Finite Group Databasebeta

← #9277

Subgroups — #9277

Order 256.

324 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 69 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#3186C211yesmaximal, characteristic
2#2364C211yesmaximal, characteristic, complemented
3#1964C211yesmaximal, characteristic, complemented
4#1519C211yesmaximal, characteristic, complemented
5C2×#327C211yesmaximal, characteristic, complemented
6#1294C211yesmaximal, characteristic, complemented
7#1282C211yesmaximal, characteristic, complemented
8C2×#121C2211yescharacteristic
9C2×#121C2211yescharacteristic
10C2×C4×D4—21no—
11#480—21no—
12#480C412yescomplemented
13#466—22no—
14#447—21no—
15#447C412yescomplemented
16C2×#99—21no—
17C2×#99—21no—
18C2×C22⋊C8C2211yescharacteristic
19C2×C22⋊C8C2211yescharacteristic
20#388—21no—
21#387—21no—
22#386—21no—
23#386—21no—
24#380C2211yescharacteristic
25C4×C22⋊C4—21no—
26C4×C22⋊C4C2211yescharacteristic
27C2×#95C2211yescharacteristic
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×C4—21no—
37C23×C4C2311yescharacteristic
38C2×D8—22no—
39C22×C8—21no—
40C22×C8—21no—
41#121—21no—
42#121—21no—
43#121—21no—
44#121—21no—
45#121—21no—
46#121—21no—
47#121C2×C412yes—
48#121D412yes—
49C4×D4—42no—
50C4×D4—42no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4C2×C411yescharacteristic, complemented
54C2×C4⋊C4D411yescharacteristic
55C2×C22⋊C4—21no—
56C2×C22⋊C4—21no—
57C2×C22⋊C4—21no—
58C2×C22⋊C4—21no—
59C2×C22⋊C4—21no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4D411yescharacteristic
62C2×C22⋊C4D411yescharacteristic
63C2×C42—21no—
64C2×C42—21no—
65#107—22no—
66#102—22no—
67#102—22no—
68#99—42no—
69#99—42no—
70C22⋊C8—21no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8C2×C412yescomplemented
74#95—21no—
75#95—21no—
76#95—21no—
77#95—21no—
78#95—21no—
79#95D411yescharacteristic
80#95D411yescharacteristic
81C24—21no—
82C24—21no—
83C24C2×D411yescharacteristic
84C2×D4—22no—
85C2×D4—22no—
86C2×D4—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—41no—
90C2×D4—41no—
91C2×D4—41no—
92C2×D4—41no—
93C2×D4—41no—
94C2×D4—41no—
95C2×D4—41no—
96C2×D4—41no—
97C2×D4—41no—
98C2×D4—41no—
99C2×D4—81no—
100C2×D4—81no—
101C2×D4C22⋊C412yes—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—81no—
115C22×C4—81no—
116C22×C4—81no—
117C22×C4C22×C411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4C2×D411yescharacteristic
121C22×C4#4111yescharacteristic
122C22×C4#4111yescharacteristic
123D8—82no—
124C2×C8—22no—
125C2×C8—41no—
126C2×C8—81no—
127C2×C8—81no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—42no—
132C4⋊C4—81no—
133C4⋊C4C22⋊C412yes—
134C22⋊C4—21no—
135C22⋊C4—22no—
136C22⋊C4—22no—
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—42no—
140C22⋊C4—42no—
141C22⋊C4—41no—
142C22⋊C4—41no—
143C22⋊C4—41no—
144C22⋊C4—41no—
145C22⋊C4—81no—
146C22⋊C4D812yescomplemented
147C42—42no—
148C42—81no—
149C23—22no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—81no—
162C23—81no—
163C23C4×D411yescharacteristic
164C23#12011yescharacteristic
165C23#12111yescharacteristic
166C23C2×D811yescharacteristic
167C23#13511yescharacteristic
168C23AGL(1,ℤ/8)11yescharacteristic
169C23AGL(1,ℤ/8)11yescharacteristic
170D4—42no—
171D4—42no—
172D4—42no—
173D4—42no—
174D4—42no—
175D4—42no—
176D4—81no—
177D4—81no—
178D4—81no—
179D4—81no—
180D4—81no—
181D4—81no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
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—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—84no—
207C2×C4—82no—
208C2×C4—82no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4C2×C22⋊C411yescharacteristic
213C2×C4C4×D411yescharacteristic
214C2×C4#12111yescharacteristic
215C2×C4#12311yescharacteristic
216C8—42no—
217C8—81no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—41no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—161no—
239C22—161no—
240C22#38611yescharacteristic
241C22C4×D811yescharacteristic
242C22#44211yescharacteristic
243C22#44711yescharacteristic
244C22#44911yescharacteristic
245C22#45911yescharacteristic
246C22#46311yescharacteristic
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—42no—
253C4—42no—
254C4—81no—
255C4—81no—
256C4—81no—
257C4—161no—
258C2—21no—
259C2—21no—
260C2—21no—
261C2—21no—
262C2—21no—
263C2—21no—
264C2—82no—
265C2—161no—
266C2#187911yescharacteristic
267C2#190611yescharacteristic
268C2#191311yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radical#3186C2
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. #2364C2semidirect
3. #1964C2semidirect
4. #1519C2semidirect
5. C2×#327C2semidirect
6. #1294C2semidirect
7. #1282C2semidirect
12. #480C4semidirect
15. #447C4semidirect
33. C22×D4C2×C4semidirect
53. C2×C4⋊C4C2×C4semidirect
73. C22⋊C8C2×C4semidirect
146. C22⋊C4D8semidirect

Maximal subgroups

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

subgroupindex
#31862
#23642
#19642
#15192
C2×#3272
#12942
#12822