Finite Group Databasebeta

← #24621

Subgroups — #24621

Order 256.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#3095C211yesmaximal, characteristic
2C2×#483C211yesmaximal, characteristic
3#2924C211yesmaximal, characteristic
4C2×#428C211yesmaximal, characteristic
5#2090C212yesmaximal, complemented
6#2064C212yesmaximal, complemented
7#1951C212yesmaximal, complemented
8#1930C212yesmaximal, complemented
9#1901C211yesmaximal, characteristic
10#1855C211yesmaximal, characteristic
11#1821C211yesmaximal, characteristic
12C22×M16C2211yescharacteristic
13C2×C22⋊Q8C2211yescharacteristic
14C2×#117C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16#485—21no—
17#485—21no—
18#484—21no—
19#483—21no—
20#483—21no—
21#483—21no—
22#483—22no—
23#433—22no—
24#432—21no—
25#431—21no—
26C2×#108C2212yes—
27#428—21no—
28#428—21no—
29#428C2212yes—
30#428C2211yescharacteristic
31#428C2211yescharacteristic
32#428C2212yes—
33C2×#107—21no—
34C2×#106—21no—
35C2×#106C2212yes—
36#424C2212yes—
37#419C2211yescharacteristic
38#419C2211yescharacteristic
39C2×#103C2212yes—
40C2×#101C2212yes—
41#407C2211yescharacteristic
42#407C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44#405C2212yes—
45#343C2212yes—
46#343C2212yes—
47#343C2212yes—
48#341C2212yes—
49C22×Q8C2311yescharacteristic
50C23×C4—21no—
51C23×C4C2311yescharacteristic
52C2×M16—21no—
53C2×M16—24no—
54C2×M16C2311yescharacteristic
55C2×M16C2312yes—
56C2×M16C2311yescharacteristic
57C2×M16C2312yes—
58C22×C8—21no—
59C22×C8C2312yes—
60C22⋊Q8—21no—
61C22⋊Q8—21no—
62C22⋊Q8—21no—
63C22⋊Q8—21no—
64#117—21no—
65#117—21no—
66#117D411yescharacteristic
67#117D411yescharacteristic
68#117C2311yescharacteristic
69#117C2311yescharacteristic
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4D411yescharacteristic
77C2×C4⋊C4D411yescharacteristic
78C2×C4⋊C4C2312yes—
79C2×C4⋊C4C2311yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C42—21no—
83#108—24no—
84#107—22no—
85#107—22no—
86#107—21no—
87#107—21no—
88#106—22no—
89#106—22no—
90#106—22no—
91#106—22no—
92#106—21no—
93#106—21no—
94C4⋊C8—22no—
95C4⋊C8—22no—
96#103—22no—
97#103—22no—
98#103—21no—
99#103—21no—
100#103—21no—
101#103—21no—
102#101—24no—
103C22⋊C8—21no—
104C22⋊C8—21no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107C22⋊C8D411yescharacteristic
108C22⋊C8D411yescharacteristic
109C22⋊C8D412yes—
110#97—22no—
111C4×C8—22no—
112C24#4111yescharacteristic
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—22no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—41no—
125C22×C4—41no—
126C22×C4—82no—
127C22×C4C2×D411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4#4111yescharacteristic
132C22×C4C2411yescharacteristic
133M16—24no—
134M16—22no—
135M16—22no—
136M16—22no—
137M16—22no—
138M16—22no—
139M16#4114yes—
140C2×C8—24no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8C2×D412yes—
148C2×C8C2×D412yes—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—21no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4C2×D411yescharacteristic
160C4⋊C4C2×D411yescharacteristic
161C4⋊C4C2×D411yescharacteristic
162C4⋊C4C2×D411yescharacteristic
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—21no—
173C23—22no—
174C23—22no—
175C23—41no—
176C23#12111yescharacteristic
177C23#12111yescharacteristic
178C23C2×#4111yescharacteristic
179Q8—41no—
180Q8—41no—
181Q8—42no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—42no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4—82no—
201C2×C4—81no—
202C2×C4—82no—
203C2×C4#12111yescharacteristic
204C2×C4#12111yescharacteristic
205C2×C4C22×D411yescharacteristic
206C2×C4C22×D411yescharacteristic
207C2×C4C2×#4112yes—
208C2×C42−1+412yes—
209C8—24no—
210C8—22no—
211C8—22no—
212C8—41no—
213C8—41no—
214C8—42no—
215C22—21no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—44no—
223C22#47112yes—
224C22C2×#12111yescharacteristic
225C22#55212yes—
226C4—21no—
227C4—21no—
228C4—41no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—81no—
233C4—82no—
234C4—81no—
235C4#52511yescharacteristic
236C4#53611yescharacteristic
237C4#54712yes—
238C2—22no—
239C2—22no—
240C2—41no—
241C2#241311yescharacteristic
242C2C2×#47111yescharacteristic
243C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radicalC24#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #2090C2semidirect
6. #2064C2semidirect
7. #1951C2semidirect
8. #1930C2semidirect

Maximal subgroups

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

subgroupindex
#30952
C2×#4832
#29242
C2×#4282
#20902
#20642
#19512
#19302
#19012
#18552
#18212