Finite Group Databasebeta

← #25667

Subgroups — #25667

Order 256.

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

normal subgroups only

This group has 510 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#2862C211yesmaximal, characteristic, complemented
5#2040C214yesmaximal, complemented
6#1997C211yesmaximal, characteristic, complemented
7#1962C211yesmaximal, characteristic, complemented
8#1860C214yesmaximal, 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⋊Q8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23#520C2212yes—
24C4×#41C2211yescharacteristic
25#489—22no—
26#487—21no—
27#486—21no—
28#477—21no—
29#476—21no—
30#475—21no—
31#474—21no—
32#428C2212yes—
33#423C2211yescharacteristic
34C2×#104C2214yes—
35C2×#104C2212yes—
36#417C2212yes—
37#417C2214yes—
38C4×M16C2211yescharacteristic
39#391—22no—
40#391—22no—
41#382C2214yes—
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⋊Q8D411yescharacteristic
61#127D411yescharacteristic
62#126—24no—
63#125—22no—
64#124—22no—
65#124—24no—
66#124D412yes—
67#123—24no—
68C22⋊Q8—22no—
69C22⋊Q8—24no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72#121—22no—
73#121—22no—
74#121—22no—
75#121—21no—
76#121—21no—
77#121—24no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×Q8D411yescharacteristic
81C4×Q8D411yescharacteristic
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—21no—
85C4×D4—21no—
86C4×D4—24no—
87C4×D4—21no—
88C4×D4—21no—
89C4×D4D411yescharacteristic
90C4×D4D411yescharacteristic
91#117—22no—
92#117—22no—
93#117—21no—
94#117C2311yescharacteristic
95C2×C4⋊C4—22no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4D412yes—
98C2×C4⋊C4C2312yes—
99C2×C4⋊C4C2312yes—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42C2312yes—
104C2×C42C2311yescharacteristic
105#107—22no—
106#106—22no—
107C4⋊C8—21no—
108C4⋊C8—21no—
109#104—28no—
110#104—24no—
111#103—22no—
112#103—24no—
113#102—22no—
114#102—24no—
115#97—21no—
116C4×C8—21no—
117#95—28no—
118#41—24no—
119#41—22no—
120#41—24no—
121#41—22no—
122#41—84no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—44no—
126C2×Q8—44no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8C2×D412yes—
130C2×Q8C2×D411yescharacteristic
131C2×D4—21no—
132C2×D4—22no—
133C2×D4—24no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4C2×D412yes—
137C2×D4C2×D411yescharacteristic
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D414yes—
148C22×C4C2411yescharacteristic
149M16—24no—
150M16—22no—
151M16—22no—
152C2×C8—22no—
153C2×C8—22no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4C2×D411yescharacteristic
169C4⋊C4C2×D411yescharacteristic
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—22no—
177C22⋊C4—22no—
178C42—24no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—22no—
183C42—42no—
184C42—42no—
185C42C2×D411yescharacteristic
186C42C2×D411yescharacteristic
187C42C2×D411yescharacteristic
188C42C2×D411yescharacteristic
189C42#4114yescomplemented
190C23—21no—
191C23—22no—
192C23C22×D411yescharacteristic
193Q8—21no—
194Q8—21no—
195Q8—24no—
196Q8—44no—
197Q8—82no—
198Q8—82no—
199Q8—82no—
200D4—21no—
201D4—21no—
202D4—24no—
203D4—44no—
204D4—42no—
205D4—42no—
206D4—42no—
207D4—84no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—24no—
215C2×C4—24no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—44no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—84no—
227C2×C4#12011yescharacteristic
228C2×C4#12011yescharacteristic
229C2×C4#12011yescharacteristic
230C2×C4#12011yescharacteristic
231C2×C4C22×D411yescharacteristic
232C2×C4C22×D411yescharacteristic
233C2×C4C2×#4112yes—
234C2×C42−1+412yes—
235C8—42no—
236C8—42no—
237C22—21no—
238C22—21no—
239C22—42no—
240C22—41no—
241C22—41no—
242C22—42no—
243C22—41no—
244C22—42no—
245C22C2×#12011yescharacteristic
246C22#54812yes—
247C4—21no—
248C4—21no—
249C4—44no—
250C4—42no—
251C4—41no—
252C4—41no—
253C4—44no—
254C4—41no—
255C4—82no—
256C4#45412yes—
257C4#45512yes—
258C4#53211yescharacteristic
259C4#53611yescharacteristic
260C4#54712yes—
261C2—22no—
262C2—81no—
263C2—82no—
264C2#243211yescharacteristic
265C2C2×#45511yescharacteristic
266C2C2×#45411yescharacteristic
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. #2862C2semidirect
5. #2040C2semidirect
6. #1997C2semidirect
7. #1962C2semidirect
8. #1860C2semidirect
9. C4×#104C2semidirect
189. 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
#28622
#20402
#19972
#19622
#18602
C4×#1042