Finite Group Databasebeta

← #24079

Subgroups — #24079

Order 256.

515 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 123 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#3492C211yesmaximal, characteristic
2#3093C212yesmaximal, complemented
3#2317C211yesmaximal, characteristic, complemented
4#2061C211yesmaximal, characteristic, complemented
5#2060C211yesmaximal, characteristic, complemented
6#2040C211yesmaximal, characteristic, complemented
7#2039C211yesmaximal, characteristic, complemented
8#1897C212yesmaximal, complemented
9#1860C212yesmaximal, complemented
10#1859C212yesmaximal, complemented
11#1821C211yesmaximal, characteristic, complemented
12#554—22no—
13#553—22no—
14#552—22no—
15#547—22no—
16#544C2211yescharacteristic
17#539—22no—
18#537C2211yescharacteristic
19#525—22no—
20#525C2212yes—
21#520C2212yes—
22C2×#117C2211yescharacteristic
23C2×#117C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#485—22no—
26#485—22no—
27#482—22no—
28#482—22no—
29#428C2212yes—
30#428C2212yes—
31C2×#104C2212yes—
32C2×#104C2212yes—
33#417C2212yes—
34#417C2212yes—
35#407C2212yes—
36#389—22no—
37#385—22no—
38#384C2211yescharacteristic
39#384C2211yescharacteristic
40#382C2211yescharacteristic
41#382C2211yescharacteristic
42#343C2211yescharacteristic
43#343C2211yescharacteristic
44#343C2212yes—
45#339C2212yes—
46#339C2212yes—
47C2×#41—22no—
48C2×#41C2312yes—
49C23×C4D412yes—
50C23×C4C2311yescharacteristic
51C2×M16C2312yes—
52C2×M16C2312yes—
53C4⋊Q8—22no—
54#126—24no—
55#125—24no—
56#125—22no—
57#123—24no—
58#123—24no—
59#123—22no—
60C22⋊Q8—22no—
61C22⋊Q8—24no—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C22⋊Q8D412yes—
65C22⋊Q8D412yes—
66#121—22no—
67#121—22no—
68#121—22no—
69#121D412yes—
70#121D412yes—
71#120—22no—
72C4×Q8—22no—
73C4×D4—24no—
74C4×D4—22no—
75C4×D4—24no—
76C4×D4—22no—
77C4×D4—22no—
78#117—22no—
79#117—22no—
80#117—22no—
81#117—22no—
82#117—22no—
83#117—22no—
84#117C2312yes—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—22no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4D412yes—
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C42—21no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—22no—
102#107—22no—
103#106—22no—
104#106—22no—
105#104—24no—
106#104—24no—
107#103—22no—
108#103—22no—
109#102—22no—
110#102—22no—
111C22⋊C8—22no—
112C22⋊C8—22no—
113#95—22no—
114#95—22no—
115C24C2×D411yescharacteristic
116#41—22no—
117#41—22no—
118#41—22no—
119#41—22no—
120#41—84no—
121C2×Q8—22no—
122C2×Q8C2×D412yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4C2×D412yes—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—24no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—82no—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D412yes—
142C22×C4C2×D412yes—
143C22×C4C2×D412yes—
144C22×C4C2×D412yes—
145C22×C4C2411yescharacteristic
146M16—22no—
147M16—22no—
148M16—22no—
149M16—22no—
150C2×C8—22no—
151C2×C8—22no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—82no—
165C4⋊C4—82no—
166C4⋊C4C2×D412yes—
167C4⋊C4C2×D412yes—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—24no—
171C22⋊C4—22no—
172C22⋊C4—22no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—41no—
177C22⋊C4—41no—
178C42—24no—
179C42—22no—
180C42—21no—
181C42—21no—
182C42—42no—
183C42#4112yescomplemented
184C42#4112yescomplemented
185C23—21no—
186C23—21no—
187C23—22no—
188C23—41no—
189C23—41no—
190C23#12011yescharacteristic
191C23#12011yescharacteristic
192C23C22×D411yescharacteristic
193Q8—22no—
194Q8—22no—
195Q8—82no—
196D4—22no—
197D4—22no—
198D4—44no—
199D4—44no—
200D4—42no—
201D4—42no—
202D4—82no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—44no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—82no—
230C2×C4—84no—
231C2×C4#12012yes—
232C2×C4C22×D412yes—
233C2×C4C2×#4111yescharacteristic
234C2×C4C2×#4111yescharacteristic
235C2×C42−1+412yes—
236C8—42no—
237C8—42no—
238C22—21no—
239C22—21no—
240C22—21no—
241C22—21no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—42no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—42no—
251C22C2×#12011yescharacteristic
252C22#52511yescharacteristic
253C22#53611yescharacteristic
254C4—22no—
255C4—42no—
256C4—42no—
257C4—42no—
258C4—42no—
259C4—42no—
260C4—42no—
261C4—42no—
262C4—81no—
263C4—81no—
264C4#54712yes—
265C4#54712yes—
266C2—21no—
267C2—21no—
268C2—41no—
269C2—41no—
270C2—82no—
271C2#240611yescharacteristic
272C2#301911yescharacteristic
273C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3492C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3093C2semidirect
3. #2317C2semidirect
4. #2061C2semidirect
5. #2060C2semidirect
6. #2040C2semidirect
7. #2039C2semidirect
8. #1897C2semidirect
9. #1860C2semidirect
10. #1859C2semidirect
11. #1821C2semidirect
183. C42#41semidirect
184. C42#41semidirect

Maximal subgroups

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

subgroupindex
#34922
#30932
#23172
#20612
#20602
#20402
#20392
#18972
#18602
#18592
#18212