Finite Group Databasebeta

← #9949

Subgroups — #9949

Order 256.

393 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 393 conjugacy classes of subgroups — too many to draw. Only its 63 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#3187C211yesmaximal, characteristic
2C2×#448C211yesmaximal, characteristic
3#2434C211yesmaximal, characteristic, complemented
4#1964C211yesmaximal, characteristic
5#1787C211yesmaximal, characteristic, complemented
6#1520C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8C23×Q8C2211yescharacteristic
9C22×SD16—21no—
10C2×C4⋊Q8—21no—
11C2×C22⋊Q8C2211yescharacteristic
12C2×#121C2211yescharacteristic
13#483—21no—
14#481—21no—
15#465—22no—
16#450—21no—
17#448—21no—
18#448—22no—
19#448—22no—
20C2×#103—21no—
21C2×#103—22no—
22C2×#99—22no—
23C2×C22⋊C8C2211yescharacteristic
24C2×C22⋊C8C2211yescharacteristic
25#395—22no—
26#391—21no—
27#385—22no—
28#380C2211yescharacteristic
29C4×C22⋊C4C2211yescharacteristic
30#328—22no—
31C22×Q8—21no—
32C22×Q8—21no—
33C22×Q8—21no—
34C22×Q8—21no—
35C22×Q8—21no—
36C22×Q8—44no—
37C22×Q8D411yescharacteristic
38C22×Q8D412yes—
39C22×D4—21no—
40C22×D4D411yescharacteristic
41C23×C4—21no—
42C23×C4C2311yescharacteristic
43C2×SD16—24no—
44C2×SD16—22no—
45C2×SD16—44no—
46C22×C8—21no—
47C22×C8D412yes—
48C4⋊Q8—44no—
49C22⋊Q8—21no—
50C22⋊Q8—21no—
51C22⋊Q8—21no—
52C22⋊Q8—21no—
53#121—21no—
54#121—21no—
55#121—21no—
56#121—21no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4D411yescharacteristic
61C2×C4⋊C4D411yescharacteristic
62C2×C22⋊C4—21no—
63C2×C22⋊C4—21no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4D412yes—
66C2×C42—21no—
67#106—22no—
68#103—24no—
69#103—22no—
70#103—44no—
71#102—22no—
72#99—44no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8D412yes—
77#95—24no—
78#95—21no—
79#95—21no—
80C24—21no—
81C24C2×D411yescharacteristic
82C2×Q8—24no—
83C2×Q8—24no—
84C2×Q8—24no—
85C2×Q8—22no—
86C2×Q8—21no—
87C2×Q8—22no—
88C2×Q8—21no—
89C2×Q8—44no—
90C2×Q8—42no—
91C2×Q8—42no—
92C2×Q8—42no—
93C2×Q8—42no—
94C2×Q8—44no—
95C2×Q8—44no—
96C2×Q8—41no—
97C2×Q8—41no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—42no—
101C2×D4—41no—
102C2×D4—41no—
103C2×D4—42no—
104C2×D4—42no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—42no—
111C22×C4—44no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4#4111yescharacteristic
121SD16—48no—
122SD16—82no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—42no—
126C2×C8—42no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—48no—
132C4⋊C4—42no—
133C4⋊C4—41no—
134C4⋊C4—41no—
135C22⋊C4—22no—
136C22⋊C4—41no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C22⋊C4—41no—
140C22⋊C4—42no—
141C22⋊C4SD1614yescomplemented
142C42—42no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—42no—
147C23—81no—
148C23—81no—
149C23#12011yescharacteristic
150C23#12111yescharacteristic
151C23#12711yescharacteristic
152C23C2×SD1612yes—
153C23#13712yes—
154Q8—24no—
155Q8—24no—
156Q8—42no—
157Q8—42no—
158Q8—44no—
159Q8—44no—
160Q8—42no—
161Q8—81no—
162D4—42no—
163D4—42no—
164D4—82no—
165D4—81no—
166D4—81no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—44no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4#12011yescharacteristic
191C2×C4#12011yescharacteristic
192C2×C4#12111yescharacteristic
193C2×C4#12111yescharacteristic
194C8—42no—
195C8—42no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—42no—
203C22—42no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—162no—
208C22#39211yescharacteristic
209C22#44812yes—
210C22#46112yes—
211C22#49211yescharacteristic
212C22#49711yescharacteristic
213C4—21no—
214C4—21no—
215C4—21no—
216C4—44no—
217C4—44no—
218C4—81no—
219C4—81no—
220C4—81no—
221C4—161no—
222C4B(4,𝔽2)12yes—
223C2—22no—
224C2—21no—
225C2—22no—
226C2—21no—
227C2—161no—
228C2#200911yescharacteristic
229C2#201311yescharacteristic
230C2#202111yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC23×C4C23
prime radicalC2×#121C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2434C2semidirect
5. #1787C2semidirect
6. #1520C2semidirect
7. #1282C2semidirect
141. C22⋊C4SD16semidirect

Maximal subgroups

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

subgroupindex
#31872
C2×#4482
#24342
#19642
#17872
#15202
#12822