Finite Group Databasebeta

← #18925

Subgroups — #18925

Order 256.

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

normal subgroups only

This group has 429 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#3502C211yesmaximal, characteristic
2#3230C211yesmaximal, characteristic
3#3209C211yesmaximal, characteristic
4C2×#475C211yesmaximal, characteristic
5#1648C214yesmaximal, complemented
6#1645C214yesmaximal, complemented
7#1542C211yesmaximal, characteristic
8#1481C211yesmaximal, characteristic
9#1479C211yesmaximal, characteristic
10Q82—21no—
11Q82C2212yes—
12#557—21no—
13#557—22no—
14#557—22no—
15#557C2212yes—
16#544—22no—
17#544—21no—
18#533—22no—
19#533C2211yescharacteristic
20#533C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22#519—21no—
23C2×C4×Q8C2211yescharacteristic
24#500C2212yes—
25#494—21no—
26#487—21no—
27#487C2212yes—
28#484—22no—
29#477—22no—
30#475—22no—
31#475C2214yes—
32#462C2214yes—
33#451—22no—
34#431C2211yescharacteristic
35#431C2211yescharacteristic
36C2×#106—21no—
37C2×C4⋊C8C2211yescharacteristic
38C2×#103—21no—
39#333C2212yes—
40#332C2212yes—
41Q8⋊C8C2214yes—
42Q8⋊C8C2214yes—
43C22×Q8—21no—
44C22×Q8D411yescharacteristic
45C2×Q16—24no—
46C22×C8—21no—
47C4⋊Q8—24no—
48C4⋊Q8—22no—
49C4⋊Q8—22no—
50C4⋊Q8—24no—
51C4⋊Q8—22no—
52C4⋊Q8—22no—
53C4⋊Q8—21no—
54C4⋊Q8—21no—
55C4⋊Q8—21no—
56C4⋊Q8D411yescharacteristic
57C4⋊Q8D411yescharacteristic
58C4⋊Q8C2312yes—
59C4⋊Q8C2311yescharacteristic
60C4⋊Q8C2311yescharacteristic
61#125—24no—
62#125—24no—
63#125—22no—
64#125—24no—
65#125—22no—
66#125—21no—
67#125—21no—
68C22⋊Q8—24no—
69C22⋊Q8—21no—
70C22⋊Q8—21no—
71C22⋊Q8D412yes—
72C22⋊Q8D412yes—
73C4×Q8—22no—
74C4×Q8—22no—
75C4×Q8—24no—
76C4×Q8—22no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×Q8—21no—
81C4×Q8—21no—
82C4×Q8—21no—
83C4×Q8—21no—
84C4×Q8D412yes—
85C4×Q8D412yes—
86C4×Q8C2312yes—
87C4×Q8C2312yes—
88#117—24no—
89#117—21no—
90#117—21no—
91C2×C4⋊C4—21no—
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—
110C2×Q8—24no—
111C2×Q8—21no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—41no—
117C2×Q8—82no—
118C2×Q8Q1614yescomplemented
119C2×Q8C2×D412yes—
120C2×Q8C2×D412yes—
121C2×Q8C2×D412yes—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—41no—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128Q16—84no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—42no—
133C4⋊C4—24no—
134C4⋊C4—24no—
135C4⋊C4—24no—
136C4⋊C4—24no—
137C4⋊C4—24no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4C2×D412yes—
156C4⋊C4C2×D412yes—
157C4⋊C4C2×D412yes—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C42—24no—
161C42—24no—
162C42—22no—
163C42—22no—
164C42—22no—
165C42—22no—
166C42C2×D411yescharacteristic
167C42C2×D411yescharacteristic
168C42C2×D411yescharacteristic
169C42C2411yescharacteristic
170C23#12011yescharacteristic
171Q8—22no—
172Q8—24no—
173Q8—24no—
174Q8—42no—
175Q8—42no—
176Q8—44no—
177Q8—81no—
178Q8—82no—
179Q8—84no—
180Q8C2×Q1614yes—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—81no—
192C2×C4—82no—
193C2×C4#12011yescharacteristic
194C2×C4#12011yescharacteristic
195C2×C4#12011yescharacteristic
196C2×C4C2×Q1612yes—
197C2×C4#13712yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C22×D411yescharacteristic
201C8—42no—
202C8—42no—
203C8—42no—
204C22—21no—
205C22—21no—
206C22#45112yes—
207C22C2×#12011yescharacteristic
208C4—24no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—42no—
213C4—42no—
214C4—42no—
215C4—42no—
216C4—42no—
217C4—82no—
218C4#45112yes—
219C4C22×Q1611yescharacteristic
220C4C2×#13711yescharacteristic
221C4#57712yes—
222C2—22no—
223C2C2×#45111yescharacteristic
224C2#301011yescharacteristic
225C2#302211yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC23#120
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
5. #1648C2semidirect
6. #1645C2semidirect
118. C2×Q8Q16semidirect

Maximal subgroups

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

subgroupindex
#35022
#32302
#32092
C2×#4752
#16482
#16452
#15422
#14812
#14792