Finite Group Databasebeta

← #19172

Subgroups — #19172

Order 256.

314 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 123 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#3165C211yesmaximal, characteristic
2#3104C211yesmaximal, characteristic
3C2×#462C211yesmaximal, characteristic
4Q8×M16C211yesmaximal, characteristic
5#1694C212yesmaximal, complemented
6#1692C212yesmaximal, complemented
7#1674C212yesmaximal, complemented
8#1672C212yesmaximal, complemented
9#1565C211yesmaximal, characteristic
10#1550C211yesmaximal, characteristic
11#1549C211yesmaximal, characteristic
12C22×Q16—21no—
13#533C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16#501—22no—
17#500C2212yes—
18#498C2212yes—
19#484—21no—
20#477C2211yescharacteristic
21#477C2211yescharacteristic
22#462—21no—
23#462—21no—
24#462C2211yescharacteristic
25#462C2211yescharacteristic
26#462C2212yes—
27#462C2212yes—
28#451—21no—
29#446—22no—
30#446C2212yes—
31C8×Q8C2212yes—
32C4×Q16—21no—
33C4×Q16—21no—
34#431C2211yescharacteristic
35#428—21no—
36#428—21no—
37#423—21no—
38#423C2211yescharacteristic
39C2×C4⋊C8C2211yescharacteristic
40C2×#103—21no—
41C4×M16—21no—
42C4×M16C2211yescharacteristic
43#335C2212yes—
44#334C2212yes—
45#333C2212yes—
46#332C2212yes—
47#332C2212yes—
48#332C2212yes—
49C22×Q8—21no—
50C22×Q8D411yescharacteristic
51C2×Q16—22no—
52C2×Q16—22no—
53C2×Q16—21no—
54C2×Q16—21no—
55C2×Q16—44no—
56C2×M16—21no—
57C2×M16D411yescharacteristic
58C2×M16D411yescharacteristic
59C22×C8—21no—
60C4⋊Q8—21no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8C2312yes—
64C4⋊Q8C2311yescharacteristic
65C4⋊Q8C2311yescharacteristic
66#125—21no—
67C22⋊Q8—21no—
68C22⋊Q8—21no—
69C4×Q8—21no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×Q8—21no—
73C4×Q8D411yescharacteristic
74C4×Q8D411yescharacteristic
75C4×Q8C2311yescharacteristic
76C4×Q8C2311yescharacteristic
77#117—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4D411yescharacteristic
82C2×C42—21no—
83C2×C42C2311yescharacteristic
84#107—22no—
85#107—22no—
86#107—21no—
87#107—21no—
88#106—22no—
89#106—22no—
90C4⋊C8—22no—
91C4⋊C8—22no—
92C4⋊C8—21no—
93C4⋊C8—21no—
94C4⋊C8C2312yes—
95C4⋊C8C2311yescharacteristic
96C4⋊C8C2311yescharacteristic
97C4⋊C8C2312yes—
98#103—21no—
99#103—21no—
100#103—22no—
101#103—22no—
102#103—21no—
103#103—21no—
104C22⋊C8—21no—
105#97—22no—
106#97D412yes—
107C4×C8—22no—
108C4×C8—21no—
109C4×C8C2312yes—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—21no—
115C2×Q8—41no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—41no—
119C2×Q8C2×D411yescharacteristic
120C2×Q8C2×D411yescharacteristic
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—41no—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4#4111yescharacteristic
127Q16—42no—
128Q16—44no—
129Q16—42no—
130Q16—44no—
131M16—22no—
132M16—42no—
133M16Q1614yescomplemented
134C2×C8—22no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—22no—
138C2×C8—42no—
139C2×C8C2×D412yes—
140C2×C8C2×D412yes—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—22no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4C2×D411yescharacteristic
157C4⋊C4C2×D411yescharacteristic
158C22⋊C4—41no—
159C42—21no—
160C42—21no—
161C42—21no—
162C42—21no—
163C42—41no—
164C42C2×D411yescharacteristic
165C42C2×D411yescharacteristic
166C42#4111yescharacteristic
167C42C2411yescharacteristic
168C23#12111yescharacteristic
169Q8—21no—
170Q8—21no—
171Q8—21no—
172Q8—42no—
173Q8—42no—
174Q8—41no—
175Q8—41no—
176Q8—42no—
177Q8—81no—
178Q8—82no—
179Q8AGL(1,ℤ/8)12yes—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—81no—
192C2×C4—82no—
193C2×C4#12111yescharacteristic
194C2×C4#12111yescharacteristic
195C2×C4#12111yescharacteristic
196C2×C4C2×Q1612yes—
197C2×C4#13712yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C8—22no—
202C8—42no—
203C8—41no—
204C8—42no—
205C8—41no—
206C8C2×Q1614yescomplemented
207C22—21no—
208C22—21no—
209C22#46212yes—
210C22C2×#12111yescharacteristic
211C4—21no—
212C4—22no—
213C4—21no—
214C4—21no—
215C4—41no—
216C4—41no—
217C4—41no—
218C4—82no—
219C4—81no—
220C4—81no—
221C4#46212yes—
222C4C22×Q1611yescharacteristic
223C4C2×AGL(1,ℤ/8)11yescharacteristic
224C4C2×#13711yescharacteristic
225C4#57511yescharacteristic
226C2—22no—
227C2C2×#46211yescharacteristic
228C2#305511yescharacteristic
229C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radicalC23#121
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. #1694C2semidirect
6. #1692C2semidirect
7. #1674C2semidirect
8. #1672C2semidirect
133. M16Q16semidirect
206. C8C2×Q16semidirect

Maximal subgroups

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

subgroupindex
#31652
#31042
C2×#4622
Q8×M162
#16942
#16922
#16742
#16722
#15652
#15502
#15492