Finite Group Databasebeta

← #29960

Subgroups — #29960

Order 256.

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

normal subgroups only

This group has 399 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#2628C211yesmaximal, characteristic
2#2310C211yesmaximal, characteristic
3#2097C212yesmaximal, complemented
4#2068C212yesmaximal, complemented
5#2066C212yesmaximal, complemented
6#2024C212yesmaximal, complemented
7#1854C212yesmaximal
8#1798C211yesmaximal, characteristic
9#1787C212yesmaximal
10C23×Q8C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×#117C2211yescharacteristic
13C2×#117C2211yescharacteristic
14C2×#107C2212yes—
15C2×#106C2212yes—
16C2×C4⋊C8C2211yescharacteristic
17C2×C4⋊C8C2211yescharacteristic
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×C22⋊C8C2212yes—
23#400—21no—
24#399—21no—
25#395—21no—
26#393—21no—
27#391—21no—
28#391—21no—
29#391C2212yes—
30#391C2212yes—
31#389C2212yes—
32#382C2211yescharacteristic
33#381—21no—
34#381—21no—
35#380C2212yes—
36#340C2212yes—
37#340C2212yes—
38#336C2212yes—
39C22×Q8—21no—
40C22×Q8—21no—
41C22×Q8—22no—
42C22×Q8—41no—
43C22×Q8—41no—
44C22×Q8—41no—
45C22×Q8—41no—
46C22×Q8Q814yescomplemented
47C22×Q8C2312yes—
48C23×C4—21no—
49C23×C4C2311yescharacteristic
50C22×C8C2312yes—
51C22×C8C2312yes—
52C4⋊Q8—41no—
53C4⋊Q8—41no—
54#117—41no—
55#117—41no—
56#117—41no—
57#117—41no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4C2312yes—
63C2×C4⋊C4C2312yes—
64C2×C4⋊C4C2312yes—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4D411yescharacteristic
68C2×C22⋊C4D411yescharacteristic
69C2×C42—21no—
70C2×C42C2311yescharacteristic
71C2×C42C2311yescharacteristic
72#107—42no—
73#106—42no—
74C4⋊C8—41no—
75C4⋊C8—41no—
76#103—42no—
77#103—42no—
78#103—42no—
79#103—42no—
80C22⋊C8—24no—
81#95—22no—
82#95—22no—
83#95—22no—
84#95—22no—
85#95—22no—
86#95—21no—
87#95—21no—
88#95—22no—
89C24C2×D411yescharacteristic
90C2×Q8—24no—
91C2×Q8—22no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—44no—
97C2×Q8—44no—
98C2×Q8—42no—
99C2×Q8—42no—
100C2×Q8—44no—
101C2×Q8—44no—
102C2×Q8—42no—
103C2×Q8—82no—
104C2×Q8C2×Q814yes—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—24no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—42no—
115C22×C4—42no—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4C2×Q812yes—
122C22×C4#4112yes—
123C22×C4C2411yescharacteristic
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4—42no—
139C4⋊C4—82no—
140C22⋊C4—41no—
141C22⋊C4—41no—
142C22⋊C4—42no—
143C22⋊C4—82no—
144C42—21no—
145C42—21no—
146C42—21no—
147C42—21no—
148C42—42no—
149C23—21no—
150C23—21no—
151C23—22no—
152C23—22no—
153C23C22⋊Q812yes—
154C23C22×D411yescharacteristic
155Q8—24no—
156Q8—24no—
157Q8—48no—
158Q8—42no—
159Q8—42no—
160Q8—84no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—48no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4C22⋊Q812yes—
189C2×C4C22×Q811yescharacteristic
190C2×C4C2×#4111yescharacteristic
191C2×C4H(5,𝔽2)11yescharacteristic
192C2×C4H(5,𝔽2)11yescharacteristic
193C2×C4H(5,𝔽2)11yescharacteristic
194C2×C4H(5,𝔽2)11yescharacteristic
195C8—42no—
196C8—42no—
197C22—24no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22C2×C22⋊Q811yescharacteristic
203C22#53411yescharacteristic
204C22#53511yescharacteristic
205C22#57712yes—
206C22#57812yes—
207C4—22no—
208C4—44no—
209C4—41no—
210C4—41no—
211C4—41no—
212C4—41no—
213C4—82no—
214C4—82no—
215C4—82no—
216C4#53811yescharacteristic
217C4#53811yescharacteristic
218C4#54311yescharacteristic
219C4#54311yescharacteristic
220C2—42no—
221C2#262611yescharacteristic
222C2#308212yes—
223C2#320011yescharacteristic
224C2#320211yescharacteristic
225C2#322512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
3. #2097C2semidirect
4. #2068C2semidirect
5. #2066C2semidirect
6. #2024C2semidirect
46. C22×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#26282
#23102
#20972
#20682
#20662
#20242
#18542
#17982
#17872