Finite Group Databasebeta

← #14843

Subgroups — #14843

Order 256.

1023 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 1023 conjugacy classes of subgroups — too many to draw. Only its 231 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
1C23×C4⋊C4C211yesmaximal, characteristic
2C2×#400C212yesmaximal, complemented
3C2×#385C211yesmaximal, characteristic
4#1445C212yesmaximal, complemented
5#1442C212yesmaximal, complemented
6#1441C212yesmaximal, complemented
7C22⋊#95C211yesmaximal, characteristic
8C22⋊#95C212yesmaximal
9C22⋊#95C212yesmaximal
10C24×C4C2211yescharacteristic
11C24×C4C2212yes—
12C22×C4⋊C4—24no—
13C22×C4⋊C4—28no—
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2212yes—
16C22×C4⋊C4C2211yescharacteristic
17C22×C4⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2211yescharacteristic
20C22×C22⋊C4C2211yescharacteristic
21#400—28no—
22#385—24no—
23C2×#95C2214yes—
24C2×#95C2214yes—
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2211yescharacteristic
28C2×#95C2211yescharacteristic
29C2×#95C2214yes—
30C2×#95C2214yes—
31C25C2311yescharacteristic
32C23×C4—24no—
33C23×C4—24no—
34C23×C4—24no—
35C23×C4—24no—
36C23×C4—28no—
37C23×C4—24no—
38C23×C4—24no—
39C23×C4—44no—
40C23×C4D414yes—
41C23×C4C2311yescharacteristic
42C23×C4C2314yes—
43C23×C4C2311yescharacteristic
44C23×C4C2314yes—
45C23×C4C2311yescharacteristic
46C23×C4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C23×C4C2311yescharacteristic
49C2×C4⋊C4—232no—
50C2×C4⋊C4—28no—
51C2×C4⋊C4—24no—
52C2×C4⋊C4—24no—
53C2×C4⋊C4—24no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—24no—
57C2×C4⋊C4—24no—
58C2×C4⋊C4D418yes—
59C2×C22⋊C4—22no—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4Q814yes—
67#95—22no—
68#95—22no—
69#95—28no—
70#95—24no—
71#95—28no—
72#95—24no—
73#95—44no—
74#95—44no—
75C24—22no—
76C24—22no—
77C24—22no—
78C24—22no—
79C24C2×D412yes—
80C24C2×Q812yes—
81C24#4112yes—
82C24#4114yes—
83C24#4114yes—
84C24C2411yescharacteristic
85C22×C4—232no—
86C22×C4—28no—
87C22×C4—22no—
88C22×C4—22no—
89C22×C4—22no—
90C22×C4—22no—
91C22×C4—28no—
92C22×C4—28no—
93C22×C4—216no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—24no—
98C22×C4—24no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—22no—
102C22×C4—24no—
103C22×C4—24no—
104C22×C4—22no—
105C22×C4—24no—
106C22×C4—24no—
107C22×C4—22no—
108C22×C4—24no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—24no—
113C22×C4—22no—
114C22×C4—22no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—24no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—24no—
121C22×C4—22no—
122C22×C4—24no—
123C22×C4—416no—
124C22×C4—416no—
125C22×C4—48no—
126C22×C4—48no—
127C22×C4C2×D418yes—
128C22×C4C2×D412yes—
129C22×C4C2×D414yes—
130C22×C4C2×D412yes—
131C22×C4C2×Q812yes—
132C22×C4C2×Q812yes—
133C22×C4#4114yes—
134C4⋊C4—216no—
135C4⋊C4—216no—
136C4⋊C4—44no—
137C4⋊C4—44no—
138C22⋊C4—28no—
139C22⋊C4—44no—
140C22⋊C4—48no—
141C23—22no—
142C23—22no—
143C23—28no—
144C23—22no—
145C23—22no—
146C23—28no—
147C23—28no—
148C23—24no—
149C23—24no—
150C23—28no—
151C23—24no—
152C23—24no—
153C23C22⋊Q814yes—
154C23#12314yes—
155C23#12414yes—
156C23#12518yes—
157C23#12618yes—
158C23C22×D411yescharacteristic
159C23C22×D412yes—
160C23C22×Q811yescharacteristic
161C23C2×#4111yescharacteristic
162C23C2×#4112yes—
163C23C2×#4112yes—
164C23C2×#4112yes—
165C23H(5,𝔽2)12yes—
166C232−1+412yes—
167C2×C4—232no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—28no—
173C2×C4—28no—
174C2×C4—28no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—28no—
178C2×C4—28no—
179C2×C4—24no—
180C2×C4—24no—
181C2×C4—432no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—44no—
185C2×C4—41no—
186C2×C4—44no—
187C2×C4—41no—
188C2×C4—44no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—44no—
197C2×C4—44no—
198C2×C4—44no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—44no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—44no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4#12014yes—
208C2×C4#12314yes—
209C22—24no—
210C22—24no—
211C22—24no—
212C22—24no—
213C22—216no—
214C22—216no—
215C22—24no—
216C22—24no—
217C22#40018yescomplemented
218C22C2×#12011yescharacteristic
219C22C2×C22⋊Q811yescharacteristic
220C22C2×#12311yescharacteristic
221C22C2×#12311yescharacteristic
222C22#52514yes—
223C22C2×#12411yescharacteristic
224C22C2×#12512yes—
225C22C2×#12612yes—
226C22#53511yescharacteristic
227C22#53611yescharacteristic
228C22#53914yes—
229C22#54614yes—
230C22#54812yes—
231C22D4×Q812yes—
232C22#55214yes—
233C22#55312yes—
234C22#55412yes—
235C4—24no—
236C4—24no—
237C4—42no—
238C4—48no—
239C4—42no—
240C4—81no—
241C4—81no—
242C4—84no—
243C2—28no—
244C2C2×#40011yescharacteristic
245C2#240911yescharacteristic
246C2#243511yescharacteristic
247C2#247314yes—
248C2#248114yes—
249C2#248711yescharacteristic
250C2#250311yescharacteristic
251C2#262311yescharacteristic
252C2#284911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC25C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#400C2semidirect
4. #1445C2semidirect
5. #1442C2semidirect
6. #1441C2semidirect
217. C22#400semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊C42
C2×#4002
C2×#3852
#14452
#14422
#14412
C22⋊#952