Finite Group Databasebeta

← #19245

Subgroups — #19245

Order 256.

303 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 303 conjugacy classes of subgroups — too many to draw. Only its 113 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#3246C211yesmaximal, characteristic
2#3209C211yesmaximal, characteristic, complemented
3#3141C211yesmaximal, characteristic
4#2962C211yesmaximal, characteristic, complemented
5#1702C212yesmaximal, complemented
6#1700C212yesmaximal, complemented
7#1682C212yesmaximal, complemented
8#1680C212yesmaximal, complemented
9#1565C211yesmaximal, characteristic, complemented
10#1540C211yesmaximal, characteristic
11C2×#332C211yesmaximal, characteristic, complemented
12#533C2211yescharacteristic
13#532C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15#494—21no—
16#489—22no—
17#487—21no—
18#487C2212yes—
19#486C2212yes—
20#476—21no—
21#475C2212yes—
22#465—22no—
23#462C2212yes—
24#462C2212yes—
25#461—21no—
26#460C2212yes—
27#451—22no—
28#446—22no—
29#446C2212yes—
30C8×Q8C2212yes—
31#431C2211yescharacteristic
32#424—21no—
33C2×C4⋊C8C2211yescharacteristic
34C2×C4⋊C8C2211yescharacteristic
35#417—21no—
36#417—21no—
37#405—21no—
38C4×M16C2211yescharacteristic
39#335C2212yes—
40#334C2212yes—
41#332C2212yes—
42#332C2212yes—
43#332C2211yescharacteristic
44#332C2211yescharacteristic
45#331C2211yescharacteristic
46#331C2211yescharacteristic
47C2×#41—21no—
48C22×Q8—21no—
49C2×Q16—22no—
50C2×Q16—22no—
51C2×SD16—22no—
52C2×M16D411yescharacteristic
53C2×M16D411yescharacteristic
54C22×C8—21no—
55C22×C8—21no—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8D411yescharacteristic
60C4⋊Q8D411yescharacteristic
61C4⋊Q8C2312yes—
62C4⋊Q8C2311yescharacteristic
63#127C2311yescharacteristic
64#125—21no—
65#124—21no—
66C22⋊Q8—21no—
67C22⋊Q8D412yes—
68#121—21no—
69#121—21no—
70C4×Q8—21no—
71C4×Q8C2312yes—
72C4×D4—21no—
73C4×D4—21no—
74#117—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C42C2311yescharacteristic
78#106—22no—
79C4⋊C8—22no—
80C4⋊C8—22no—
81C4⋊C8—21no—
82C4⋊C8—21no—
83C4⋊C8C2311yescharacteristic
84C4⋊C8C2311yescharacteristic
85C4⋊C8C2311yescharacteristic
86C4⋊C8C2311yescharacteristic
87C4⋊C8C2312yes—
88#103—22no—
89#103—22no—
90#103—22no—
91#103—22no—
92#103—22no—
93#102—22no—
94#102—22no—
95#102—22no—
96C22⋊C8—21no—
97#97—22no—
98#97D412yes—
99C4×C8—22no—
100C4×C8—21no—
101C4×C8C2312yes—
102#41—42no—
103#41—42no—
104C2×Q8—21no—
105C2×Q8—21no—
106C2×Q8—22no—
107C2×Q8—41no—
108C2×Q8—81no—
109C2×Q8—81no—
110C2×Q8C2×D412yes—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—41no—
115C22×C4—41no—
116C22×C4—41no—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4#4111yescharacteristic
120Q16—82no—
121Q16—82no—
122SD16—82no—
123M16—22no—
124M16—22no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—42no—
131C2×C8—81no—
132C2×C8C2×D412yes—
133C2×C8C2×D412yes—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—21no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4C2×D412yes—
146C22⋊C4—22no—
147C22⋊C4—41no—
148C22⋊C4—41no—
149C42—22no—
150C42C2×D411yescharacteristic
151C42C2×D411yescharacteristic
152C42#4111yescharacteristic
153C42C2411yescharacteristic
154C23—41no—
155C23#12111yescharacteristic
156Q8—22no—
157Q8—22no—
158Q8—41no—
159Q8—41no—
160Q8—42no—
161Q8—42no—
162Q8—81no—
163Q8—82no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—41no—
168D4—81no—
169D4—82no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4#12111yescharacteristic
180C2×C4#12111yescharacteristic
181C2×C4#12111yescharacteristic
182C2×C4#13512yes—
183C2×C4#13712yes—
184C2×C4C22×D411yescharacteristic
185C2×C4C22×D411yescharacteristic
186C2×C4C2×#4111yescharacteristic
187C8—22no—
188C8—22no—
189C8—41no—
190C8—41no—
191C8—42no—
192C8—41no—
193C8—41no—
194C22—21no—
195C22—21no—
196C22—81no—
197C22—81no—
198C22—81no—
199C22#46412yes—
200C22C2×#12111yescharacteristic
201C4—21no—
202C4—21no—
203C4—21no—
204C4—42no—
205C4—42no—
206C4—81no—
207C4—82no—
208C4#46412yes—
209C4C2×#13511yescharacteristic
210C4C2×#13711yescharacteristic
211C4#57611yescharacteristic
212C4#57711yescharacteristic
213C2—22no—
214C2—161no—
215C2C2×#46411yescharacteristic
216C2#306211yescharacteristic
217C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#532C22
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
2. #3209C2semidirect
4. #2962C2semidirect
5. #1702C2semidirect
6. #1700C2semidirect
7. #1682C2semidirect
8. #1680C2semidirect
9. #1565C2semidirect
11. C2×#332C2semidirect

Maximal subgroups

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

subgroupindex
#32462
#32092
#31412
#29622
#17022
#17002
#16822
#16802
#15652
#15402
C2×#3322