Finite Group Databasebeta

← #28510

Subgroups — #28510

Order 256.

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

normal subgroups only

This group has 441 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×#477C211yesmaximal, characteristic, complemented
2C2×#475C211yesmaximal, characteristic, complemented
3C2×#476C211yesmaximal, characteristic
4C2×#474C211yesmaximal, characteristic
5#2862C211yesmaximal, characteristic, complemented
6C4×#124C211yesmaximal, characteristic
7#2097C212yesmaximal, complemented
8#2092C212yesmaximal
9#1958C212yesmaximal
10#1846C212yesmaximal, complemented
11C4×C4⋊C8C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×#124C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17#477—24no—
18#476—24no—
19#475—24no—
20#474—24no—
21C2×#107C2212yes—
22C2×#107C2212yes—
23C2×#106C2212yes—
24C2×#106C2212yes—
25C2×C4⋊C8C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×C4×C8C2212yes—
32#391—22no—
33#391—22no—
34#391C2211yescharacteristic
35#390C2211yescharacteristic
36#388—22no—
37#382C2212yes—
38#382C2212yes—
39#382C2211yescharacteristic
40C4×C22⋊C4—22no—
41C43C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—21no—
45C22×Q8C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—22no—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54#124—22no—
55#124Q814yescomplemented
56C4×Q8—22no—
57C4×Q8—22no—
58C4×D4—22no—
59C4×D4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C22⋊C4—22no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C42—21no—
71C2×C42D412yes—
72C2×C42D412yes—
73C2×C42C2311yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76#107—24no—
77#107—24no—
78#106—24no—
79#106—24no—
80C4⋊C8D414yes—
81C4⋊C8D414yes—
82#103—24no—
83#103—24no—
84#102—24no—
85#102—24no—
86C4×C8—24no—
87#95—22no—
88#95—22no—
89#95—24no—
90#95—24no—
91C24—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—44no—
98C2×Q8—42no—
99C2×Q8—42no—
100C2×Q8—44no—
101C2×Q8C2×Q812yes—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C2×Q812yes—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—21no—
113C22×C4—42no—
114C22×C4—44no—
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2411yescharacteristic
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8C2×D414yes—
125C2×C8C2×D414yes—
126C4⋊C4—22no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—22no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C22⋊C4—28no—
138C22⋊C4—44no—
139C22⋊C4—42no—
140C42—24no—
141C42—21no—
142C42—21no—
143C42—42no—
144C42—42no—
145C42C2×D412yes—
146C42C2×D412yes—
147C42C2×Q812yes—
148C42#4112yes—
149C23—22no—
150C23—41no—
151C23—41no—
152C23—42no—
153C23—42no—
154C23C22×D411yescharacteristic
155Q8—22no—
156Q8—22no—
157Q8—41no—
158Q8—41no—
159Q8—84no—
160Q8—84no—
161D4—22no—
162D4—22no—
163D4—41no—
164D4—41no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—44no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—84no—
183C2×C4C22⋊Q814yes—
184C2×C4#12712yes—
185C2×C4#12712yes—
186C2×C4#13514yes—
187C2×C4C22×D411yescharacteristic
188C2×C4C22×D411yescharacteristic
189C2×C4C22×Q811yescharacteristic
190C2×C4C2×#4111yescharacteristic
191C2×C42−1+411yescharacteristic
192C2×C42−1+411yescharacteristic
193C8—24no—
194C8—24no—
195C22—44no—
196C22—42no—
197C22—81no—
198C22—81no—
199C22—82no—
200C22C2×C22⋊Q811yescharacteristic
201C22C2×#12711yescharacteristic
202C22#53611yescharacteristic
203C22C2×#13512yes—
204C22#57512yes—
205C4—24no—
206C4—22no—
207C4—42no—
208C4—42no—
209C4—81no—
210C4—82no—
211C4—81no—
212C4—82no—
213C4#49514yes—
214C4#54711yescharacteristic
215C4#54711yescharacteristic
216C4D4×Q811yescharacteristic
217C4D4×Q811yescharacteristic
218C2—82no—
219C2#254211yescharacteristic
220C2#307612yes—
221C2C2×#49511yescharacteristic
222C2#315311yescharacteristic
223C2#323312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#477C2semidirect
2. C2×#475C2semidirect
5. #2862C2semidirect
7. #2097C2semidirect
10. #1846C2semidirect
11. C4×C4⋊C8C2semidirect
55. #124Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×#4752
C2×#4762
C2×#4742
#28622
C4×#1242
#20972
#20922
#19582
#18462
C4×C4⋊C82