Finite Group Databasebeta

← #18647

Subgroups — #18647

Order 256.

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

normal subgroups only

This group has 464 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#3511C211yesmaximal, characteristic
2#3227C211yesmaximal, characteristic, complemented
3C2×#487C211yesmaximal, characteristic, complemented
4#3079C211yesmaximal, characteristic, complemented
5#1638C214yesmaximal, complemented
6#1635C214yesmaximal, complemented
7#1549C211yesmaximal, characteristic, complemented
8#1492C211yesmaximal, characteristic, complemented
9#1476C211yesmaximal, characteristic, complemented
10#556C2212yes—
11#555—21no—
12#554—21no—
13#553—22no—
14#548—22no—
15#547C2212yes—
16#542—22no—
17#537—21no—
18#533—21no—
19#532—21no—
20#532C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23#520—21no—
24C4×#41C2211yescharacteristic
25#498—21no—
26#487—22no—
27#487C2214yes—
28#486—21no—
29#481—22no—
30#477C2212yes—
31#476C2212yes—
32#476C2214yes—
33#475—21no—
34#474—21no—
35#431C2211yescharacteristic
36#428—21no—
37#423C2211yescharacteristic
38#417—21no—
39C2×#103—22no—
40C2×C4×C8C2211yescharacteristic
41#332C2214yes—
42Q8⋊C8C2212yes—
43D4⋊C8C2212yes—
44D4⋊C8C2214yes—
45C2×#41—22no—
46C2×#41—21no—
47C2×#41D411yescharacteristic
48C22×Q8—21no—
49C22×Q8—21no—
50C2×M16—21no—
51C22×C8—21no—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8—21no—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8D412yes—
58C4⋊Q8D411yescharacteristic
59C4⋊Q8C2312yes—
60C4⋊Q8C2312yes—
61#127D411yescharacteristic
62#126—24no—
63#125—22no—
64#124—24no—
65#124—22no—
66#124—21no—
67#123—24no—
68C22⋊Q8—22no—
69C22⋊Q8—24no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72#121—22no—
73#121—22no—
74#121—24no—
75#121—22no—
76#121—21no—
77#121D412yes—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×Q8D411yescharacteristic
81C4×Q8C2311yescharacteristic
82C4×D4—22no—
83C4×D4—24no—
84C4×D4—22no—
85C4×D4—21no—
86C4×D4—21no—
87C4×D4—21no—
88C4×D4D411yescharacteristic
89C4×D4C2311yescharacteristic
90C4×D4C2312yes—
91#117—22no—
92#117—22no—
93#117—21no—
94#117D411yescharacteristic
95C2×C4⋊C4—21no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4D412yes—
100C2×C42—21no—
101C2×C42C2311yescharacteristic
102#107—22no—
103#106—22no—
104#106—24no—
105C4⋊C8—21no—
106C4⋊C8C2312yes—
107C4⋊C8C2312yes—
108#103—22no—
109#103—28no—
110#102—22no—
111#102—24no—
112C22⋊C8—21no—
113C4×C8—21no—
114C4×C8—21no—
115C4×C8C2312yes—
116#41—22no—
117#41—22no—
118#41—42no—
119#41—42no—
120#41—84no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—44no—
126C2×Q8—41no—
127C2×Q8—82no—
128C2×Q8C2×D411yescharacteristic
129C2×D4—21no—
130C2×D4—22no—
131C2×D4—24no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4C2×D411yescharacteristic
135C2×D4C2×D412yes—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—41no—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146M16—42no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—42no—
151C4⋊C4—24no—
152C4⋊C4—24no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—22no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—41no—
162C4⋊C4—41no—
163C4⋊C4C2×D414yes—
164C4⋊C4C2×D411yescharacteristic
165C4⋊C4C2×D412yes—
166C4⋊C4C2×D411yescharacteristic
167C22⋊C4—24no—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—22no—
173C22⋊C4—21no—
174C22⋊C4—22no—
175C42—24no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42C2×D411yescharacteristic
180C42C2×D411yescharacteristic
181C42C2×D411yescharacteristic
182C42C2411yescharacteristic
183C23—21no—
184C23—22no—
185C23#12011yescharacteristic
186Q8—21no—
187Q8—21no—
188Q8—44no—
189Q8—44no—
190Q8—42no—
191Q8—82no—
192Q8—82no—
193D4—21no—
194D4—21no—
195D4—24no—
196D4—42no—
197D4—44no—
198D4—42no—
199D4—42no—
200D4—84no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—24no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—82no—
215C2×C4—84no—
216C2×C4—82no—
217C2×C4#12011yescharacteristic
218C2×C4#12011yescharacteristic
219C2×C4#12011yescharacteristic
220C2×C4#13512yes—
221C2×C4#13712yes—
222C2×C4C22×D411yescharacteristic
223C2×C4C22×D411yescharacteristic
224C2×C4C22×D411yescharacteristic
225C8—42no—
226C8—42no—
227C8—42no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—42no—
232C22—41no—
233C22—42no—
234C22—41no—
235C22—42no—
236C22#45212yes—
237C22C2×#12011yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—42no—
242C4—41no—
243C4—44no—
244C4—41no—
245C4—82no—
246C4#45212yes—
247C4#45412yes—
248C4C2×#13511yescharacteristic
249C4C2×#13711yescharacteristic
250C4#57712yes—
251C2—22no—
252C2—81no—
253C2—82no—
254C2C2×#45211yescharacteristic
255C2#300911yescharacteristic
256C2C2×#45411yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radical#3511C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3227C2semidirect
3. C2×#487C2semidirect
4. #3079C2semidirect
5. #1638C2semidirect
6. #1635C2semidirect
7. #1549C2semidirect
8. #1492C2semidirect
9. #1476C2semidirect

Maximal subgroups

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

subgroupindex
#35112
#32272
C2×#4872
#30792
#16382
#16352
#15492
#14922
#14762