Finite Group Databasebeta

← #9343

Subgroups — #9343

Order 256.

288 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 59 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#3192C211yesmaximal, characteristic
2#2377C211yesmaximal, characteristic, complemented
3#1964C211yesmaximal, characteristic, complemented
4#1510C211yesmaximal, characteristic, complemented
5#1507C211yesmaximal, characteristic, complemented
6#1305C211yesmaximal, characteristic, complemented
7#1284C211yesmaximal, characteristic, complemented
8C2×#121C2211yescharacteristic
9C2×#121C2211yescharacteristic
10#481—21no—
11#481C412yescomplemented
12#469—22no—
13#447—21no—
14#447C412yescomplemented
15C2×#99—21no—
16C2×#99—21no—
17#407C2211yescharacteristic
18#407C2211yescharacteristic
19#390—21no—
20#388—21no—
21#387—21no—
22#386—21no—
23#386—21no—
24#385—21no—
25#381C2211yescharacteristic
26#380C2211yescharacteristic
27C4×C22⋊C4C2211yescharacteristic
28#327—22no—
29#327—21no—
30#327—21no—
31C22×D4—21no—
32C22×D4—21no—
33C22×D4C2×C411yescharacteristic, complemented
34C22×D4D411yescharacteristic
35C23×C4—21no—
36C23×C4C2311yescharacteristic
37C2×D8—22no—
38C2×M16—21no—
39C2×M16—21no—
40#121—21no—
41#121—21no—
42#121—21no—
43#121—21no—
44#121—21no—
45#121—21no—
46#121C2×C412yes—
47#121D412yes—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4C2×C411yescharacteristic, complemented
51C2×C4⋊C4D411yescharacteristic
52C2×C22⋊C4—21no—
53C2×C22⋊C4—21no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4D411yescharacteristic
60C2×C22⋊C4D411yescharacteristic
61C2×C42—21no—
62C2×C42—21no—
63#106—22no—
64#102—22no—
65#102—22no—
66#99—42no—
67#99—42no—
68C22⋊C8—21no—
69C22⋊C8—21no—
70C22⋊C8—21no—
71C22⋊C8C2×C412yescomplemented
72#95—21no—
73#95—21no—
74#95—21no—
75#95—21no—
76#95—21no—
77#95D411yescharacteristic
78#95D411yescharacteristic
79C24—21no—
80C24—21no—
81C24C2×D411yescharacteristic
82C2×D4—22no—
83C2×D4—22no—
84C2×D4—21no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—41no—
88C2×D4—41no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—41no—
93C2×D4—81no—
94C2×D4—81no—
95C2×D4—81no—
96C2×D4—81no—
97C2×D4C22⋊C412yes—
98C22×C4—21no—
99C22×C4—21no—
100C22×C4—21no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—41no—
105C22×C4—81no—
106C22×C4—81no—
107C22×C4—81no—
108C22×C4—81no—
109C22×C4C22×C411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4#4111yescharacteristic
114C22×C4#4111yescharacteristic
115D8—82no—
116M16—81no—
117M16—81no—
118C2×C8—22no—
119C2×C8—41no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—81no—
124C4⋊C4—81no—
125C4⋊C4C22⋊C412yes—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—22no—
129C22⋊C4—41no—
130C22⋊C4—41no—
131C22⋊C4—41no—
132C22⋊C4—41no—
133C22⋊C4—41no—
134C22⋊C4—81no—
135C22⋊C4—81no—
136C22⋊C4—81no—
137C22⋊C4—81no—
138C42—81no—
139C42—81no—
140C23—21no—
141C23—21no—
142C23—22no—
143C23—41no—
144C23—41no—
145C23—41no—
146C23—41no—
147C23—41no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—81no—
155C23—81no—
156C23C4×D411yescharacteristic
157C23#12011yescharacteristic
158C23#12111yescharacteristic
159D4—42no—
160D4—42no—
161D4—81no—
162D4—81no—
163D4—82no—
164D4—82no—
165D4—81no—
166D4—81no—
167D4—81no—
168D4—81no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—84no—
197C2×C4C2×C22⋊C411yescharacteristic
198C2×C4C4×D411yescharacteristic
199C2×C4#12111yescharacteristic
200C2×C4#12311yescharacteristic
201C8—42no—
202C8—81no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22—81no—
223C22—161no—
224C22—161no—
225C22#38611yescharacteristic
226C22#45311yescharacteristic
227C22#45411yescharacteristic
228C4—21no—
229C4—21no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—81no—
234C4—81no—
235C4—81no—
236C4—81no—
237C4—82no—
238C4—81no—
239C2—21no—
240C2—21no—
241C2—41no—
242C2—41no—
243C2—82no—
244C2—161no—
245C2#188311yescharacteristic
246C2#190411yescharacteristic
247C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#3192C2
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
2. #2377C2semidirect
3. #1964C2semidirect
4. #1510C2semidirect
5. #1507C2semidirect
6. #1305C2semidirect
7. #1284C2semidirect
11. #481C4semidirect
14. #447C4semidirect
33. C22×D4C2×C4semidirect
50. C2×C4⋊C4C2×C4semidirect
71. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#31922
#23772
#19642
#15102
#15072
#13052
#12842