Finite Group Databasebeta

← #28456

Subgroups — #28456

Order 256.

524 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 143 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×#497C211yesmaximal, characteristic
2C2×#494C211yesmaximal, characteristic, complemented
3C2×#464C212yesmaximal
4C2×#462C212yesmaximal, complemented
5#2658C211yesmaximal, characteristic
6#2327C211yesmaximal, characteristic, complemented
7#2016C212yesmaximal, complemented
8#2015C212yesmaximal
9#1960C211yesmaximal, characteristic, complemented
10#1958C211yesmaximal, characteristic
11#1768C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×Q16C2212yes—
14C22×Q16C2212yes—
15C22×SD16C2212yes—
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×C4⋊Q8C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C4×Q8C2212yes—
24#497—24no—
25#494—24no—
26#464—28no—
27#462—28no—
28C2×C4⋊C8C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#102C2212yes—
33C2×#97C2211yescharacteristic
34C2×C4×C8C2211yescharacteristic
35#399—24no—
36#391C2212yes—
37#387—24no—
38#382C2211yescharacteristic
39#376C2211yescharacteristic
40#336C2212yes—
41C22×Q8C2311yescharacteristic
42C22×Q8C2311yescharacteristic
43C22×Q8C2311yescharacteristic
44C22×Q8C2312yes—
45C22×D4C2311yescharacteristic
46C2×Q16—24no—
47C2×Q16—24no—
48C2×Q16—24no—
49C2×Q16—24no—
50C2×Q16—24no—
51C2×Q16—24no—
52C2×Q16—42no—
53C2×Q16—42no—
54C2×Q16—42no—
55C2×SD16—24no—
56C2×SD16—24no—
57C2×SD16—42no—
58C22×C8C2312yes—
59C22×C8C2312yes—
60C4⋊Q8—24no—
61C4⋊Q8—22no—
62C4⋊Q8—22no—
63C4⋊Q8—22no—
64C4⋊Q8—42no—
65C4⋊Q8D414yes—
66#124—24no—
67#124—22no—
68#124—22no—
69#124—42no—
70C4×Q8—24no—
71C4×Q8—24no—
72C2×C4⋊C4—22no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4C2312yes—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C42—22no—
82C2×C42C2311yescharacteristic
83C2×C42C2311yescharacteristic
84C2×C42C2311yescharacteristic
85C4⋊C8D418yes—
86#103—24no—
87#103—24no—
88#103—24no—
89#102—24no—
90#97—22no—
91C4×C8—22no—
92#95—22no—
93#95—24no—
94#95D414yes—
95C24—21no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—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×Q8C2×D414yes—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—22no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4C2×D412yes—
124C22×C4C2×D412yes—
125C22×C4C2×D412yes—
126C22×C4C2411yescharacteristic
127Q16—48no—
128Q16—48no—
129Q16—48no—
130SD16—48no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8C2×D414yes—
136C2×C8C2×D414yes—
137C4⋊C4—28no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—44no—
141C4⋊C4—44no—
142C4⋊C4—44no—
143C4⋊C4—44no—
144C22⋊C4—44no—
145C22⋊C4—44no—
146C22⋊C4—44no—
147C42—22no—
148C42—44no—
149C42C2×D412yes—
150C42C2×D414yes—
151C42#4112yes—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—44no—
156C23C22×D411yescharacteristic
157Q8—24no—
158Q8—24no—
159Q8—42no—
160Q8—42no—
161Q8—42no—
162Q8—42no—
163Q8—42no—
164Q8—42no—
165Q8—42no—
166Q8—42no—
167D4—42no—
168D4—42no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—22no—
173C2×C4—24no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—44no—
187C2×C4#12114yes—
188C2×C4#12714yes—
189C2×C4C22×D411yescharacteristic
190C2×C4C22×D412yes—
191C2×C4C2×#4111yescharacteristic
192C2×C4H(5,𝔽2)12yes—
193C8—24no—
194C8—24no—
195C22—82no—
196C22—82no—
197C22—82no—
198C22—81no—
199C22C2×#12111yescharacteristic
200C22C2×#12711yescharacteristic
201C22#53511yescharacteristic
202C22#57814yes—
203C4—24no—
204C4—24no—
205C4—42no—
206C4—44no—
207C4—82no—
208C4—81no—
209C4—81no—
210C4—81no—
211C4D4212yes—
212C4#55012yes—
213C2—161no—
214C2#254111yescharacteristic
215C2#304412yes—
216C2#315612yes—
217C2#322312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#494C2semidirect
4. C2×#462C2semidirect
6. #2327C2semidirect
7. #2016C2semidirect
9. #1960C2semidirect
11. #1768C2semidirect

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×#4642
C2×#4622
#26582
#23272
#20162
#20152
#19602
#19582
#17682