Finite Group Databasebeta

← #29995

Subgroups — #29995

Order 256.

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

normal subgroups only

This group has 369 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, complemented
2#2293C211yesmaximal, characteristic
3#2088C212yesmaximal, complemented
4#2087C212yesmaximal, complemented
5#2067C212yesmaximal
6#2065C212yesmaximal
7#1948C211yesmaximal, characteristic, complemented
8#1947C211yesmaximal, characteristic, complemented
9#1842C211yesmaximal, characteristic, complemented
10#1802C211yesmaximal, characteristic
11#1802C211yesmaximal, characteristic
12C2×#125—21no—
13C2×C4×D4C2211yescharacteristic
14C2×#117—21no—
15C2×#107C2212yes—
16C2×#106C2212yes—
17C2×C4⋊C8C2211yescharacteristic
18C2×C4⋊C8C2211yescharacteristic
19C2×C4⋊C8C2212yes—
20C2×C4⋊C8C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23#390C2211yescharacteristic
24#390C2211yescharacteristic
25#389—21no—
26#389—21no—
27#389C2212yes—
28#389C2212yes—
29#389C2211yescharacteristic
30#389C2211yescharacteristic
31#388—21no—
32#388—21no—
33#384C2211yescharacteristic
34#384C2211yescharacteristic
35#382—21no—
36C4×C4⋊C4C2211yescharacteristic
37C4×C22⋊C4—21no—
38C43C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43C22×D4C2311yescharacteristic
44C23×C4—21no—
45C22×C8C2312yes—
46C22×C8C2312yes—
47#125—44no—
48C4×D4—22no—
49C4×D4Q814yes—
50#117—44no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2311yescharacteristic
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C42—21no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#107—42no—
78#106—42no—
79C4⋊C8—22no—
80C4⋊C8—22no—
81C4⋊C8—42no—
82C4⋊C8—42no—
83#102—42no—
84#102—42no—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—22no—
89#95—22no—
90C24—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—21no—
96C2×D4C2×Q812yes—
97C22×C4—21no—
98C22×C4—24no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—42no—
104C22×C4—42no—
105C22×C4C2×D411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2411yescharacteristic
112C2×C8—22no—
113C2×C8—22no—
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—42no—
128C4⋊C4—42no—
129C4⋊C4—42no—
130C4⋊C4—42no—
131C4⋊C4—44no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—44no—
135C4⋊C4C2×Q812yes—
136C22⋊C4—24no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C22⋊C4—42no—
141C42—22no—
142C42—22no—
143C42—22no—
144C42—22no—
145C42—21no—
146C42—21no—
147C42—21no—
148C42—21no—
149C42—42no—
150C42C2×Q812yes—
151C42#4112yes—
152C23—22no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23C22×D411yescharacteristic
160D4—22no—
161D4—22no—
162D4—41no—
163D4—41no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—48no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—41no—
192C2×C4C22⋊Q812yes—
193C2×C4C22⋊Q812yes—
194C2×C4#13512yes—
195C2×C4AGL(1,ℤ/8)12yes—
196C2×C4AGL(1,ℤ/8)12yes—
197C2×C4#13712yes—
198C2×C4C22×Q811yescharacteristic
199C2×C4C2×#4111yescharacteristic
200C2×C4H(5,𝔽2)11yescharacteristic
201C2×C4H(5,𝔽2)11yescharacteristic
202C2×C42−1+411yescharacteristic
203C2×C42−1+411yescharacteristic
204C8—42no—
205C8—42no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22C2×C22⋊Q811yescharacteristic
214C22#53711yescharacteristic
215C22#53711yescharacteristic
216C22C2×#13511yescharacteristic
217C22C2×AGL(1,ℤ/8)11yescharacteristic
218C22C2×AGL(1,ℤ/8)11yescharacteristic
219C22C2×#13711yescharacteristic
220C4—22no—
221C4—22no—
222C4—22no—
223C4—42no—
224C4—41no—
225C4—41no—
226C4—82no—
227C4—81no—
228C4—82no—
229C4—81no—
230C4#47814yes—
231C4#54011yescharacteristic
232C4#54111yescharacteristic
233C4#54411yescharacteristic
234C4#54411yescharacteristic
235C2—82no—
236C2#262911yescharacteristic
237C2C2×#47811yescharacteristic
238C2#307911yescharacteristic
239C2#324711yescharacteristic
240C2#324811yescharacteristic
241C2#325311yescharacteristic
242C2#325411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #2859C2semidirect
3. #2088C2semidirect
4. #2087C2semidirect
7. #1948C2semidirect
8. #1947C2semidirect
9. #1842C2semidirect

Maximal subgroups

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

subgroupindex
#28592
#22932
#20882
#20872
#20672
#20652
#19482
#19472
#18422
#18022