Finite Group Databasebeta

← #12717

Subgroups — #12717

Order 256.

472 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 73 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#3499C211yesmaximal, characteristic
2#2132C211yesmaximal, characteristic, complemented
3#2126C211yesmaximal, characteristic, complemented
4#2123C212yesmaximal, complemented
5#1796C212yesmaximal, complemented
6C2×2−1+4C411yescharacteristic, complemented
7C2×2−1+4C2211yescharacteristic
8D4×Q8—22no—
9#547—22no—
10#547—24no—
11#536—21no—
12#536—21no—
13#536C411yescharacteristic, complemented
14#536C2211yescharacteristic
15#532—21no—
16#532C2211yescharacteristic
17C2×C4⋊Q8—21no—
18C2×C4⋊Q8C411yescharacteristic, complemented
19C4×#41C411yescharacteristic, complemented
20#421—22no—
21#419—22no—
22#413C2212yes—
23#410C2212yes—
24#356—21no—
25#355—21no—
26#354—21no—
27#353—21no—
28#352—24no—
292−1+4—42no—
302−1+4—42no—
312−1+4—42no—
322−1+4—42no—
33C2×#41—21no—
34C2×#41—21no—
35C2×#41—21no—
36C2×#41—22no—
37C2×#41—22no—
38C2×#41C2×C411yescharacteristic
39C2×#41C2×C411yescharacteristic
40C2×#41D412yes—
41C2×#41C2311yescharacteristic
42C22×Q8—21no—
43C22×Q8—21no—
44C22×Q8—21no—
45C22×Q8C2×C411yescharacteristic
46C22×Q8C2×C411yescharacteristic
47C22×Q8D412yes—
48C2×M16—22no—
49C4⋊Q8—24no—
50C4⋊Q8—21no—
51C4⋊Q8—22no—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8—21no—
55C4⋊Q8D411yescharacteristic
56#127D411yescharacteristic
57#124—21no—
58#124—21no—
59#124D411yescharacteristic
60#124D411yescharacteristic
61#123—24no—
62#123—24no—
63#123—22no—
64#123D414yes—
65C22⋊Q8—24no—
66C22⋊Q8—24no—
67C22⋊Q8—22no—
68C22⋊Q8—22no—
69#121—24no—
70#121—22no—
71C4×Q8—21no—
72C4×D4—24no—
73C4×D4—21no—
74C4×D4—22no—
75C4×D4—22no—
76#117—21no—
77#117—22no—
78#117C2×C411yescharacteristic
79C2×C4⋊C4—24no—
80C2×C4⋊C4—22no—
81C2×C42—21no—
82C2×C42C2×C411yescharacteristic
83#104—44no—
84#103—44no—
85#101—22no—
86#100—22no—
87#99—22no—
88#99—22no—
89#41—22no—
90#41—22no—
91#41—24no—
92#41—41no—
93#41—41no—
94#41—42no—
95#41—44no—
96#41—44no—
97#41—41no—
98#41—41no—
99#41—48no—
100#41—48no—
101#41—42no—
102#41—42no—
103C2×Q8—22no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—24no—
109C2×Q8—41no—
110C2×Q8—41no—
111C2×Q8—42no—
112C2×Q8—41no—
113C2×Q8—41no—
114C2×Q8—44no—
115C2×Q8—44no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8C22⋊C411yescharacteristic
119C2×Q8C22⋊C412yes—
120C2×Q8C22⋊C411yescharacteristic
121C2×Q8C2×D411yescharacteristic
122C2×D4—21no—
123C2×D4—22no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4C22⋊C411yescharacteristic
128C2×D4C22⋊C411yescharacteristic
129C2×D4C2×D411yescharacteristic
130C2×D4C2×D412yes—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—22no—
140C22×C4C22⋊C412yes—
141C22×C4C22×C411yescharacteristic
142C22×C4C2×D412yes—
143M16—42no—
144M16—42no—
145C2×C8—42no—
146C4⋊C4—24no—
147C4⋊C4—24no—
148C4⋊C4—24no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—22no—
152C4⋊C4—24no—
153C4⋊C4—42no—
154C22⋊C4—22no—
155C22⋊C4—24no—
156C22⋊C4—22no—
157C22⋊C4—24no—
158C22⋊C4—22no—
159C22⋊C4—42no—
160C22⋊C4C22⋊C412yes—
161C42—22no—
162C42—21no—
163C42—21no—
164C42—42no—
165C42C22⋊C411yescharacteristic
166C42C22⋊C411yescharacteristic
167C23—21no—
168C23—21no—
169C23C2×C22⋊C411yescharacteristic
170C23#12012yes—
171Q8—21no—
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—44no—
176Q8—41no—
177Q8—42no—
178Q8—42no—
179Q8—44no—
180Q8—41no—
181Q8—41no—
182Q8#9912yes—
183D4—24no—
184D4—21no—
185D4—42no—
186D4—41no—
187D4—41no—
188D4—44no—
189D4—44no—
190D4—42no—
191D4—44no—
192D4#9912yes—
193C2×C4—24no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—44no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—44no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—44no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—82no—
216C2×C4C2×C22⋊C411yescharacteristic
217C2×C4C2×C22⋊C411yescharacteristic
218C2×C4#12012yes—
219C8—82no—
220C22—21no—
221C22—22no—
222C22—22no—
223C22—41no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—41no—
228C22#37911yescharacteristic
229C4—22no—
230C4—22no—
231C4—21no—
232C4—22no—
233C4—42no—
234C4—41no—
235C4—42no—
236C4—41no—
237C4—42no—
238C4—162no—
239C4C2×#9911yescharacteristic
240C4C2×#9911yescharacteristic
241C2—21no—
242C2—42no—
243C2—41no—
244C2—42no—
245C2—81no—
246C2#178111yescharacteristic
subgroupG/H
center, socleC2#1781
commutatorC22×C4C22×C4
Frattini, cosocleC2×#41C23
prime radical#3499C2
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
2. #2132C2semidirect
3. #2126C2semidirect
4. #2123C2semidirect
5. #1796C2semidirect
6. C2×2−1+4C4semidirect
13. #536C4semidirect
18. C2×C4⋊Q8C4semidirect
19. C4×#41C4semidirect

Maximal subgroups

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

subgroupindex
#34992
#21322
#21262
#21232
#17962