30#388—22no—
31#384C2212yes—
32#382C2212yes—
33#381—21no—
34#380—21no—
35C4×C4⋊C4C2211yescharacteristic
36#376C2211yescharacteristic
37#340C2212yes—
38#340C2212yes—
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×D4C2311yescharacteristic
45C23×C4—21no—
46C22×C8C2312yes—
47C22×C8C2311yescharacteristic
48C22×C8C2311yescharacteristic
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51#125—22no—
52#125—22no—
53C4×D4—22no—
54C4×D4D414yes—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C4⋊C4C2311yescharacteristic
66C2×C22⋊C4—21no—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—22no—
69C2×C42—21no—
70C2×C42C2312yes—
71C2×C42C2311yescharacteristic
72#107—22no—
73#106—24no—
74#106—22no—
75C4⋊C8—22no—
76C4⋊C8—22no—
77#102—24no—
78#102—22no—
79#102—22no—
80#95—21no—
81#95—22no—
82#95—22no—
83#95—22no—
84#95D411yescharacteristic
85#95D411yescharacteristic
86#95D412yes—
87C24—21no—
88C2×Q8—42no—
89C2×Q8—81no—
90C2×Q8—81no—
91C2×D4—21no—
92C2×D4—22no—
93C2×D4—22no—
94C2×D4C2×D412yes—
95C22×C4—21no—
96C22×C4—24no—
97C22×C4—21no—
98C22×C4—22no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—82no—
102C22×C4C2×D412yes—
103C22×C4C2×D412yes—
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2411yescharacteristic
107C2×C8—22no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—24no—
111C2×C8—22no—
112C2×C8—22no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—22no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—82no—
126C4⋊C4—82no—
127C4⋊C4—82no—
128C4⋊C4C2×D412yes—
129C22⋊C4—24no—
130C22⋊C4—82no—
131C22⋊C4—82no—
132C42—22no—
133C42—22no—
134C42—81no—
135C42C2×D412yes—
136C42#4112yes—
137C23—22no—
138C23—42no—
139C23—41no—
140C23—42no—
141C23—41no—
142C23C22×D411yescharacteristic
143Q8—84no—
144D4—22no—
145D4—22no—
146D4—42no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—42no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—84no—
170C2×C4#12112yes—
171C2×C4#12112yes—
172C2×C4C22×D411yescharacteristic
173C2×C4C2×#4111yescharacteristic
174C2×C42−1+412yes—
175C2×C42−1+412yes—
176C8—42no—
177C8—41no—
178C8—41no—
179C22—42no—
180C22—42no—
181C22—42no—
182C22—82no—
183C22—82no—
184C22C2×#12111yescharacteristic
185C22#53612yes—
186C22#57611yescharacteristic
187C22#57712yes—
188C22#57711yescharacteristic
189C4—22no—
190C4—22no—
191C4—42no—
192C4—42no—
193C4—42no—
194C4—82no—
195C4—81no—
196C4—81no—
197C4—82no—
198C4#53611yescharacteristic
199C4#53611yescharacteristic
200C4#54112yes—
201C2—82no—
202C2#262111yescharacteristic
203C2#304211yescharacteristic
204C2#304511yescharacteristic
205C2#324112yes—
206C2#324412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×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
3. #2660C2semidirect
5. #2085C2semidirect
6. #2084C2semidirect
9. #1989C2semidirect
10. #1988C2semidirect
11. #1832C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C2×#4762
#26602
#23192
#20852
#20842
#20732
#20712
#19892
#19882
#18322