Finite Group Databasebeta

← #18905

Subgroups — #18905

Order 256.

463 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 129 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#3501C211yesmaximal, characteristic
2#3230C211yesmaximal, characteristic, complemented
3#3209C211yesmaximal, characteristic, complemented
4C2×#474C211yesmaximal, characteristic
5#1649C214yesmaximal, complemented
6#1644C214yesmaximal, complemented
7#1542C211yesmaximal, characteristic, complemented
8#1481C211yesmaximal, characteristic, complemented
9#1478C211yesmaximal, characteristic
10#554—21no—
11#554—22no—
12#554—22no—
13#554C2212yes—
14D4×Q8—21no—
15D4×Q8C2212yes—
16#544—21no—
17#543—22no—
18#533C2211yescharacteristic
19#533C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×C22⋊Q8—22no—
22#518—21no—
23C2×C4×D4C2211yescharacteristic
24#500C2212yescomplemented
25#494—21no—
26#487—21no—
27#487C2212yescomplemented
28#484—22no—
29#477—22no—
30#474—22no—
31#474C2214yes—
32#462C2214yescomplemented
33#451—22no—
34#431C2211yescharacteristic
35#431C2211yescharacteristic
36C2×#107—21no—
37C2×C4⋊C8C2211yescharacteristic
38C2×#102—21no—
39#333C2212yescomplemented
40#332C2212yescomplemented
41Q8⋊C8C2214yescomplemented
42D4⋊C8C2214yes—
43C22×Q8—22no—
44C22×Q8—21no—
45C22×D4D411yescharacteristic
46C23×C4—21no—
47C2×Q16—24no—
48C22×C8—21no—
49C4⋊Q8—22no—
50C4⋊Q8—22no—
51C4⋊Q8—21no—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8D411yescharacteristic
55C4⋊Q8D411yescharacteristic
56C4⋊Q8C2312yes—
57C4⋊Q8C2311yescharacteristic
58C4⋊Q8C2311yescharacteristic
59#125—22no—
60#125—22no—
61#125—21no—
62#125—21no—
63C22⋊Q8—24no—
64C22⋊Q8—24no—
65C22⋊Q8—24no—
66C22⋊Q8—24no—
67C22⋊Q8—22no—
68C22⋊Q8—24no—
69C22⋊Q8—21no—
70C22⋊Q8—21no—
71C22⋊Q8—22no—
72C22⋊Q8—24no—
73C22⋊Q8D412yes—
74C22⋊Q8D412yes—
75C4×Q8—21no—
76C4×Q8—21no—
77C4×Q8D412yes—
78C4×Q8C2312yes—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—21no—
84C4×D4—21no—
85C4×D4D412yes—
86C4×D4C2312yes—
87#117—21no—
88#117—21no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4D411yescharacteristic
94C2×C22⋊C4—24no—
95C2×C22⋊C4—21no—
96C2×C42C2311yescharacteristic
97#107—24no—
98#107—24no—
99C4⋊C8—21no—
100C4⋊C8—21no—
101C4⋊C8C2312yes—
102C4⋊C8C2312yes—
103#103—24no—
104#103—24no—
105#102—24no—
106C22⋊C8—21no—
107C22⋊C8—21no—
108C4×C8—21no—
109C4×C8C2312yes—
110C24—21no—
111C2×Q8—24no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—22no—
115C2×Q8—41no—
116C2×Q8—82no—
117C2×Q8—84no—
118C2×Q8C2×D412yes—
119C2×Q8C2×D412yes—
120C2×D4—21no—
121C2×D4—22no—
122C2×D4Q1614yescomplemented
123C2×D4C2×D412yes—
124C22×C4—24no—
125C22×C4—24no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—41no—
130C22×C4—82no—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134Q16—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—22no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4C2×D412yes—
158C4⋊C4C2×D412yes—
159C4⋊C4C2×D412yes—
160C22⋊C4—24no—
161C22⋊C4—24no—
162C22⋊C4—22no—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C42—22no—
167C42—22no—
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C42C2×D411yescharacteristic
171C42C2411yescharacteristic
172C23—22no—
173C23—24no—
174C23—81no—
175C23—81no—
176C23#12011yescharacteristic
177Q8—24no—
178Q8—42no—
179Q8—42no—
180Q8—44no—
181Q8—81no—
182Q8—82no—
183Q8—84no—
184D4—22no—
185D4—24no—
186D4C2×Q1614yes—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—41no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4—84no—
199C2×C4—84no—
200C2×C4—82no—
201C2×C4#12011yescharacteristic
202C2×C4#12011yescharacteristic
203C2×C4#12011yescharacteristic
204C2×C4C2×Q1612yes—
205C2×C4#13712yes—
206C2×C4C22×D411yescharacteristic
207C2×C4C22×D411yescharacteristic
208C2×C4C22×D411yescharacteristic
209C8—42no—
210C8—42no—
211C8—42no—
212C22—21no—
213C22—21no—
214C22—24no—
215C22—42no—
216C22—42no—
217C22—84no—
218C22#45112yes—
219C22C2×#12011yescharacteristic
220C4—21no—
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—42no—
227C4—42no—
228C4—82no—
229C4#45112yes—
230C4C22×Q1611yescharacteristic
231C4C2×#13711yescharacteristic
232C4#57612yes—
233C2—22no—
234C2—44no—
235C2C2×#45111yescharacteristic
236C2#300811yescharacteristic
237C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC22×D4D4
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
3. #3209C2semidirect
5. #1649C2semidirect
6. #1644C2semidirect
7. #1542C2semidirect
8. #1481C2semidirect
24. #500C22semidirect
27. #487C22semidirect
32. #462C22semidirect
39. #333C22semidirect
40. #332C22semidirect
41. Q8⋊C8C22semidirect
122. C2×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#35012
#32302
#32092
C2×#4742
#16492
#16442
#15422
#14812
#14782