Finite Group Databasebeta

← #18205

Subgroups — #18205

Order 256.

491 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 129 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#3501C211yesmaximal, characteristic
2C2×#484C212yesmaximal, complemented
3#2487C211yesmaximal, characteristic
4#1617C214yesmaximal, complemented
5#1614C214yesmaximal, complemented
6#1520C212yesmaximal, complemented
7#1459C211yesmaximal, characteristic
8#554—24no—
9#554—22no—
10D4×Q8—22no—
11#544C2212yes—
12#543—21no—
13#543C2212yes—
14#533—21no—
15C2×C22⋊Q8—21no—
16C2×C22⋊Q8C2212yes—
17#518—21no—
18C2×C4×D4C2211yescharacteristic
19C22×C4⋊C4C2211yescharacteristic
20#484—24no—
21#484C2218yes—
22C2×#107—22no—
23C2×#103—22no—
24C2×C22⋊C8C2212yes—
25#400C2212yes—
26#399C2212yes—
27#389—21no—
28#386—21no—
29#385—21no—
30#380—21no—
31#380C2211yescharacteristic
32#328C2218yes—
33#324C2212yes—
34#323C2212yes—
35C22×Q8—21no—
36C22×Q8D412yes—
37C22×D4D411yescharacteristic
38C23×C4—21no—
39C23×C4D411yescharacteristic
40C23×C4C2311yescharacteristic
41C22×C8—22no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8—24no—
45#125—22no—
46#125—24no—
47C22⋊Q8—22no—
48C22⋊Q8—22no—
49C22⋊Q8—24no—
50C22⋊Q8—24no—
51C22⋊Q8—24no—
52C22⋊Q8—22no—
53C22⋊Q8—22no—
54C22⋊Q8—22no—
55C22⋊Q8—22no—
56C22⋊Q8—24no—
57C22⋊Q8D414yes—
58C22⋊Q8C2314yes—
59C4×Q8—22no—
60C4×Q8—22no—
61C4×D4—24no—
62C4×D4—24no—
63C4×D4—22no—
64C4×D4—22no—
65#117—22no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4D412yes—
75C2×C4⋊C4C2312yes—
76C2×C4⋊C4C2311yescharacteristic
77C2×C22⋊C4—22no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4D411yescharacteristic
82C2×C22⋊C4C2311yescharacteristic
83C2×C42D411yescharacteristic
84#107—28no—
85#103—28no—
86C22⋊C8—22no—
87C22⋊C8C2314yes—
88#95—22no—
89#95—22no—
90#95C2312yes—
91C24C2×D411yescharacteristic
92C24C2×D411yescharacteristic
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—44no—
97C2×Q8—82no—
98C2×Q8C2×D414yes—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4Q1614yes—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—82no—
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D412yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D412yes—
120C22×C4C2411yescharacteristic
121C2×C8—24no—
122C2×C8—44no—
123C4⋊C4—22no—
124C4⋊C4—24no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—24no—
130C4⋊C4—22no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—24no—
134C4⋊C4—42no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—82no—
138C4⋊C4C2×D414yes—
139C22⋊C4—24no—
140C22⋊C4—24no—
141C22⋊C4—22no—
142C22⋊C4—24no—
143C22⋊C4—22no—
144C22⋊C4—42no—
145C22⋊C4—82no—
146C42—24no—
147C42—21no—
148C42—21no—
149C23—21no—
150C23—21no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23#12012yes—
156C23C2×Q1612yes—
157C23C2×Q1612yes—
158C23#13712yes—
159C23C22×D411yescharacteristic
160Q8—44no—
161Q8—44no—
162Q8—82no—
163Q8—82no—
164D4—28no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—21no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—81no—
185C2×C4—82no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—82no—
189C2×C4—84no—
190C2×C4#12011yescharacteristic
191C2×C4#12011yescharacteristic
192C2×C4C2×Q1612yes—
193C2×C4C22×D412yes—
194C8—44no—
195C22—21no—
196C22—22no—
197C22—21no—
198C22—22no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22#45114yes—
207C22C2×#12011yescharacteristic
208C22C22×Q1611yescharacteristic
209C22C2×#13711yescharacteristic
210C4—22no—
211C4—22no—
212C4—21no—
213C4—41no—
214C4—44no—
215C4—41no—
216C4—44no—
217C4—82no—
218C4—82no—
219C4#45812yes—
220C2—21no—
221C2—21no—
222C2—24no—
223C2—42no—
224C2C2×#45111yescharacteristic
225C2#302111yescharacteristic
226C2C2×#45811yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#484C2semidirect
4. #1617C2semidirect
5. #1614C2semidirect
6. #1520C2semidirect

Maximal subgroups

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

subgroupindex
#35012
C2×#4842
#24872
#16172
#16142
#15202
#14592