Finite Group Databasebeta

← #13101

Subgroups — #13101

Order 256.

315 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 61 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#3011C211yesmaximal, characteristic
2#2884C211yesmaximal, characteristic
3#2129C211yesmaximal, characteristic, complemented
4#2123C211yesmaximal, characteristic, complemented
5#1796C211yesmaximal, characteristic
6#1395C211yesmaximal, characteristic, complemented
7#1391C211yesmaximal, characteristic, complemented
8C2×2−1+4C2211yescharacteristic
9C2×#137C411yescharacteristic, complemented
10C22×Q16C411yescharacteristic, complemented
11#536C2211yescharacteristic
12#519—21no—
13#452—21no—
14#452—21no—
15#451—21no—
16#451—21no—
17#421—21no—
18#419—21no—
19#419C411yescharacteristic, complemented
20#416C411yescharacteristic, complemented
21#413C2211yescharacteristic
22#410—21no—
23#410—22no—
24#410—22no—
25#410C2211yescharacteristic
26#410C2211yescharacteristic
27#410C2211yescharacteristic
28#408C2211yescharacteristic
29#354—22no—
30#352—22no—
312−1+4—42no—
322−1+4—42no—
33C2×#41—21no—
34C2×#41—21no—
35C2×#41—21no—
36C2×#41C2×C411yescharacteristic
37C2×#41D411yescharacteristic
38C2×#41D411yescharacteristic
39C2×#41C2311yescharacteristic
40C22×Q8—21no—
41C22×Q8C2×C411yescharacteristic
42C22×Q8C2×C411yescharacteristic
43C22×Q8D411yescharacteristic
44C22×Q8D411yescharacteristic
45#137—42no—
46C2×Q16—21no—
47C2×Q16—21no—
48C2×Q16—21no—
49C2×Q16—42no—
50C2×SD16—21no—
51C2×M16—21no—
52C2×M16C2×C411yescharacteristic
53C22×C8C2×C411yescharacteristic
54C4⋊Q8—21no—
55#124—21no—
56#123—21no—
57#123—21no—
58C22⋊Q8—21no—
59C22⋊Q8—21no—
60C4×Q8—42no—
61C4×Q8—42no—
62#117—24no—
63#117—21no—
64#117—21no—
65#117—21no—
66#117C2×C411yescharacteristic
67#104—42no—
68#103—21no—
69#103—21no—
70#103—21no—
71#103—42no—
72#102—21no—
73#101—21no—
74#100—21no—
75#99—24no—
76#99—21no—
77#99—21no—
78#99—21no—
79#99—21no—
80#99D412yes—
81#99D412yes—
82C22⋊C8—21no—
83C22⋊C8—21no—
84#41—22no—
85#41—22no—
86#41—41no—
87#41—41no—
88#41—41no—
89#41—41no—
90#41—41no—
91#41—41no—
92#41—44no—
93#41—44no—
94#41—44no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—41no—
105C2×Q8—41no—
106C2×Q8—41no—
107C2×Q8—41no—
108C2×Q8—44no—
109C2×Q8C22⋊C411yescharacteristic
110C2×Q8C22⋊C411yescharacteristic
111C2×Q8C22⋊C411yescharacteristic
112C2×Q8#4111yescharacteristic
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4C22⋊C411yescharacteristic
117C2×D4C2×D411yescharacteristic
118C2×D4C2×D411yescharacteristic
119C2×D4#4111yescharacteristic
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4C22×C411yescharacteristic
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130Q16—42no—
131Q16—42no—
132Q16—42no—
133SD16—42no—
134M16—41no—
135M16—41no—
136M16—41no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—41no—
140C2×C8—41no—
141C4⋊C4—21no—
142C4⋊C4—44no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—41no—
147C4⋊C4—41no—
148C22⋊C4—21no—
149C22⋊C4—22no—
150C22⋊C4—22no—
151C22⋊C4—41no—
152C22⋊C4—41no—
153C22⋊C4—41no—
154C42—21no—
155C42—44no—
156C42—41no—
157C42—41no—
158C42—41no—
159C23—21no—
160C23C4×D411yescharacteristic
161C23#12011yescharacteristic
162C23#12111yescharacteristic
163Q8—24no—
164Q8—21no—
165Q8—21no—
166Q8—21no—
167Q8—21no—
168Q8—41no—
169Q8—42no—
170Q8—42no—
171Q8—41no—
172Q8—41no—
173Q8—81no—
174Q8#9912yes—
175Q8#9912yes—
176D4—21no—
177D4—21no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—42no—
182D4—42no—
183D4—42no—
184C2×C4—24no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4C2×C22⋊C411yescharacteristic
206C2×C4C4×D411yescharacteristic
207C2×C4#12111yescharacteristic
208C2×C4#12311yescharacteristic
209C8—41no—
210C8—41no—
211C8—81no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—81no—
220C22#38611yescharacteristic
221C4—21no—
222C4—22no—
223C4—22no—
224C4—21no—
225C4—21no—
226C4—41no—
227C4—81no—
228C4—82no—
229C4—82no—
230C4—81no—
231C4—161no—
232C4C2×#9911yescharacteristic
233C4C2×#9911yescharacteristic
234C2—21no—
235C2—41no—
236C2—41no—
237C2—41no—
238C2—81no—
239C2#189811yescharacteristic
subgroupG/H
center, socleC2#1898
commutatorC22×C4C22×C4
Frattini, cosocleC2×#41C23
prime radicalC2×2−1+4C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G
LECSG ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2129C2semidirect
4. #2123C2semidirect
6. #1395C2semidirect
7. #1391C2semidirect
9. C2×#137C4semidirect
10. C22×Q16C4semidirect
19. #419C4semidirect
20. #416C4semidirect

Maximal subgroups

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

subgroupindex
#30112
#28842
#21292
#21232
#17962
#13952
#13912