Finite Group Databasebeta

← #9997

Subgroups — #9997

Order 256.

311 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 311 conjugacy classes of subgroups — too many to draw. Only its 59 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#3100C211yesmaximal, characteristic
2#3037C211yesmaximal, characteristic, complemented
3#2536C211yesmaximal, characteristic, complemented
4#1517C211yesmaximal, characteristic, complemented
5#1486C211yesmaximal, characteristic, complemented
6#1479C211yesmaximal, characteristic, complemented
7#1286C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#532C2211yescharacteristic
10C2×C22⋊Q8—21no—
11C2×C4×Q8C2211yescharacteristic
12C2×C4×D4C2211yescharacteristic
13#480—21no—
14#474—21no—
15#464—22no—
16#462—21no—
17#460—21no—
18#459—21no—
19#447—21no—
20C4×D8—22no—
21#431C2211yescharacteristic
22#431C2211yescharacteristic
23#423C2211yescharacteristic
24#417—21no—
25#398—21no—
26#394—21no—
27#388—22no—
28#382C2211yescharacteristic
29#380—21no—
30#327—22no—
31Q8⋊C8—22no—
32D4⋊C8—22no—
33C2×#41—21no—
34C22×Q8D411yescharacteristic
35C22×D4D411yescharacteristic
36C23×C4—21no—
37#137—42no—
38#137—42no—
39C2×Q16—22no—
40C2×SD16—22no—
41C2×D8—22no—
42C2×M16—21no—
43C4⋊Q8D411yescharacteristic
44#127D411yescharacteristic
45#124—21no—
46C22⋊Q8—42no—
47C22⋊Q8—42no—
48#121—21no—
49#121—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×Q8—21no—
54C4×D4—21no—
55C4×D4—21no—
56C4×D4—21no—
57C4×D4—21no—
58C4×D4—21no—
59C4×D4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4D411yescharacteristic
65C2×C22⋊C4—22no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—21no—
68C2×C42—21no—
69C2×C42C2311yescharacteristic
70#107—22no—
71C4⋊C8—21no—
72C4⋊C8—21no—
73C4⋊C8—21no—
74C4⋊C8D412yes—
75#103—22no—
76#102—22no—
77#102—22no—
78#102—22no—
79C22⋊C8—21no—
80C22⋊C8—21no—
81C4×C8—21no—
82C4×C8D412yes—
83#95—22no—
84#95—21no—
85#95D412yes—
86C24—21no—
87#41—41no—
88#41—41no—
89#41—41no—
90#41—41no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8D812yescomplemented
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—42no—
104C2×D4—41no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4—41no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—41no—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119Q16—44no—
120SD16—44no—
121D8—44no—
122M16—41no—
123M16—41no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—41no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—44no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C22⋊C4—44no—
139C22⋊C4—44no—
140C22⋊C4—41no—
141C22⋊C4—41no—
142C22⋊C4—41no—
143C22⋊C4—41no—
144C42—41no—
145C42—41no—
146C42C2×D411yescharacteristic
147C42C2×D411yescharacteristic
148C42C2×D411yescharacteristic
149C42#4111yescharacteristic
150C23—22no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23#12011yescharacteristic
159Q8—22no—
160Q8—22no—
161Q8—41no—
162Q8—41no—
163Q8—41no—
164Q8—41no—
165D4—22no—
166D4—22no—
167D4—41no—
168D4—41no—
169D4—42no—
170D4—41no—
171D4—41no—
172D4—81no—
173D4—81no—
174D4—81no—
175C2×C4—22no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4#12011yescharacteristic
196C2×C4#12011yescharacteristic
197C2×C4#12111yescharacteristic
198C2×C4#12111yescharacteristic
199C2×C4#12111yescharacteristic
200C2×C4#12711yescharacteristic
201C2×C4C2×D811yescharacteristic
202C2×C4#13511yescharacteristic
203C2×C4AGL(1,ℤ/8)11yescharacteristic
204C2×C4AGL(1,ℤ/8)11yescharacteristic
205C8—42no—
206C8—42no—
207C8—81no—
208C22—21no—
209C22—21no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—81no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22—81no—
220C22#39211yescharacteristic
221C22#44711yescharacteristic
222C22#44911yescharacteristic
223C4—21no—
224C4—21no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—82no—
229C4—81no—
230C4—81no—
231C4—81no—
232C4#44711yescharacteristic
233C4#44911yescharacteristic
234C4#46611yescharacteristic
235C4#46911yescharacteristic
236C2—21no—
237C2—21no—
238C2—82no—
239C2—161no—
240C2#199911yescharacteristic
241C2#200911yescharacteristic
242C2#201011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3100C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3037C2semidirect
3. #2536C2semidirect
4. #1517C2semidirect
5. #1486C2semidirect
6. #1479C2semidirect
7. #1286C2semidirect
97. C2×Q8D8semidirect

Maximal subgroups

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

subgroupindex
#31002
#30372
#25362
#15172
#14862
#14792
#12862