Finite Group Databasebeta

← #18613

Subgroups — #18613

Order 256.

421 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 421 conjugacy classes of subgroups — too many to draw. Only its 117 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#3237C211yesmaximal, characteristic
3#3185C211yesmaximal, characteristic
4#3104C211yesmaximal, characteristic
5#1653C212yesmaximal, complemented
6#1645C212yesmaximal, complemented
7#1636C212yesmaximal, complemented
8#1633C212yesmaximal, complemented
9#1554C211yesmaximal, characteristic
10#1491C211yesmaximal, characteristic
11#1485C211yesmaximal, characteristic
12Q82—21no—
13Q82C2212yes—
14#557—21no—
15#557—22no—
16#557—22no—
17#557C2212yes—
18#544—22no—
19#544—21no—
20#533—22no—
21#533C2211yescharacteristic
22#533C2211yescharacteristic
23#533C2211yescharacteristic
24#519—21no—
25C2×C4×Q8C2211yescharacteristic
26#501—21no—
27#498C2211yescharacteristic
28#494C2211yescharacteristic
29#491—21no—
30#487C2211yescharacteristic
31#487C2211yescharacteristic
32#485—22no—
33#484—21no—
34#477C2212yes—
35#477C2212yes—
36#475C2212yes—
37#470—22no—
38#462C2212yes—
39#451—21no—
40#441—22no—
41C4×Q16—22no—
42#432C2211yescharacteristic
43#431C2211yescharacteristic
44C4×M16C2211yescharacteristic
45#333C2212yes—
46#332C2212yes—
47Q8⋊C8C2212yes—
48Q8⋊C8C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51C22×Q8D411yescharacteristic
52C2×Q16—22no—
53C2×Q16—22no—
54C2×M16—21no—
55C4⋊Q8—24no—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58C4⋊Q8—24no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8D411yescharacteristic
62C4⋊Q8D411yescharacteristic
63C4⋊Q8C2311yescharacteristic
64C4⋊Q8C2311yescharacteristic
65C4⋊Q8C2311yescharacteristic
66C4⋊Q8C2311yescharacteristic
67#125—24no—
68#125—24no—
69#125—22no—
70#125—24no—
71#125—22no—
72#125—21no—
73#125—21no—
74#125—21no—
75C22⋊Q8—24no—
76C22⋊Q8—21no—
77C22⋊Q8—21no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8D412yes—
81C22⋊Q8D412yes—
82C4×Q8—22no—
83C4×Q8—22no—
84C4×Q8—24no—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8D412yes—
96C4×Q8D412yes—
97C4×Q8C2312yes—
98C4×Q8C2312yes—
99#117—24no—
100#117—21no—
101#117—21no—
102#117—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4D411yescharacteristic
105C2×C42—21no—
106C2×C42C2311yescharacteristic
107#107—22no—
108#107—22no—
109#106—22no—
110#106—22no—
111C4⋊C8C2312yes—
112C4⋊C8C2312yes—
113#103—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#103—22no—
118#103—22no—
119C22⋊C8—21no—
120C22⋊C8—21no—
121#97—21no—
122#97—21no—
123C4×C8—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126C2×Q8—24no—
127C2×Q8—24no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8C2×D412yes—
134C2×Q8C2×D412yes—
135C2×Q8C2×D412yes—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141Q16—44no—
142Q16—44no—
143M16—42no—
144C2×C8—21no—
145C2×C8—22no—
146C2×C8—21no—
147C2×C8—22no—
148C4⋊C4—24no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4C2×D412yes—
173C4⋊C4C2×D412yes—
174C4⋊C4C2×D412yes—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C22⋊C4—41no—
178C42—24no—
179C42—24no—
180C42—22no—
181C42—22no—
182C42—22no—
183C42—22no—
184C42—41no—
185C42C2×D411yescharacteristic
186C42C2×D411yescharacteristic
187C42C2×D411yescharacteristic
188C42C2411yescharacteristic
189C23#12011yescharacteristic
190Q8—24no—
191Q8—22no—
192Q8—24no—
193Q8—42no—
194Q8—42no—
195Q8—42no—
196Q8—42no—
197Q8—81no—
198Q8—84no—
199Q8#13714yes—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4#12011yescharacteristic
214C2×C4#12011yescharacteristic
215C2×C4#12011yescharacteristic
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D411yescharacteristic
218C2×C4C22×D411yescharacteristic
219C8—22no—
220C8—41no—
221C8—42no—
222C8—42no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22C2×#12011yescharacteristic
227C4—24no—
228C4—21no—
229C4—21no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—42no—
234C4—42no—
235C4—42no—
236C4—42no—
237C4—81no—
238C4—81no—
239C4C2×#13712yes—
240C4#57812yes—
241C2—41no—
242C2#300411yescharacteristic
243C2#301111yescharacteristic
244C2#301811yescharacteristic
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. #1653C2semidirect
6. #1645C2semidirect
7. #1636C2semidirect
8. #1633C2semidirect

Maximal subgroups

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

subgroupindex
#35022
#32372
#31852
#31042
#16532
#16452
#16362
#16332
#15542
#14912
#14852