Finite Group Databasebeta

← #29964

Subgroups — #29964

Order 256.

419 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 119 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#2629C211yesmaximal, characteristic, complemented
2#2311C211yesmaximal, characteristic
3#2095C211yesmaximal, characteristic, complemented
4#2093C211yesmaximal, characteristic, complemented
5#2068C212yesmaximal, complemented
6#2067C212yesmaximal, complemented
7#2026C211yesmaximal, characteristic, complemented
8#2025C211yesmaximal, characteristic, complemented
9#1852C212yesmaximal, complemented
10#1798C211yesmaximal, characteristic, complemented
11#1788C212yesmaximal
12C22×#41C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×#117C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C2×#107C2212yes—
17C2×#107C2212yes—
18C2×C4⋊C8C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×C22⋊C8C2212yes—
24#398—22no—
25#396—22no—
26#391C2211yescharacteristic
27#391C2211yescharacteristic
28#390C2211yescharacteristic
29#390C2211yescharacteristic
30#389C2212yes—
31#386—22no—
32#384C2211yescharacteristic
33#380—22no—
34#380C2212yes—
35#340C2212yes—
36#340C2212yes—
37#336C2211yescharacteristic
38#336C2211yescharacteristic
39C2×#41—22no—
40C2×#41—42no—
41C2×#41—42no—
42C2×#41Q814yes—
43C22×Q8C2311yescharacteristic
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C23×C4—21no—
48C23×C4C2311yescharacteristic
49C22×C8C2312yes—
50C22×C8C2312yes—
51#125—42no—
52#117—42no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—42no—
58C2×C4⋊C4D412yes—
59C2×C4⋊C4C2311yescharacteristic
60C2×C4⋊C4C2311yescharacteristic
61C2×C4⋊C4C2312yes—
62C2×C4⋊C4C2312yes—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4D412yes—
67C2×C42C2311yescharacteristic
68C2×C42C2311yescharacteristic
69#107—42no—
70#107—42no—
71C4⋊C8—42no—
72#103—42no—
73#103—42no—
74#102—42no—
75#102—42no—
76C22⋊C8—24no—
77#95—22no—
78#95—22no—
79#95—22no—
80#95—22no—
81C24—21no—
82C24C2×D411yescharacteristic
83#41—48no—
84#41—48no—
85C2×Q8—21no—
86C2×Q8—22no—
87C2×Q8—22no—
88C2×Q8C2×Q812yes—
89C2×D4—24no—
90C2×D4—21no—
91C2×D4—22no—
92C2×D4—22no—
93C2×D4—42no—
94C2×D4—42no—
95C2×D4C2×Q812yes—
96C22×C4—22no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—24no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—42no—
107C22×C4—82no—
108C22×C4C2×D412yes—
109C22×C4C2×D412yes—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×Q812yes—
112C22×C4#4112yes—
113C22×C4C2411yescharacteristic
114C2×C8—22no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—42no—
127C4⋊C4—42no—
128C4⋊C4—44no—
129C4⋊C4—82no—
130C4⋊C4—82no—
131C22⋊C4—42no—
132C22⋊C4—82no—
133C22⋊C4—82no—
134C22⋊C4—82no—
135C42—22no—
136C42—22no—
137C23—21no—
138C23—21no—
139C23—22no—
140C23—22no—
141C23—22no—
142C23—41no—
143C23—41no—
144C23—42no—
145C23—42no—
146C23C22⋊Q812yes—
147C23C22×D411yescharacteristic
148Q8—24no—
149Q8—42no—
150D4—24no—
151D4—48no—
152D4—42no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—24no—
156C2×C4—24no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—48no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—84no—
178C2×C4C22⋊Q812yes—
179C2×C4C22×Q811yescharacteristic
180C2×C4C2×#4111yescharacteristic
181C2×C4H(5,𝔽2)12yes—
182C2×C4H(5,𝔽2)12yes—
183C8—42no—
184C8—42no—
185C22—24no—
186C22—22no—
187C22—42no—
188C22—41no—
189C22—41no—
190C22—44no—
191C22—42no—
192C22—82no—
193C22—81no—
194C22—81no—
195C22C2×C22⋊Q811yescharacteristic
196C22#53411yescharacteristic
197C22#53511yescharacteristic
198C22#57612yes—
199C22#57812yes—
200C4—22no—
201C4—42no—
202C4—42no—
203C4—42no—
204C4—82no—
205C4—82no—
206C4—81no—
207C4—81no—
208C4#53812yes—
209C4#54312yes—
210C2—42no—
211C2—82no—
212C2#262611yescharacteristic
213C2#308312yes—
214C2#320312yes—
215C2#322011yescharacteristic
216C2#322311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2629C2semidirect
3. #2095C2semidirect
4. #2093C2semidirect
5. #2068C2semidirect
6. #2067C2semidirect
7. #2026C2semidirect
8. #2025C2semidirect
9. #1852C2semidirect
10. #1798C2semidirect

Maximal subgroups

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

subgroupindex
#26292
#23112
#20952
#20932
#20682
#20672
#20262
#20252
#18522
#17982
#17882