Finite Group Databasebeta

← #29997

Subgroups — #29997

Order 256.

331 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 131 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#2859C211yesmaximal, characteristic
2#2295C211yesmaximal, characteristic
3#2088C212yesmaximal
4#2086C212yesmaximal
5#2068C212yesmaximal
6#2066C212yesmaximal
7#1948C211yesmaximal, characteristic
8#1947C211yesmaximal, characteristic
9#1842C211yesmaximal, characteristic
10#1803C211yesmaximal, characteristic
11#1803C211yesmaximal, characteristic
12C2×#125—21no—
13C2×C4×Q8C2211yescharacteristic
14C2×#107C2212yes—
15C2×#106C2212yes—
16C2×C4⋊C8C2211yescharacteristic
17C2×C4⋊C8C2211yescharacteristic
18C2×C4⋊C8C2212yes—
19C2×C4⋊C8C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22#391C2211yescharacteristic
23#391C2211yescharacteristic
24#389—21no—
25#389—21no—
26#389C2212yes—
27#389C2212yes—
28#389C2211yescharacteristic
29#389C2211yescharacteristic
30#387—21no—
31#387—21no—
32#384C2211yescharacteristic
33#384C2211yescharacteristic
34#382—21no—
35C4×C4⋊C4—21no—
36C4×C4⋊C4C2211yescharacteristic
37#376—21no—
38C43C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43C22×Q8C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2312yes—
46#125—44no—
47C4×Q8—22no—
48C4×Q8Q814yes—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—22no—
53C2×C4⋊C4—22no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4C2311yescharacteristic
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2311yescharacteristic
63C2×C42—21no—
64C2×C42—21no—
65C2×C42D411yescharacteristic
66C2×C42D411yescharacteristic
67C2×C42D411yescharacteristic
68C2×C42D411yescharacteristic
69C2×C42C2311yescharacteristic
70C2×C42C2311yescharacteristic
71C2×C42C2311yescharacteristic
72#107—42no—
73#106—42no—
74C4⋊C8—22no—
75C4⋊C8—22no—
76C4⋊C8—42no—
77C4⋊C8—42no—
78#103—42no—
79#103—42no—
80#95—21no—
81#95—21no—
82#95—21no—
83#95—21no—
84#95—21no—
85#95—21no—
86#95—21no—
87#95—22no—
88#95—22no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8C2×Q812yes—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—21no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4C2×D411yescharacteristic
102C22×C4C2×D411yescharacteristic
103C22×C4C2×D411yescharacteristic
104C22×C4C2×D411yescharacteristic
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2411yescharacteristic
108C2×C8—22no—
109C2×C8—22no—
110C2×C8—22no—
111C2×C8—22no—
112C2×C8—22no—
113C2×C8—22no—
114C4⋊C4—24no—
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—42no—
125C4⋊C4—42no—
126C4⋊C4—44no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—44no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4C2×Q812yes—
133C42—24no—
134C42—22no—
135C42—22no—
136C42—22no—
137C42—22no—
138C42—21no—
139C42—21no—
140C42—21no—
141C42—21no—
142C42—42no—
143C42C2×Q812yes—
144C42#4112yes—
145C23C22×D411yescharacteristic
146Q8—22no—
147Q8—22no—
148Q8—41no—
149Q8—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4C22⋊Q812yes—
181C2×C4C22⋊Q812yes—
182C2×C4#13512yes—
183C2×C4AGL(1,ℤ/8)12yes—
184C2×C4#13712yes—
185C2×C4#13712yes—
186C2×C4C22×Q811yescharacteristic
187C2×C4C2×#4111yescharacteristic
188C2×C4H(5,𝔽2)11yescharacteristic
189C2×C4H(5,𝔽2)11yescharacteristic
190C2×C42−1+411yescharacteristic
191C2×C42−1+411yescharacteristic
192C8—42no—
193C8—42no—
194C22C2×C22⋊Q811yescharacteristic
195C22#53711yescharacteristic
196C22#53711yescharacteristic
197C22C2×#13511yescharacteristic
198C22C2×AGL(1,ℤ/8)11yescharacteristic
199C22C2×#13711yescharacteristic
200C22C2×#13711yescharacteristic
201C4—22no—
202C4—22no—
203C4—22no—
204C4—42no—
205C4—42no—
206C4—41no—
207C4—41no—
208C4—81no—
209C4—82no—
210C4—81no—
211C4—82no—
212C4#47914yes—
213C4#54011yescharacteristic
214C4#54111yescharacteristic
215C4#54411yescharacteristic
216C4#54411yescharacteristic
217C2#262911yescharacteristic
218C2C2×#47911yescharacteristic
219C2#307911yescharacteristic
220C2#324711yescharacteristic
221C2#324811yescharacteristic
222C2#325411yescharacteristic
223C2#325511yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ 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
#28592
#22952
#20882
#20862
#20682
#20662
#19482
#19472
#18422
#18032