Finite Group Databasebeta

← #25846

Subgroups — #25846

Order 256.

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

normal subgroups only

This group has 482 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×#557C211yesmaximal, characteristic
2C2×#501C211yesmaximal, characteristic
3C2×#479C211yesmaximal, characteristic
4#2584C211yesmaximal, characteristic
5#2059C212yesmaximal
6#2059C212yesmaximal
7#1987C211yesmaximal, characteristic
8#1985C211yesmaximal, characteristic
9#1865C212yesmaximal
10#1863C212yesmaximal
11#1763C211yesmaximal, characteristic
12#557—24no—
13#557—24no—
14C2×C4⋊Q8—21no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125—22no—
17C2×#125C2211yescharacteristic
18C2×#125C2212yes—
19C2×C4×Q8—21no—
20C2×C4×Q8C2212yes—
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23#501—24no—
24#479—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2212yes—
27C2×#106C2211yescharacteristic
28C2×#106C2212yes—
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2212yes—
33C2×#97C2211yescharacteristic
34#398—21no—
35#395—21no—
36#389—22no—
37#389C2212yes—
38#389C2212yes—
39C4×C4⋊C4C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8D411yescharacteristic
47C22×Q8C2311yescharacteristic
48C22×C8C2311yescharacteristic
49C22×C8C2311yescharacteristic
50C22×C8C2312yes—
51C4⋊Q8—28no—
52C4⋊Q8—22no—
53C4⋊Q8D414yes—
54#125—216no—
55#125—24no—
56#125—24no—
57#125—22no—
58#125—22no—
59C4×Q8—28no—
60C4×Q8—24no—
61C4×Q8—24no—
62C4×Q8—22no—
63C4×Q8—22no—
64C4×Q8—22no—
65C4×Q8D414yes—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4D411yescharacteristic
76C2×C4⋊C4D412yes—
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C4⋊C4C2312yes—
81C2×C4⋊C4C2312yes—
82C2×C42—22no—
83C2×C42—21no—
84C2×C42—21no—
85C2×C42C2312yes—
86C2×C42C2311yescharacteristic
87#107—22no—
88#107—24no—
89#106—22no—
90#106—24no—
91C4⋊C8—22no—
92#103—22no—
93#103—22no—
94#103Q818yescomplemented
95#97—22no—
96#95—21no—
97#95—22no—
98#95—22no—
99C2×Q8—22no—
100C2×Q8—24no—
101C2×Q8—21no—
102C2×Q8—22no—
103C2×Q8—42no—
104C2×Q8—44no—
105C2×Q8C2×D412yes—
106C2×Q8C2×Q814yes—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4C2×D412yes—
114C22×C4C2×D412yes—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2411yescharacteristic
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8C2×Q814yes—
125C4⋊C4—28no—
126C4⋊C4—28no—
127C4⋊C4—28no—
128C4⋊C4—24no—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—24no—
132C4⋊C4—21no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—44no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—82no—
143C4⋊C4C2×D414yes—
144C4⋊C4C2×D412yes—
145C4⋊C4C2×Q814yes—
146C42—28no—
147C42—24no—
148C42—22no—
149C42—22no—
150C42—81no—
151C42C2×D412yes—
152C42C2×D412yes—
153C23C22×D411yescharacteristic
154Q8—22no—
155Q8—24no—
156Q8—44no—
157Q8—48no—
158Q8C4⋊Q814yes—
159C2×C4—22no—
160C2×C4—24no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—42no—
174C2×C4—44no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4#12012yes—
184C2×C4#12012yes—
185C2×C4C22×D411yescharacteristic
186C2×C4C22×D411yescharacteristic
187C2×C4C22×Q812yes—
188C2×C42−1+412yes—
189C8—24no—
190C8—41no—
191C8—41no—
192C22C2×#12011yescharacteristic
193C22D4×Q812yes—
194C22#57512yes—
195C22#57711yescharacteristic
196C22#57811yescharacteristic
197C4—24no—
198C4—22no—
199C4—22no—
200C4—44no—
201C4—42no—
202C4—44no—
203C4—42no—
204C4—42no—
205C4—81no—
206C4—81no—
207C4C2×C4⋊Q811yescharacteristic
208C4#53611yescharacteristic
209C4D4×Q812yes—
210C2#243411yescharacteristic
211C2#300511yescharacteristic
212C2#301311yescharacteristic
213C2#338812yes—
214C2#339012yes—
subgroupG/H
center, socle, prime radicalC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
94. #103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#5012
C2×#4792
#25842
#20592
#19872
#19852
#18652
#18632
#17632