Finite Group Databasebeta

← #30223

Subgroups — #30223

Order 256.

352 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 111 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#2658C211yesmaximal, characteristic
2#2647C211yesmaximal, characteristic, complemented
3#2043C212yesmaximal, complemented
4#2043C212yesmaximal, complemented
5#2042C212yesmaximal
6#2042C212yesmaximal
7#1960C211yesmaximal, characteristic, complemented
8#1959C211yesmaximal, characteristic
9#1958C211yesmaximal, characteristic
10#1956C211yesmaximal, characteristic, complemented
11#1746C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×#124C2211yescharacteristic
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#102C2212yes—
25C2×#102C2212yes—
26C2×#97C2211yescharacteristic
27C2×C4×C8C2211yescharacteristic
28#400—22no—
29#399—22no—
30#399—22no—
31#398—22no—
32#389C2212yes—
33#389C2212yes—
34#384C2211yescharacteristic
35#382C2211yescharacteristic
36#376C2211yescharacteristic
37#336C2212yes—
38#336C2212yes—
39C22×Q8C2311yescharacteristic
40C22×Q8C2311yescharacteristic
41C22×Q8C2311yescharacteristic
42C22×D4C2311yescharacteristic
43C22×C8C2312yes—
44C22×C8C2312yes—
45C4⋊Q8—22no—
46C4⋊Q8—22no—
47C4⋊Q8—42no—
48C4⋊Q8—42no—
49#124—22no—
50#124—22no—
51#124—42no—
52#124—42no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C22⋊C4—21no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—22no—
66C2×C42C2311yescharacteristic
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69#103—42no—
70#103—42no—
71#103—42no—
72#103—42no—
73#103—42no—
74#103—42no—
75#102—42no—
76#102—42no—
77#97—22no—
78C4×C8—22no—
79#95—22no—
80#95—22no—
81#95—22no—
82#95D412yes—
83#95D412yes—
84C24—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—22no—
100C2×D4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D412yes—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D412yes—
110C22×C4C2411yescharacteristic
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—44no—
124C4⋊C4—44no—
125C4⋊C4—82no—
126C4⋊C4—82no—
127C4⋊C4—82no—
128C22⋊C4—42no—
129C22⋊C4—42no—
130C22⋊C4—44no—
131C22⋊C4—44no—
132C42—22no—
133C42—22no—
134C42#4112yes—
135C42#4112yes—
136C23—41no—
137C23—41no—
138C23—41no—
139C23—42no—
140C23—42no—
141C23C22×D411yescharacteristic
142Q8—42no—
143Q8—42no—
144Q8—42no—
145Q8—42no—
146Q8—42no—
147Q8—42no—
148D4—42no—
149D4—42no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4#12414yes—
175C2×C4C2×#4111yescharacteristic
176C2×C4C2×#4111yescharacteristic
177C2×C4H(5,𝔽2)12yes—
178C2×C4H(5,𝔽2)12yes—
179C8—42no—
180C8—42no—
181C22—82no—
182C22—81no—
183C22—81no—
184C22—81no—
185C22—82no—
186C22C2×#12411yescharacteristic
187C22#53511yescharacteristic
188C22#53511yescharacteristic
189C22#57712yes—
190C22#57812yes—
191C4—24no—
192C4—41no—
193C4—41no—
194C4—42no—
195C4—81no—
196C4—81no—
197C4—81no—
198C4—82no—
199C4—82no—
200C4#54012yes—
201C4#54012yes—
202C2—161no—
203C2#264111yescharacteristic
204C2#313712yes—
205C2#322111yescharacteristic
206C2#322311yescharacteristic
207C2#322512yes—
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
2. #2647C2semidirect
3. #2043C2semidirect
4. #2043C2semidirect
7. #1960C2semidirect
10. #1956C2semidirect
11. #1746C2semidirect

Maximal subgroups

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

subgroupindex
#26582
#26472
#20432
#20422
#19602
#19592
#19582
#19562
#17462