Finite Group Databasebeta

← #8984

Subgroups — #8984

Order 256.

296 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 85 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×#474C211yesmaximal, characteristic
2#2354C211yesmaximal, characteristic
3#1990C211yesmaximal, characteristic, complemented
4#1981C211yesmaximal, characteristic, complemented
5C2×D4⋊C8C211yesmaximal, characteristic
6#1296C211yesmaximal, characteristic, complemented
7#1279C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10C2×#117—21no—
11#474—22no—
12#474C414yescomplemented
13C2×#107—21no—
14C2×C4⋊C8C2211yescharacteristic
15C2×C4⋊C8C2211yescharacteristic
16C2×#103—21no—
17C2×#103—21no—
18C2×#102—21no—
19C2×C4×C8C2211yescharacteristic
20#390—21no—
21#388—22no—
22#384C2211yescharacteristic
23#380—21no—
24C4×C4⋊C4—21no—
25C4×C4⋊C4C2211yescharacteristic
26#340—22no—
27D4⋊C8—22no—
28C22×Q8—21no—
29C22×D4D411yescharacteristic
30C23×C4—21no—
31C22×C8—21no—
32C22×C8—21no—
33C22×C8—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8C2×C412yes—
38C4×D4—21no—
39C4×D4—21no—
40C4×D4C2×C412yes—
41C4×D4D412yes—
42#117—42no—
43#117—42no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4D411yescharacteristic
49C2×C4⋊C4D411yescharacteristic
50C2×C4⋊C4D411yescharacteristic
51C2×C4⋊C4D412yes—
52C2×C22⋊C4—21no—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—22no—
55C2×C42—22no—
56C2×C42—21no—
57C2×C42C2311yescharacteristic
58#107—24no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8C2×C412yes—
63#103—42no—
64#103—42no—
65#102—24no—
66C4×C8—21no—
67C4×C8—21no—
68#95—22no—
69#95—21no—
70C24—21no—
71C2×Q8—22no—
72C2×Q8—41no—
73C2×Q8—41no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×Q8—41no—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4—21no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4C22⋊C412yes—
83C22×C4—21no—
84C22×C4—21no—
85C22×C4—22no—
86C22×C4—22no—
87C22×C4—21no—
88C22×C4—22no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—41no—
100C2×C8—41no—
101C2×C8—41no—
102C2×C8—41no—
103C2×C8—41no—
104C2×C8—41no—
105C2×C8—41no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—22no—
109C4⋊C4—22no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—42no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4C22⋊C412yes—
120C4⋊C4D812yes—
121C22⋊C4—22no—
122C22⋊C4—22no—
123C22⋊C4—44no—
124C22⋊C4—82no—
125C42—44no—
126C42—41no—
127C42—41no—
128C42C22×C411yescharacteristic
129C42C2×D411yescharacteristic
130C42#4111yescharacteristic
131C42#4111yescharacteristic
132C23—22no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23#12011yescharacteristic
140Q8—42no—
141Q8—42no—
142Q8—81no—
143Q8—81no—
144D4—22no—
145D4—22no—
146D4—41no—
147D4—41no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—41no—
166C2×C4—42no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4C2×C22⋊C411yescharacteristic
170C2×C4C4×D411yescharacteristic
171C2×C4C4×D411yescharacteristic
172C2×C4#12111yescharacteristic
173C2×C4#12111yescharacteristic
174C2×C4#12311yescharacteristic
175C2×C4C2×D811yescharacteristic
176C2×C4#13511yescharacteristic
177C2×C4#13511yescharacteristic
178C2×C4#13511yescharacteristic
179C2×C4AGL(1,ℤ/8)11yescharacteristic
180C2×C4AGL(1,ℤ/8)11yescharacteristic
181C2×C4#13711yescharacteristic
182C2×C4#13711yescharacteristic
183C8—42no—
184C8—42no—
185C8—81no—
186C22—42no—
187C22—42no—
188C22—42no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22#38611yescharacteristic
194C22#44711yescharacteristic
195C22#44911yescharacteristic
196C22#45211yescharacteristic
197C22#45211yescharacteristic
198C22#45511yescharacteristic
199C22#45611yescharacteristic
200C4—21no—
201C4—21no—
202C4—42no—
203C4—42no—
204C4—82no—
205C4—82no—
206C4—81no—
207C4#41611yescharacteristic
208C4#41911yescharacteristic
209C4C4×D811yescharacteristic
210C4#44211yescharacteristic
211C4#45911yescharacteristic
212C4#46311yescharacteristic
213C4#48511yescharacteristic
214C4#48511yescharacteristic
215C2—82no—
216C2#164111yescharacteristic
217C2#164611yescharacteristic
218C2#164711yescharacteristic
219C2#165211yescharacteristic
220C2#187311yescharacteristic
221C2#187911yescharacteristic
222C2#188211yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1990C2semidirect
4. #1981C2semidirect
6. #1296C2semidirect
7. #1279C2semidirect
12. #474C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4742
#23542
#19902
#19812
C2×D4⋊C82
#12962
#12792