Finite Group Databasebeta

← #25831

Subgroups — #25831

Order 256.

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

normal subgroups only

This group has 473 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
1#3504C211yesmaximal, characteristic
2#3166C211yesmaximal, characteristic, complemented
3#3080C211yesmaximal, characteristic
4#2859C211yesmaximal, characteristic, complemented
5#2061C212yesmaximal, complemented
6#2060C212yesmaximal, complemented
7#1998C211yesmaximal, characteristic, complemented
8#1994C211yesmaximal, characteristic, complemented
9#1860C212yesmaximal
10#1859C212yesmaximal
11C4×#104C211yesmaximal, characteristic
12#557—22no—
13#554—22no—
14#554—24no—
15#544—22no—
16#544C2212yes—
17#533—21no—
18#533C2211yescharacteristic
19C2×#125—21no—
20C2×#125C2211yescharacteristic
21C2×#125C2211yescharacteristic
22#520C2212yes—
23C4×#41C2211yescharacteristic
24#501—22no—
25#499—21no—
26#499—21no—
27#479—22no—
28#478—22no—
29#428C2212yes—
30#428C2212yes—
31#428C2212yes—
32#423C2211yescharacteristic
33C2×#104C2211yescharacteristic
34C2×#104C2211yescharacteristic
35#417C2212yes—
36C4×M16C2211yescharacteristic
37#389—24no—
38#384C2212yes—
39#382C2212yes—
40C43C2211yescharacteristic
41#343C2212yes—
42#343C2212yes—
43#339C2211yescharacteristic
44#339C2212yes—
45#339C2211yescharacteristic
46#339C2212yes—
47C2×#41C2311yescharacteristic
48C2×M16C2312yes—
49C2×M16C2311yescharacteristic
50C2×M16C2311yescharacteristic
51C4⋊Q8—24no—
52C4⋊Q8—22no—
53C4⋊Q8—22no—
54C4⋊Q8D412yes—
55#125—24no—
56#125—24no—
57#125—22no—
58#125—22no—
59#125—21no—
60#125—21no—
61#125—22no—
62#125—22no—
63#125—21no—
64#125—21no—
65#125D411yescharacteristic
66#125D411yescharacteristic
67C22⋊Q8—24no—
68C22⋊Q8—24no—
69C22⋊Q8—24no—
70C22⋊Q8—22no—
71C22⋊Q8—24no—
72C4×Q8—24no—
73C4×Q8—22no—
74C4×Q8—22no—
75C4×Q8D412yes—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—24no—
79C4×D4D412yes—
80#117—22no—
81#117—21no—
82#117C2311yescharacteristic
83#117C2312yes—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—24no—
88C2×C4⋊C4D414yes—
89C2×C4⋊C4C2312yes—
90C2×C4⋊C4C2312yes—
91C2×C42—21no—
92C2×C42—22no—
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95C2×C42C2311yescharacteristic
96#107—22no—
97#107—22no—
98#107—22no—
99#106—22no—
100#106—22no—
101#106—22no—
102C4⋊C8—22no—
103#104Q814yescomplemented
104#104Q814yescomplemented
105#103—22no—
106#102—22no—
107#97—21no—
108C4×C8—21no—
109#95—24no—
110#41—21no—
111#41—21no—
112#41C2×Q812yes—
113#41C2×Q812yes—
114C2×Q8—24no—
115C2×Q8—22no—
116C2×Q8C2×D411yescharacteristic
117C2×D4—21no—
118C2×D4C2×D411yescharacteristic
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4C2×D414yes—
126C22×C4C2×D412yes—
127C22×C4C2411yescharacteristic
128M16—21no—
129M16—21no—
130M16—22no—
131M16—22no—
132M16C2×Q812yes—
133M16C2×Q812yes—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—22no—
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—24no—
140C4⋊C4—24no—
141C4⋊C4—22no—
142C4⋊C4—24no—
143C4⋊C4—24no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—48no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4C2×D412yes—
155C4⋊C4C2×D412yes—
156C4⋊C4C2×D412yes—
157C22⋊C4—24no—
158C22⋊C4—22no—
159C22⋊C4—21no—
160C22⋊C4—22no—
161C42—24no—
162C42—22no—
163C42—21no—
164C42—22no—
165C42—21no—
166C42—21no—
167C42—44no—
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C42C2×D412yes—
171C42C2×Q812yes—
172C42C2×Q812yes—
173C23—21no—
174C23C22×D411yescharacteristic
175Q8—21no—
176Q8—82no—
177Q8—84no—
178Q8C4⋊Q812yes—
179D4—22no—
180D4—22no—
181D4—21no—
182D4C4⋊Q812yes—
183C2×C4—24no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—44no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—82no—
199C2×C4—84no—
200C2×C4#12012yes—
201C2×C4#12012yes—
202C2×C4C22×D411yescharacteristic
203C2×C4C22×D411yescharacteristic
204C2×C4C22×Q811yescharacteristic
205C2×C4C22×Q811yescharacteristic
206C2×C42−1+411yescharacteristic
207C2×C42−1+411yescharacteristic
208C8—22no—
209C8—22no—
210C8—42no—
211C22—22no—
212C22—22no—
213C22—42no—
214C22C2×#12011yescharacteristic
215C22D4×Q811yescharacteristic
216C22D4×Q811yescharacteristic
217C4—24no—
218C4—22no—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—44no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4C2×C4⋊Q811yescharacteristic
228C4#53611yescharacteristic
229C4D4×Q811yescharacteristic
230C4D4×Q811yescharacteristic
231C2—21no—
232C2—21no—
233C2—42no—
234C2#243411yescharacteristic
235C2#301812yes—
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3166C2semidirect
4. #2859C2semidirect
5. #2061C2semidirect
6. #2060C2semidirect
7. #1998C2semidirect
8. #1994C2semidirect
103. #104Q8semidirect
104. #104Q8semidirect

Maximal subgroups

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

subgroupindex
#35042
#31662
#30802
#28592
#20612
#20602
#19982
#19942
#18602
#18592
C4×#1042