Finite Group Databasebeta

← #27738

Subgroups — #27738

Order 256.

315 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 123 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#2571C211yesmaximal, characteristic
2#2567C211yesmaximal, characteristic
3#2099C212yesmaximal
4#2074C212yesmaximal
5#1990C211yesmaximal, characteristic
6#1987C211yesmaximal, characteristic
7#1935C212yesmaximal
8#1836C212yesmaximal
9#1803C211yesmaximal, characteristic
10#1770C211yesmaximal, characteristic
11C4×#103C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C2×C4⋊C8C2211yescharacteristic
15C2×#103C2212yes—
16C2×#103C2211yescharacteristic
17C2×#103C2211yescharacteristic
18C2×#97C2212yes—
19C2×C4×C8C2211yescharacteristic
20#400—22no—
21#395—22no—
22#391C2212yes—
23#389—21no—
24#389—21no—
25#389—21no—
26#389C2212yes—
27#384C2212yes—
28#384C2211yescharacteristic
29#382—21no—
30#382C2212yes—
31C4×C4⋊C4C2211yescharacteristic
32C4×C4⋊C4C2211yescharacteristic
33#340C2212yes—
34#340C2212yes—
35#340C2212yes—
36#340C2211yescharacteristic
37#340C2212yes—
38#340C2211yescharacteristic
39#340C2212yes—
40#336C2212yes—
41C22×Q8—21no—
42C22×Q8C2311yescharacteristic
43C22×C8C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2311yescharacteristic
46C4⋊Q8—22no—
47C4⋊Q8—22no—
48C4×Q8—22no—
49C4×Q8—22no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2312yes—
64C2×C42—21no—
65C2×C42—21no—
66C2×C42C2312yes—
67C2×C42C2311yescharacteristic
68C4⋊C8—22no—
69#103—24no—
70#103—22no—
71#103—22no—
72#97—24no—
73C4×C8—22no—
74#95—21no—
75#95—22no—
76#95—22no—
77#95—22no—
78#95—22no—
79#95—22no—
80#95D412yes—
81C2×Q8—21no—
82C2×Q8—21no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—42no—
87C2×Q8—81no—
88C2×Q8—81no—
89C2×Q8#4112yes—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—22no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4C2×D412yes—
97C22×C4C2×D412yes—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2411yescharacteristic
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—21no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—21no—
107C2×C8—22no—
108C2×C8—21no—
109C2×C8#4112yes—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—22no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—22no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4—81no—
126C4⋊C4—82no—
127C4⋊C4—82no—
128C4⋊C4—82no—
129C42—22no—
130C42—22no—
131C42—42no—
132C42—42no—
133C42#4112yes—
134C42#4112yes—
135C23C22×D411yescharacteristic
136Q8—22no—
137Q8—22no—
138Q8—41no—
139Q8—41no—
140Q8—84no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—42no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4#12412yes—
168C2×C4#12412yes—
169C2×C4#13712yes—
170C2×C4#13712yes—
171C2×C4C2×#4111yescharacteristic
172C2×C4C2×#4111yescharacteristic
173C2×C4C2×#4111yescharacteristic
174C2×C4C2×#4111yescharacteristic
175C2×C42−1+411yescharacteristic
176C2×C42−1+411yescharacteristic
177C8—22no—
178C8—42no—
179C8—41no—
180C22C2×#12411yescharacteristic
181C22#54711yescharacteristic
182C22#54811yescharacteristic
183C22C2×#13711yescharacteristic
184C22C2×#13711yescharacteristic
185C22#57611yescharacteristic
186C22#57711yescharacteristic
187C4—22no—
188C4—42no—
189C4—42no—
190C4—42no—
191C4—81no—
192C4—81no—
193C4—82no—
194C4—81no—
195C4—82no—
196C4#49114yes—
197C4#52911yescharacteristic
198C4#54111yescharacteristic
199C4#55211yescharacteristic
200C4#55211yescharacteristic
201C2#248911yescharacteristic
202C2C2×#49111yescharacteristic
203C2#313611yescharacteristic
204C2#335611yescharacteristic
205C2#335711yescharacteristic
206C2#336911yescharacteristic
207C2#337511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#25712
#25672
#20992
#20742
#19902
#19872
#19352
#18362
#18032
#17702
C4×#1032