Finite Group Databasebeta

← #28529

Subgroups — #28529

Order 256.

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

normal subgroups only

This group has 435 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×#498C211yesmaximal, characteristic, complemented
2C2×#477C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic
4C2×C4×SD16C211yesmaximal, characteristic
5#2568C211yesmaximal, characteristic, complemented
6#2563C211yesmaximal, characteristic
7#2097C212yesmaximal, complemented
8#2092C212yesmaximal
9#1968C212yesmaximal
10#1846C212yesmaximal, complemented
11C8×C4⋊C4C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16C2×C4×D4C2211yescharacteristic
17#498—24no—
18#477—24no—
19#474—24no—
20C4×SD16—24no—
21C2×#107C2212yes—
22C2×#107C2212yes—
23C2×#106C2212yes—
24C2×#106C2211yescharacteristic
25C2×#106C2212yes—
26C2×C4⋊C8C2211yescharacteristic
27C2×#103C2212yes—
28C2×#103C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2212yes—
31C2×C4×C8C2212yes—
32C2×C4×C8C2211yescharacteristic
33#400—22no—
34#393—22no—
35#391—22no—
36#391C2212yes—
37#390C2212yes—
38#382C2212yes—
39#382C2212yes—
40#380—22no—
41C4×C4⋊C4C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8—21no—
45C22×Q8C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C2×SD16—22no—
49C2×SD16—22no—
50C2×SD16Q814yescomplemented
51C22×C8C2311yescharacteristic
52C22×C8C2312yes—
53C22×C8C2311yescharacteristic
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4×Q8—22no—
59C4×Q8—22no—
60C4×D4—22no—
61C4×D4—22no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4D412yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C4⋊C4C2312yes—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—24no—
72C2×C42—21no—
73C2×C42C2312yes—
74C2×C42C2311yescharacteristic
75#107—24no—
76#107—24no—
77#106—22no—
78#106—24no—
79#106—24no—
80C4⋊C8D414yes—
81#103—22no—
82#103—24no—
83#102—22no—
84#102—24no—
85C4×C8—24no—
86C4×C8D414yes—
87#95—24no—
88#95—24no—
89#95—24no—
90#95D412yes—
91C24—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—42no—
98C2×Q8—42no—
99C2×Q8—44no—
100C2×Q8—82no—
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—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—42no—
114C22×C4—44no—
115C22×C4C2×D414yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2411yescharacteristic
119SD16—28no—
120C2×C8—22no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8C2×D412yes—
125C2×C8C2×D414yes—
126C2×C8C2×D412yes—
127C2×C8C2×Q812yes—
128C2×C8#4112yes—
129C4⋊C4—24no—
130C4⋊C4—21no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—42no—
139C4⋊C4—42no—
140C4⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—84no—
143C42—22no—
144C42—22no—
145C42—42no—
146C42C2×D412yes—
147C42C2×D412yes—
148C23—22no—
149C23—41no—
150C23—41no—
151C23—42no—
152C23—42no—
153C23C22×D411yescharacteristic
154Q8—22no—
155Q8—22no—
156Q8—41no—
157Q8—41no—
158Q8—84no—
159Q8—84no—
160D4—22no—
161D4—22no—
162D4—41no—
163D4—41no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—44no—
177C2×C4—44no—
178C2×C4—42no—
179C2×C4—84no—
180C2×C4#12712yes—
181C2×C4#12712yes—
182C2×C4#13514yes—
183C2×C4C22×D411yescharacteristic
184C2×C4C22×D411yescharacteristic
185C2×C4C22×Q811yescharacteristic
186C2×C4C2×#4111yescharacteristic
187C2×C42−1+411yescharacteristic
188C2×C42−1+411yescharacteristic
189C8—22no—
190C8—24no—
191C8C22⋊Q814yes—
192C22—44no—
193C22—42no—
194C22—82no—
195C22—81no—
196C22—81no—
197C22C2×#12711yescharacteristic
198C22#54711yescharacteristic
199C22D4×Q811yescharacteristic
200C22C2×#13512yes—
201C22#57712yes—
202C4—22no—
203C4—44no—
204C4—42no—
205C4—81no—
206C4—82no—
207C4—81no—
208C4—82no—
209C4#49514yes—
210C4C2×C22⋊Q811yescharacteristic
211C4#53611yescharacteristic
212C4#54711yescharacteristic
213C4D4×Q811yescharacteristic
214C2—82no—
215C2#254211yescharacteristic
216C2C2×#49511yescharacteristic
217C2#315411yescharacteristic
218C2#334012yes—
219C2#338112yes—
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
1. C2×#498C2semidirect
2. C2×#477C2semidirect
5. #2568C2semidirect
7. #2097C2semidirect
10. #1846C2semidirect
11. C8×C4⋊C4C2semidirect
50. C2×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#4982
C2×#4772
C2×#4742
C2×C4×SD162
#25682
#25632
#20972
#20922
#19682
#18462
C8×C4⋊C42