Finite Group Databasebeta

← #28471

Subgroups — #28471

Order 256.

541 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 541 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×#497C212yesmaximal
2C2×#494C211yesmaximal, characteristic, complemented
3C2×#462C211yesmaximal, characteristic, complemented
4C2×#461C211yesmaximal, characteristic
5C2×C4×SD16C211yesmaximal, characteristic
6#2547C211yesmaximal, characteristic
7#2546C211yesmaximal, characteristic, complemented
8#2016C212yesmaximal, complemented
9#2015C212yesmaximal
10#1971C212yesmaximal, complemented
11#1770C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×Q16C2212yes—
14C22×Q16C2212yes—
15C22×SD16C2212yes—
16C22×SD16C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C4⋊Q8C2212yes—
20C2×#124C2212yes—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23#497—28no—
24#494—24no—
25#462—24no—
26#461—24no—
27C4×SD16—24no—
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#102C2211yescharacteristic
34C2×#97C2212yes—
35C2×C4×C8C2211yescharacteristic
36#398—22no—
37#397—22no—
38#391C2212yes—
39#391C2212yes—
40#389—22no—
41#380—22no—
42C4×C4⋊C4C2211yescharacteristic
43#336C2212yes—
44C22×Q8C2312yes—
45C22×Q8C2312yes—
46C22×Q8C2311yescharacteristic
47C22×D4C2311yescharacteristic
48C23×C4—21no—
49C2×Q16—24no—
50C2×Q16—24no—
51C2×Q16—24no—
52C2×Q16—24no—
53C2×Q16—24no—
54C2×Q16—24no—
55C2×Q16—42no—
56C2×Q16—42no—
57C2×Q16—42no—
58C2×SD16—24no—
59C2×SD16—22no—
60C2×SD16—24no—
61C2×SD16—22no—
62C2×SD16—42no—
63C2×SD16D414yes—
64C22×C8C2312yes—
65C22×C8C2311yescharacteristic
66C22×C8C2311yescharacteristic
67C4⋊Q8—24no—
68C4⋊Q8—22no—
69C4⋊Q8—22no—
70C4⋊Q8—22no—
71C4⋊Q8—22no—
72C4⋊Q8—42no—
73#124—24no—
74#124—42no—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×D4—22no—
78C4×D4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—24no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D412yes—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—24no—
90C2×C42—21no—
91C2×C42C2312yes—
92C2×C42C2311yescharacteristic
93#106—22no—
94C4⋊C8D414yes—
95#103—22no—
96#103—24no—
97#103—24no—
98#102—22no—
99#97—24no—
100C4×C8D414yes—
101#95—24no—
102#95—24no—
103#95D412yes—
104C24—21no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8C2×D412yes—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4C2×D412yes—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—42no—
135C22×C4—44no—
136C22×C4C2×D414yes—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2411yescharacteristic
140Q16—48no—
141Q16—48no—
142Q16—48no—
143SD16—28no—
144SD16—48no—
145C2×C8—22no—
146C2×C8—22no—
147C2×C8—21no—
148C2×C8—21no—
149C2×C8C2×D412yes—
150C2×C8C2×D414yes—
151C2×C8C2×D412yes—
152C2×C8C2×D412yes—
153C2×C8#4112yes—
154C4⋊C4—24no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—48no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C22⋊C4—48no—
166C22⋊C4—42no—
167C42—22no—
168C42—22no—
169C42—42no—
170C42C2×D412yes—
171C42C2×D412yes—
172C23—22no—
173C23—41no—
174C23—41no—
175C23—42no—
176C23—42no—
177C23C22×D411yescharacteristic
178Q8—22no—
179Q8—22no—
180Q8—41no—
181Q8—41no—
182Q8—42no—
183Q8—42no—
184Q8—42no—
185Q8—42no—
186Q8—42no—
187Q8—42no—
188Q8—42no—
189Q8—42no—
190D4—22no—
191D4—22no—
192D4—41no—
193D4—41no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—24no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—84no—
212C2×C4#12712yes—
213C2×C4#12712yes—
214C2×C4#13714yes—
215C2×C4C22×D411yescharacteristic
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D411yescharacteristic
218C2×C4C2×#4111yescharacteristic
219C2×C4H(5,𝔽2)11yescharacteristic
220C2×C4H(5,𝔽2)11yescharacteristic
221C8—22no—
222C8—24no—
223C8#12114yes—
224C22—44no—
225C22—42no—
226C22—82no—
227C22—81no—
228C22—81no—
229C22C2×#12711yescharacteristic
230C22D4211yescharacteristic
231C22#55011yescharacteristic
232C22C2×#13712yes—
233C22#57812yes—
234C4—22no—
235C4—44no—
236C4—42no—
237C4—81no—
238C4—81no—
239C4—82no—
240C4—82no—
241C4#49714yescomplemented
242C4C2×#12111yescharacteristic
243C4#53511yescharacteristic
244C4D4211yescharacteristic
245C4#55011yescharacteristic
246C2—82no—
247C2#254111yescharacteristic
248C2C2×#49711yescharacteristic
249C2#315611yescharacteristic
250C2#327612yes—
251C2#339812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#494C2semidirect
3. C2×#462C2semidirect
7. #2546C2semidirect
8. #2016C2semidirect
10. #1971C2semidirect
11. #1770C2semidirect
241. C4#497semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4942
C2×#4622
C2×#4612
C2×C4×SD162
#25472
#25462
#20162
#20152
#19712
#17702