Finite Group Databasebeta

← #25845

Subgroups — #25845

Order 256.

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

normal subgroups only

This group has 520 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×#554C211yesmaximal, characteristic
2C2×#501C211yesmaximal, characteristic, complemented
3C2×#478C211yesmaximal, characteristic
4#2584C211yesmaximal, characteristic, complemented
5#2059C212yesmaximal, complemented
6#2059C212yesmaximal, complemented
7#1988C211yesmaximal, characteristic, complemented
8#1985C211yesmaximal, characteristic, complemented
9#1864C212yesmaximal
10#1862C212yesmaximal
11#1762C211yesmaximal, characteristic
12#554—24no—
13#554—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×#125C2212yes—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8—22no—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4C2212yes—
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4—21no—
23#501—24no—
24#478—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2212yes—
27C2×#106C2211yescharacteristic
28C2×#106C2212yes—
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2212yes—
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×D4C2311yescharacteristic
48C23×C4—21no—
49C23×C4—22no—
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C22×C8C2312yes—
53C4⋊Q8—22no—
54C4⋊Q8D414yes—
55#125—24no—
56#125—24no—
57#125—22no—
58#125—22no—
59C22⋊Q8—28no—
60C22⋊Q8—216no—
61C4×Q8—22no—
62C4×Q8—22no—
63C4×D4—24no—
64C4×D4—24no—
65C4×D4—22no—
66C4×D4D414yes—
67C2×C4⋊C4—28no—
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×C22⋊C4—22no—
83C2×C22⋊C4—21no—
84C2×C42—21no—
85C2×C42C2312yes—
86C2×C42C2311yescharacteristic
87#107—22no—
88#107—24no—
89#106—22no—
90#106—24no—
91C4⋊C8—22no—
92#102—22no—
93#102—22no—
94#102Q818yescomplemented
95#97—22no—
96#95—21no—
97#95—22no—
98#95—22no—
99C24—21no—
100C2×Q8—22no—
101C2×Q8—24no—
102C2×Q8—42no—
103C2×Q8—81no—
104C2×Q8—81no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4C2×D412yes—
108C2×D4C2×Q814yes—
109C22×C4—22no—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—28no—
113C22×C4—24no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—44no—
117C22×C4—82no—
118C22×C4—82no—
119C22×C4C2×D412yes—
120C22×C4C2×D412yes—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2411yescharacteristic
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8C2×Q814yes—
131C4⋊C4—28no—
132C4⋊C4—28no—
133C4⋊C4—22no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—44no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—82no—
147C4⋊C4C2×D414yes—
148C4⋊C4C2×D412yes—
149C4⋊C4C2×Q814yes—
150C22⋊C4—28no—
151C22⋊C4—24no—
152C42—22no—
153C42—22no—
154C42—81no—
155C42C2×D412yes—
156C42C2×D412yes—
157C23—22no—
158C23—24no—
159C23—42no—
160C23—41no—
161C23—41no—
162C23C22×D411yescharacteristic
163Q8—44no—
164Q8—84no—
165D4—22no—
166D4—24no—
167D4C4⋊Q814yes—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—24no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—48no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—88no—
192C2×C4#12012yes—
193C2×C4#12012yes—
194C2×C4C22×D411yescharacteristic
195C2×C4C22×D411yescharacteristic
196C2×C4C22×Q812yes—
197C2×C42−1+412yes—
198C8—24no—
199C8—41no—
200C8—41no—
201C22—24no—
202C22—44no—
203C22—44no—
204C22—42no—
205C22—42no—
206C22C2×#12011yescharacteristic
207C22D4×Q812yes—
208C22#57512yes—
209C22#57611yescharacteristic
210C22#57711yescharacteristic
211C4—22no—
212C4—22no—
213C4—44no—
214C4—42no—
215C4—42no—
216C4—42no—
217C4—44no—
218C4—81no—
219C4—81no—
220C4C2×C4⋊Q811yescharacteristic
221C4#53611yescharacteristic
222C4D4×Q812yes—
223C2—44no—
224C2#243411yescharacteristic
225C2#300511yescharacteristic
226C2#301211yescharacteristic
227C2#338812yes—
228C2#338912yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
2. C2×#501C2semidirect
4. #2584C2semidirect
5. #2059C2semidirect
6. #2059C2semidirect
7. #1988C2semidirect
8. #1985C2semidirect
94. #102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#5012
C2×#4782
#25842
#20592
#19882
#19852
#18642
#18622
#17622