Finite Group Databasebeta

← #18951

Subgroups — #18951

Order 256.

395 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 395 conjugacy classes of subgroups — too many to draw. Only its 113 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#3556C211yesmaximal, characteristic
2#3249C211yesmaximal, characteristic
3#3237C211yesmaximal, characteristic
4#3231C211yesmaximal, characteristic
5#1653C212yesmaximal, complemented
6#1648C212yesmaximal, complemented
7#1636C212yesmaximal, complemented
8#1636C212yesmaximal, complemented
9#1552C211yesmaximal, characteristic
10#1497C211yesmaximal, characteristic
11#1481C211yesmaximal, characteristic
12#564—22no—
13#557—21no—
14#557—21no—
15#557—21no—
16#557—21no—
17#557C2212yes—
18#557C2212yes—
19#541—22no—
20#541—22no—
21#541—22no—
22#533C2211yescharacteristic
23#533C2211yescharacteristic
24#533C2211yescharacteristic
25#533C2211yescharacteristic
26#501—21no—
27#500C2211yescharacteristic
28#498C2211yescharacteristic
29#491—21no—
30#487C2211yescharacteristic
31#487C2211yescharacteristic
32#485—22no—
33#485—22no—
34#479—22no—
35#477—21no—
36#477C2212yes—
37#477C2212yes—
38#475C2212yes—
39#467—22no—
40#462—21no—
41#462C2212yes—
42#432C2211yescharacteristic
43#431C2211yescharacteristic
44C2×C4⋊C8C2211yescharacteristic
45#332C2212yes—
46#332C2212yes—
47Q8⋊C8C2212yes—
48Q8⋊C8C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51C2×Q16—22no—
52C22×C8—21no—
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4⋊Q8—22no—
56C4⋊Q8D411yescharacteristic
57C4⋊Q8D411yescharacteristic
58C4⋊Q8D411yescharacteristic
59C4⋊Q8D411yescharacteristic
60C4⋊Q8C2311yescharacteristic
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2311yescharacteristic
63C4⋊Q8C2311yescharacteristic
64#126—24no—
65#126—24no—
66#125—22no—
67#125—24no—
68#125—24no—
69#125—24no—
70#125—24no—
71#125—22no—
72#125—22no—
73#125—24no—
74#125—24no—
75#125—21no—
76#125—21no—
77#125—21no—
78#125D412yes—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84C22⋊Q8—21no—
85C22⋊Q8D412yes—
86C22⋊Q8D412yes—
87C4×Q8—22no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—21no—
94C4×Q8—21no—
95C4×Q8—21no—
96C4×Q8—21no—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8C2312yes—
100C4×Q8C2312yes—
101#117—21no—
102#117—21no—
103#117—21no—
104#117D412yes—
105C2×C42C2311yescharacteristic
106#107—22no—
107#107—22no—
108#107—22no—
109#106—22no—
110#106—22no—
111C4⋊C8—21no—
112C4⋊C8—21no—
113C4⋊C8C2312yes—
114C4⋊C8C2312yes—
115#103—22no—
116#103—22no—
117#103—22no—
118#103—22no—
119#103—22no—
120#103—22no—
121C22⋊C8—21no—
122C22⋊C8—21no—
123#97—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8C2×D412yes—
133C2×Q8C2×D412yes—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137Q16—82no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—81no—
143C4⋊C4—24no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4C2×D412yes—
170C4⋊C4C2×D412yes—
171C4⋊C4C2×D412yes—
172C4⋊C4C2×D412yes—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—41no—
177C42—24no—
178C42—24no—
179C42—22no—
180C42—22no—
181C42—22no—
182C42—41no—
183C42C2×D411yescharacteristic
184C42C2×D411yescharacteristic
185C42C2×D411yescharacteristic
186C42C2411yescharacteristic
187C23#12011yescharacteristic
188Q8—22no—
189Q8—24no—
190Q8—22no—
191Q8—42no—
192Q8—42no—
193Q8—42no—
194Q8—41no—
195Q8—42no—
196Q8—41no—
197Q8—41no—
198Q8—41no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4#12011yescharacteristic
212C2×C4#12011yescharacteristic
213C2×C4#12011yescharacteristic
214C2×C4C22×D411yescharacteristic
215C2×C4C22×D411yescharacteristic
216C2×C4C22×D411yescharacteristic
217C8—41no—
218C8—41no—
219C8—42no—
220C8—42no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22C2×#12011yescharacteristic
225C4—21no—
226C4—21no—
227C4—21no—
228C4—21no—
229C4—42no—
230C4—42no—
231C4—42no—
232C4—42no—
233C4—42no—
234C4—42no—
235C4—81no—
236C4—81no—
237C4#57711yescharacteristic
238C4#57812yes—
239C4#57811yescharacteristic
240C2—41no—
241C2#301111yescharacteristic
242C2#301311yescharacteristic
243C2#302011yescharacteristic
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. #1648C2semidirect
7. #1636C2semidirect
8. #1636C2semidirect

Maximal subgroups

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

subgroupindex
#35562
#32492
#32372
#32312
#16532
#16482
#16362
#15522
#14972
#14812