Finite Group Databasebeta

← #25666

Subgroups — #25666

Order 256.

546 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 127 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#3511C211yesmaximal, characteristic
2#3141C211yesmaximal, characteristic, complemented
3#3079C211yesmaximal, characteristic, complemented
4#2868C211yesmaximal, characteristic, complemented
5#2039C214yesmaximal, complemented
6#1997C211yesmaximal, characteristic, complemented
7#1962C211yesmaximal, characteristic, complemented
8#1859C214yesmaximal, complemented
9C4×#104C211yesmaximal, characteristic, complemented
10#556—21no—
11#555—21no—
12#554—21no—
13#553—22no—
14#548—22no—
15#547—21no—
16#542—22no—
17#537C2212yes—
18#533—21no—
19#532—21no—
20#532C2211yescharacteristic
21C2×C4⋊Q8—22no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24#520C2212yes—
25C4×#41C2211yescharacteristic
26#489—22no—
27#487—21no—
28#486—21no—
29#477—21no—
30#476—21no—
31#475—21no—
32#474—21no—
33#428C2212yes—
34#423C2211yescharacteristic
35C2×#104C2214yes—
36C2×#104C2212yes—
37#417C2212yes—
38#417C2214yes—
39C4×M16C2211yescharacteristic
40#384—22no—
41#384C2214yes—
42C43C2211yescharacteristic
43#343C2214yes—
44#339C2212yes—
45C2×#41—22no—
46C2×#41C2311yescharacteristic
47C2×#41C2312yes—
48C22×Q8—21no—
49C22×Q8D412yes—
50C2×M16C2312yes—
51C2×M16C2312yes—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8—21no—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8—48no—
61C4⋊Q8D411yescharacteristic
62#127D411yescharacteristic
63#126—24no—
64#125—22no—
65#124—22no—
66#124—24no—
67#124D412yes—
68#123—24no—
69C22⋊Q8—22no—
70C22⋊Q8—24no—
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73#121—22no—
74#121—22no—
75#121—22no—
76#121—21no—
77#121—21no—
78#121—24no—
79C4×Q8—22no—
80C4×Q8—22no—
81C4×Q8D411yescharacteristic
82C4×Q8D411yescharacteristic
83C4×D4—22no—
84C4×D4—22no—
85C4×D4—21no—
86C4×D4—21no—
87C4×D4—24no—
88C4×D4—21no—
89C4×D4—21no—
90C4×D4D411yescharacteristic
91C4×D4D411yescharacteristic
92#117—22no—
93#117—22no—
94#117—21no—
95#117C2311yescharacteristic
96C2×C4⋊C4—22no—
97C2×C4⋊C4—28no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4D412yes—
100C2×C4⋊C4C2312yes—
101C2×C4⋊C4C2312yes—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42C2312yes—
106C2×C42C2311yescharacteristic
107#107—22no—
108#106—22no—
109C4⋊C8—21no—
110C4⋊C8—21no—
111#104—28no—
112#104—24no—
113#103—22no—
114#103—24no—
115#102—22no—
116#102—24no—
117#97—21no—
118C4×C8—21no—
119#41—24no—
120#41—22no—
121#41—24no—
122#41—22no—
123#41—84no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—44no—
127C2×Q8—44no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8C2×D412yes—
131C2×Q8C2×D411yescharacteristic
132C2×D4—21no—
133C2×D4—22no—
134C2×D4—24no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4C2×D412yes—
138C2×D4C2×D411yescharacteristic
139C22×C4—22no—
140C22×C4—24no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D414yes—
149C22×C4C2411yescharacteristic
150M16—24no—
151M16—22no—
152M16—22no—
153C2×C8—22no—
154C2×C8—22no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—416no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4C2×D411yescharacteristic
171C4⋊C4C2×D411yescharacteristic
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—24no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C42—24no—
181C42—24no—
182C42—24no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—22no—
187C42C2×D411yescharacteristic
188C42C2×D411yescharacteristic
189C42C2×D411yescharacteristic
190C42C2×D411yescharacteristic
191C42#4114yescomplemented
192C23—21no—
193C23—22no—
194C23C22×D411yescharacteristic
195Q8—21no—
196Q8—21no—
197Q8—24no—
198Q8—44no—
199Q8—82no—
200Q8—82no—
201Q8—82no—
202D4—21no—
203D4—21no—
204D4—24no—
205D4—44no—
206D4—42no—
207D4—42no—
208D4—42no—
209D4—84no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—24no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—44no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4#12011yescharacteristic
230C2×C4#12011yescharacteristic
231C2×C4#12011yescharacteristic
232C2×C4#12011yescharacteristic
233C2×C4C22×D411yescharacteristic
234C2×C4C22×D411yescharacteristic
235C2×C4C2×#4112yes—
236C2×C42−1+412yes—
237C8—42no—
238C8—42no—
239C22—21no—
240C22—21no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—42no—
245C22—41no—
246C22—42no—
247C22C2×#12011yescharacteristic
248C22#54812yes—
249C4—28no—
250C4—21no—
251C4—21no—
252C4—42no—
253C4—41no—
254C4—41no—
255C4—44no—
256C4—41no—
257C4—82no—
258C4#45312yes—
259C4#45612yes—
260C4#53211yescharacteristic
261C4#53611yescharacteristic
262C4#54712yes—
263C2—22no—
264C2—81no—
265C2—82no—
266C2#243211yescharacteristic
267C2C2×#45311yescharacteristic
268C2C2×#45611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radical#3511C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3141C2semidirect
3. #3079C2semidirect
4. #2868C2semidirect
5. #2039C2semidirect
6. #1997C2semidirect
7. #1962C2semidirect
8. #1859C2semidirect
9. C4×#104C2semidirect
191. C42#41semidirect

Maximal subgroups

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

subgroupindex
#35112
#31412
#30792
#28682
#20392
#19972
#19622
#18592
C4×#1042