Finite Group Databasebeta

← #18909

Subgroups — #18909

Order 256.

441 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 121 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#3507C211yesmaximal, characteristic
2#3230C211yesmaximal, characteristic, complemented
3#3206C211yesmaximal, characteristic
4C2×#475C211yesmaximal, characteristic, complemented
5#1650C214yesmaximal, complemented
6#1643C214yesmaximal, complemented
7#1542C211yesmaximal, characteristic, complemented
8#1480C211yesmaximal, characteristic
9#1479C211yesmaximal, characteristic, complemented
10#559—21no—
11#557—21no—
12#556—22no—
13#556C2212yes—
14#554C2212yes—
15#552—22no—
16#539—22no—
17#536—21no—
18#533C2211yescharacteristic
19#532C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21#529—22no—
22#520—21no—
23C2×C4×Q8C2211yescharacteristic
24#500C2212yes—
25#492—21no—
26#487—21no—
27#487C2212yes—
28#484—22no—
29#477—22no—
30#475—22no—
31#475C2214yes—
32#461C2214yes—
33#448—22no—
34#431C2211yescharacteristic
35#431C2211yescharacteristic
36C2×#106—21no—
37C2×C4⋊C8C2211yescharacteristic
38C2×#103—21no—
39#333C2212yes—
40#332C2212yes—
41Q8⋊C8C2214yes—
42D4⋊C8C2214yes—
43C2×#41—21no—
44C22×Q8—21no—
45C22×Q8D411yescharacteristic
46C2×SD16—24no—
47C22×C8—21no—
48C4⋊Q8—22no—
49C4⋊Q8—21no—
50C4⋊Q8—21no—
51C4⋊Q8—21no—
52C4⋊Q8D411yescharacteristic
53C4⋊Q8C2312yes—
54C4⋊Q8C2311yescharacteristic
55C4⋊Q8C2311yescharacteristic
56#127D411yescharacteristic
57#126—24no—
58#126—24no—
59#125—24no—
60#125—22no—
61#125—21no—
62#124—22no—
63#124—22no—
64#124—21no—
65#123—24no—
66#123—24no—
67C22⋊Q8—22no—
68C22⋊Q8—22no—
69C22⋊Q8—24no—
70C22⋊Q8—21no—
71C22⋊Q8D412yes—
72#121—21no—
73#121D412yes—
74C4×Q8—22no—
75C4×Q8—22no—
76C4×Q8—21no—
77C4×Q8—21no—
78C4×Q8—21no—
79C4×Q8D412yes—
80C4×Q8D412yes—
81C4×Q8C2312yes—
82C4×D4—24no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—21no—
86C4×D4C2312yes—
87#117—24no—
88#117—21no—
89#117—22no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4D411yescharacteristic
95C2×C42—21no—
96C2×C42C2311yescharacteristic
97#107—24no—
98#106—24no—
99C4⋊C8—21no—
100C4⋊C8—21no—
101C4⋊C8C2312yes—
102C4⋊C8C2312yes—
103#103—24no—
104#103—24no—
105#103—24no—
106C22⋊C8—21no—
107C22⋊C8—21no—
108C4×C8—21no—
109C4×C8C2312yes—
110#41—42no—
111#41—42no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—41no—
119C2×Q8—82no—
120C2×Q8C2×D412yes—
121C2×Q8C2×D412yes—
122C2×D4—22no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4C2×D412yes—
126C22×C4—21no—
127C22×C4—24no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—41no—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134SD16—84no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—42no—
139C4⋊C4—24no—
140C4⋊C4—24no—
141C4⋊C4—24no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4C2×D412yes—
156C4⋊C4C2×D412yes—
157C4⋊C4C2×D412yes—
158C22⋊C4—24no—
159C22⋊C4—24no—
160C22⋊C4—24no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C42—24no—
165C42—22no—
166C42—22no—
167C42—22no—
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C42C2×D411yescharacteristic
171C42C2411yescharacteristic
172C23—22no—
173C23#12011yescharacteristic
174Q8—22no—
175Q8—22no—
176Q8—42no—
177Q8—42no—
178Q8—42no—
179Q8—42no—
180Q8—44no—
181Q8—82no—
182D4—24no—
183D4—44no—
184D4—42no—
185D4—81no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—81no—
198C2×C4—84no—
199C2×C4—82no—
200C2×C4#12011yescharacteristic
201C2×C4#12011yescharacteristic
202C2×C4#12011yescharacteristic
203C2×C4#13512yes—
204C2×C4#13712yes—
205C2×C4C22×D411yescharacteristic
206C2×C4C22×D411yescharacteristic
207C2×C4C22×D411yescharacteristic
208C8—42no—
209C8—42no—
210C8—42no—
211C22—21no—
212C22—21no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22#45212yes—
217C22C2×#12011yescharacteristic
218C4—21no—
219C4—21no—
220C4—21no—
221C4—42no—
222C4—42no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4#45212yes—
228C4C2×#13511yescharacteristic
229C4C2×#13711yescharacteristic
230C4#57712yes—
231C2—22no—
232C2—82no—
233C2C2×#45211yescharacteristic
234C2#301011yescharacteristic
235C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3230C2semidirect
4. C2×#475C2semidirect
5. #1650C2semidirect
6. #1643C2semidirect
7. #1542C2semidirect
9. #1479C2semidirect

Maximal subgroups

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

subgroupindex
#35072
#32302
#32062
C2×#4752
#16502
#16432
#15422
#14802
#14792