Finite Group Databasebeta

← #27757

Subgroups — #27757

Order 256.

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

normal subgroups only

This group has 341 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#2546C211yesmaximal, characteristic
3#2074C212yesmaximal
4#2043C212yesmaximal
5#1981C211yesmaximal, characteristic
6#1960C212yesmaximal
7#1956C211yesmaximal, characteristic
8#1935C212yesmaximal
9#1803C211yesmaximal, characteristic
10#1770C211yesmaximal, characteristic
11C4×#103C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2212yes—
14C2×C4×Q8C2211yescharacteristic
15C2×C4⋊C8C2211yescharacteristic
16C2×#103C2212yes—
17C2×#103C2211yescharacteristic
18C2×#103C2211yescharacteristic
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#97C2212yes—
24C2×C4×C8C2211yescharacteristic
25#400—22no—
26#398—22no—
27#391C2212yes—
28#389—21no—
29#389—21no—
30#389—21no—
31#389C2212yes—
32#384C2211yescharacteristic
33#382—21no—
34#382C2212yes—
35C4×C4⋊C4C2211yescharacteristic
36C4×C4⋊C4C2211yescharacteristic
37#340C2212yes—
38#340C2211yescharacteristic
39#340C2211yescharacteristic
40#336C2212yes—
41C22×Q8C2312yes—
42C22×Q8C2311yescharacteristic
43C22×C8C2311yescharacteristic
44C22×C8C2312yes—
45C22×C8C2311yescharacteristic
46C4⋊Q8—24no—
47C4⋊Q8—22no—
48C4⋊Q8—22no—
49C4⋊Q8—42no—
50C4×Q8—22no—
51C4×Q8—22no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C42—21no—
67C2×C42—21no—
68C2×C42C2312yes—
69C2×C42C2311yescharacteristic
70C4⋊C8—22no—
71#103—22no—
72#103—22no—
73#103—24no—
74#103—42no—
75#103—42no—
76#103—42no—
77#103—42no—
78#97—42no—
79C4×C8—22no—
80#95—21no—
81#95—22no—
82#95—22no—
83#95—22no—
84#95—22no—
85#95—22no—
86#95D412yes—
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—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8#4112yes—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4C2×D412yes—
106C22×C4C2×D412yes—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2411yescharacteristic
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—21no—
118C2×C8#4112yes—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—44no—
130C4⋊C4—44no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—81no—
136C4⋊C4—82no—
137C42—22no—
138C42—22no—
139C42—42no—
140C42—42no—
141C42#4112yes—
142C42#4112yes—
143C23C22×D411yescharacteristic
144Q8—22no—
145Q8—22no—
146Q8—41no—
147Q8—41no—
148Q8—42no—
149Q8—42no—
150Q8—42no—
151Q8—42no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—42no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4#12412yes—
179C2×C4#12412yes—
180C2×C4#13712yes—
181C2×C4#13712yes—
182C2×C4C2×#4111yescharacteristic
183C2×C4C2×#4111yescharacteristic
184C2×C4C2×#4111yescharacteristic
185C2×C4C2×#4111yescharacteristic
186C2×C4H(5,𝔽2)11yescharacteristic
187C2×C4H(5,𝔽2)11yescharacteristic
188C8—22no—
189C8—42no—
190C8—41no—
191C22C2×#12411yescharacteristic
192C22#54611yescharacteristic
193C22#55011yescharacteristic
194C22C2×#13711yescharacteristic
195C22C2×#13711yescharacteristic
196C22#57711yescharacteristic
197C22#57811yescharacteristic
198C4—22no—
199C4—42no—
200C4—42no—
201C4—42no—
202C4—81no—
203C4—81no—
204C4—81no—
205C4—82no—
206C4—82no—
207C4#49114yes—
208C4#52911yescharacteristic
209C4#54011yescharacteristic
210C4#55311yescharacteristic
211C4#55311yescharacteristic
212C2#249011yescharacteristic
213C2C2×#49111yescharacteristic
214C2#313711yescharacteristic
215C2#332811yescharacteristic
216C2#332911yescharacteristic
217C2#339811yescharacteristic
218C2#340011yescharacteristic
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
#25462
#20742
#20432
#19812
#19602
#19562
#19352
#18032
#17702
C4×#1032