Finite Group Databasebeta

← #9258

Subgroups — #9258

Order 256.

354 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#451C211yesmaximal, characteristic
2Q8×C22⋊C4C211yesmaximal, characteristic
3#1996C211yesmaximal, characteristic
4#1787C211yesmaximal, characteristic
5#1520C211yesmaximal, characteristic
6#1289C211yesmaximal, characteristic
7#1282C211yesmaximal, characteristic
8C23×Q8C2211yescharacteristic
9C22×Q16C412yescomplemented
10C2×C22⋊Q8C2211yescharacteristic
11C2×C4×Q8—21no—
12#451—21no—
13#451—22no—
14#451—21no—
15C2×#103—22no—
16C2×#103C412yescomplemented
17C2×C22⋊C8C2211yescharacteristic
18C2×C22⋊C8C2211yescharacteristic
19#391—21no—
20#391—22no—
21#380—21no—
22#380C2211yescharacteristic
23#380C2211yescharacteristic
24C4×C22⋊C4—21no—
25C4×C22⋊C4C2211yescharacteristic
26#342—21no—
27#342—21no—
28#328—22no—
29C22×Q8—21no—
30C22×Q8—21no—
31C22×Q8—21no—
32C22×Q8—44no—
33C22×Q8C2×C412yes—
34C22×Q8C2×C411yescharacteristic
35C22×Q8D411yescharacteristic
36C22×Q8D412yes—
37C22×Q8D411yescharacteristic
38C23×C4—21no—
39C23×C4C2311yescharacteristic
40C2×Q16—22no—
41C2×Q16—22no—
42C2×Q16—44no—
43C22×C8—21no—
44C22×C8C2×C412yes—
45C22⋊Q8—21no—
46C22⋊Q8—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49C4×Q8—44no—
50C2×C4⋊C4—22no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4C2×C411yescharacteristic
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4D411yescharacteristic
59C2×C22⋊C4D411yescharacteristic
60C2×C42—22no—
61C2×C42—21no—
62#103—22no—
63#103—22no—
64#103—44no—
65C22⋊C8—21no—
66C22⋊C8—21no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69#95—22no—
70#95—22no—
71#95—21no—
72#95—21no—
73#95D412yes—
74C24C2×D411yescharacteristic
75C2×Q8—24no—
76C2×Q8—22no—
77C2×Q8—22no—
78C2×Q8—21no—
79C2×Q8—21no—
80C2×Q8—24no—
81C2×Q8—24no—
82C2×Q8—21no—
83C2×Q8—21no—
84C2×Q8—42no—
85C2×Q8—42no—
86C2×Q8—42no—
87C2×Q8—44no—
88C2×Q8—42no—
89C2×Q8—44no—
90C2×Q8—44no—
91C2×Q8C22⋊C412yes—
92C2×Q8C22⋊C412yes—
93C22×C4—21no—
94C22×C4—22no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4—22no—
98C22×C4—21no—
99C22×C4—22no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—42no—
103C22×C4—44no—
104C22×C4—41no—
105C22×C4C22×C411yescharacteristic
106C22×C4C2×D411yescharacteristic
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4#4111yescharacteristic
110C22×C4#4111yescharacteristic
111Q16—44no—
112Q16—44no—
113C2×C8—22no—
114C2×C8—42no—
115C2×C8—42no—
116C2×C8—41no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—44no—
120C4⋊C4—42no—
121C4⋊C4—42no—
122C4⋊C4—41no—
123C4⋊C4—41no—
124C22⋊C4—21no—
125C22⋊C4—21no—
126C22⋊C4—21no—
127C22⋊C4—41no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—41no—
131C22⋊C4—41no—
132C22⋊C4—41no—
133C22⋊C4Q1612yescomplemented
134C42—44no—
135C42—42no—
136C23—41no—
137C23—41no—
138C23C4×D411yescharacteristic
139C23#12011yescharacteristic
140C23#12111yescharacteristic
141C23C2×Q1611yescharacteristic
142C23#13511yescharacteristic
143C23#13711yescharacteristic
144C23#13711yescharacteristic
145Q8—24no—
146Q8—22no—
147Q8—22no—
148Q8—22no—
149Q8—44no—
150Q8—44no—
151Q8—42no—
152Q8—42no—
153Q8—81no—
154Q8#9914yes—
155C2×C4—24no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—44no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—41no—
171C2×C4—42no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4—81no—
183C2×C4C2×C22⋊C411yescharacteristic
184C2×C4C4×D411yescharacteristic
185C2×C4#12111yescharacteristic
186C2×C4#12311yescharacteristic
187C8—42no—
188C8—81no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—21no—
195C22—41no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22#38611yescharacteristic
200C22C4×Q1611yescharacteristic
201C22#44111yescharacteristic
202C22#45111yescharacteristic
203C22#45211yescharacteristic
204C22#46211yescharacteristic
205C22#46411yescharacteristic
206C4—21no—
207C4—24no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—42no—
212C4—81no—
213C4—82no—
214C4—81no—
215C4—81no—
216C4—161no—
217C4C2×#9912yes—
218C2—21no—
219C2—21no—
220C2—21no—
221C2—21no—
222C2—21no—
223C2—21no—
224C2#187811yescharacteristic
225C2#189811yescharacteristic
226C2#190511yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C22×Q16C4semidirect
16. C2×#103C4semidirect
133. C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4512
Q8×C22⋊C42
#19962
#17872
#15202
#12892
#12822