Finite Group Databasebeta

← #19471

Subgroups — #19471

Order 256.

294 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 109 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#3209C211yesmaximal, characteristic
2#3113C211yesmaximal, characteristic
3C2×#475C211yesmaximal, characteristic
4#2984C211yesmaximal, characteristic
5#1700C212yesmaximal, complemented
6#1690C212yesmaximal, complemented
7#1680C212yesmaximal, complemented
8#1670C212yesmaximal, complemented
9#1573C211yesmaximal, characteristic
10#1556C211yesmaximal, characteristic
11#1552C211yesmaximal, characteristic
12#533C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15#494C2212yes—
16#487C2212yes—
17#483—21no—
18#475—21no—
19#475—21no—
20#475C2212yes—
21#475C2212yes—
22#475C2211yescharacteristic
23#475C2211yescharacteristic
24#462—22no—
25#462C2211yescharacteristic
26#462C2211yescharacteristic
27#451—21no—
28#451—21no—
29#451—21no—
30#446—22no—
31#446C2212yes—
32C8×Q8C2212yes—
33#441—21no—
34#441—21no—
35#432—21no—
36#432C2211yescharacteristic
37#432C2211yescharacteristic
38#431—21no—
39#431C2211yescharacteristic
40C2×#106—21no—
41C2×C4⋊C8C2211yescharacteristic
42C2×#103—21no—
43#335C2212yes—
44#334C2212yes—
45#332C2212yes—
46#332C2212yes—
47#332C2212yes—
48#332C2212yes—
49C22×Q8—21no—
50C22×Q8D411yescharacteristic
51C2×Q16—21no—
52C2×Q16—21no—
53C2×Q16—22no—
54C2×Q16—22no—
55C22×C8—21no—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8C2312yes—
60C4⋊Q8C2311yescharacteristic
61C4⋊Q8C2311yescharacteristic
62#125—21no—
63C22⋊Q8—21no—
64C22⋊Q8—21no—
65C4×Q8—21no—
66C4×Q8—21no—
67C4×Q8—21no—
68C4×Q8—21no—
69C4×Q8D411yescharacteristic
70C4×Q8D411yescharacteristic
71C4×Q8C2311yescharacteristic
72C4×Q8C2311yescharacteristic
73#117—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4D411yescharacteristic
78C2×C42—21no—
79C2×C42C2311yescharacteristic
80#106—22no—
81#106—22no—
82#106—21no—
83#106—21no—
84C4⋊C8—22no—
85C4⋊C8—22no—
86C4⋊C8—21no—
87C4⋊C8D412yes—
88C4⋊C8C2312yes—
89C4⋊C8C2311yescharacteristic
90C4⋊C8C2312yes—
91C4⋊C8C2311yescharacteristic
92#103—22no—
93#103—22no—
94#103—21no—
95#103—21no—
96#103—22no—
97#103—22no—
98#103—21no—
99#103—21no—
100C22⋊C8—21no—
101C22⋊C8—21no—
102C22⋊C8D411yescharacteristic
103C22⋊C8D411yescharacteristic
104#97—22no—
105#97—21no—
106#97—21no—
107C4×C8—22no—
108C4×C8C2312yes—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—21no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—41no—
115C2×Q8—41no—
116C2×Q8—82no—
117C2×Q8C2×D411yescharacteristic
118C2×Q8C2×D411yescharacteristic
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—41no—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4#4111yescharacteristic
125Q16—42no—
126Q16—42no—
127Q16—82no—
128Q16—82no—
129C2×C8—22no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—22no—
133C2×C8—42no—
134C2×C8C2×D412yes—
135C2×C8C2×D412yes—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—41no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4C2×D411yescharacteristic
152C4⋊C4C2×D411yescharacteristic
153C22⋊C4—41no—
154C42—21no—
155C42—21no—
156C42—21no—
157C42—21no—
158C42—41no—
159C42C2×D411yescharacteristic
160C42C2×D411yescharacteristic
161C42#4111yescharacteristic
162C42C2411yescharacteristic
163C23#12111yescharacteristic
164Q8—21no—
165Q8—21no—
166Q8—21no—
167Q8—42no—
168Q8—41no—
169Q8—41no—
170Q8—42no—
171Q8—42no—
172Q8—42no—
173Q8—82no—
174Q8#13712yes—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—82no—
189C2×C4#12111yescharacteristic
190C2×C4#12111yescharacteristic
191C2×C4#12111yescharacteristic
192C2×C4C22×D411yescharacteristic
193C2×C4C22×D411yescharacteristic
194C2×C4C2×#4111yescharacteristic
195C8—22no—
196C8—22no—
197C8—41no—
198C8—42no—
199C8—41no—
200C8—42no—
201C22—21no—
202C22—21no—
203C22#47312yes—
204C22C2×#12111yescharacteristic
205C4—21no—
206C4—21no—
207C4—22no—
208C4—21no—
209C4—21no—
210C4—41no—
211C4—41no—
212C4—41no—
213C4—82no—
214C4—81no—
215C4—81no—
216C4C2×#13711yescharacteristic
217C4#57511yescharacteristic
218C4#57712yes—
219C2—22no—
220C2#304311yescharacteristic
221C2#305611yescharacteristic
222C2C2×#47311yescharacteristic
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. #1700C2semidirect
6. #1690C2semidirect
7. #1680C2semidirect
8. #1670C2semidirect

Maximal subgroups

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

subgroupindex
#32092
#31132
C2×#4752
#29842
#17002
#16902
#16802
#16702
#15732
#15562
#15522