Finite Group Databasebeta

← #27749

Subgroups — #27749

Order 256.

365 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 139 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#2552C211yesmaximal, characteristic
2#2546C211yesmaximal, characteristic
3#2072C212yesmaximal
4#2043C212yesmaximal
5#1981C211yesmaximal, characteristic
6#1956C211yesmaximal, characteristic
7#1956C212yesmaximal
8#1933C212yesmaximal
9#1803C211yesmaximal, characteristic
10C8×C4⋊C4C211yesmaximal, characteristic
11C4×#103C211yesmaximal, characteristic
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2212yes—
14C2×#125—21no—
15C2×C4×Q8C2211yescharacteristic
16C2×C4⋊C8C2211yescharacteristic
17C2×#103C2212yes—
18C2×#103C2211yescharacteristic
19C2×#103C2211yescharacteristic
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×C4×C8C2211yescharacteristic
25C2×C4×C8C2212yes—
26#398—22no—
27#398—22no—
28#391C2212yes—
29#389—21no—
30#389—21no—
31#389—21no—
32#389C2212yes—
33#384C2211yescharacteristic
34#384C2212yes—
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—
50#125—44no—
51C4×Q8—22no—
52C4×Q8—22no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C42—21no—
70C2×C42—21no—
71C2×C42C2312yes—
72C2×C42C2311yescharacteristic
73C4⋊C8—22no—
74#103—22no—
75#103—22no—
76#103—24no—
77#103—42no—
78#103—42no—
79#103—42no—
80#103—42no—
81C4×C8—22no—
82C4×C8—42no—
83#95—21no—
84#95—22no—
85#95—22no—
86#95—22no—
87#95D412yes—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—21no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8#4112yes—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D412yes—
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2411yescharacteristic
111C2×C8—21no—
112C2×C8—21no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—21no—
119C2×C8#4112yes—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—44no—
129C4⋊C4—44no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—44no—
134C4⋊C4—44no—
135C4⋊C4—42no—
136C4⋊C4—82no—
137C4⋊C4—81no—
138C4⋊C4SD1614yes—
139C4⋊C4Q1614yes—
140C42—22no—
141C42—22no—
142C42—42no—
143C42—42no—
144C42#4112yes—
145C42#4112yes—
146C23C22×D411yescharacteristic
147Q8—22no—
148Q8—22no—
149Q8—41no—
150Q8—41no—
151Q8—42no—
152Q8—42no—
153Q8—42no—
154Q8—42no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4#12412yes—
180C2×C4#12412yes—
181C2×C4C2×SD1614yes—
182C2×C4C2×SD1612yes—
183C2×C4C2×Q1614yes—
184C2×C4C2×Q1612yes—
185C2×C4C2×#4111yescharacteristic
186C2×C4C2×#4111yescharacteristic
187C2×C4C2×#4111yescharacteristic
188C2×C4C2×#4111yescharacteristic
189C2×C4H(5,𝔽2)11yescharacteristic
190C2×C4H(5,𝔽2)11yescharacteristic
191C8—22no—
192C8—42no—
193C8—41no—
194C22C2×#12411yescharacteristic
195C22#54611yescharacteristic
196C22#55011yescharacteristic
197C22C22×SD1611yescharacteristic
198C22C22×Q1611yescharacteristic
199C22#57711yescharacteristic
200C22#57811yescharacteristic
201C4—22no—
202C4—24no—
203C4—24no—
204C4—42no—
205C4—81no—
206C4—82no—
207C4—81no—
208C4—82no—
209C4—81no—
210C4#48714yes—
211C4#52911yescharacteristic
212C4#54011yescharacteristic
213C4#55311yescharacteristic
214C4#55311yescharacteristic
215C2#249011yescharacteristic
216C2C2×#48711yescharacteristic
217C2#313711yescharacteristic
218C2#329811yescharacteristic
219C2#329911yescharacteristic
220C2#339211yescharacteristic
221C2#339511yescharacteristic
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
#25522
#25462
#20722
#20432
#19812
#19562
#19332
#18032
C8×C4⋊C42
C4×#1032