Finite Group Databasebeta

← #8991

Subgroups — #8991

Order 256.

289 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 73 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
2D4×M16C211yesmaximal, characteristic, complemented
3#1980C211yesmaximal, characteristic, complemented
4C4×#102C211yesmaximal, characteristic, complemented
5#1492C211yesmaximal, characteristic, complemented
6#1303C211yesmaximal, characteristic, complemented
7#1278C211yesmaximal, characteristic, complemented
8C22×M16—21no—
9#532C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#480—21no—
12#474C411yescharacteristic, complemented
13#474C411yescharacteristic, complemented
14#459C411yescharacteristic, complemented
15#459C411yescharacteristic, complemented
16#447—21no—
17C4×D8—21no—
18C4×D8—21no—
19#436—21no—
20#435—22no—
21C8×D4—21no—
22#431C2211yescharacteristic
23#423C2211yescharacteristic
24C2×#104—21no—
25C2×#104—21no—
26C2×#102—21no—
27#407—21no—
28C4×M16C2211yescharacteristic
29C2×C4×C8C2211yescharacteristic
30C4×C4⋊C4C2211yescharacteristic
31#340—21no—
32D4⋊C8—22no—
33C2×#41—21no—
34C22×D4D411yescharacteristic
35C23×C4—21no—
36C2×D8—21no—
37C2×D8—21no—
38C2×M16—22no—
39C2×M16—21no—
40C2×M16—42no—
41C2×M16—42no—
42C2×M16D411yescharacteristic
43C2×M16D411yescharacteristic
44C22×C8—22no—
45C22×C8—21no—
46C4⋊Q8C2×C411yescharacteristic
47#127C2×C411yescharacteristic
48#124—21no—
49#121—21no—
50#121—21no—
51C4×D4—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4C2×C411yescharacteristic
56C4×D4C2×C411yescharacteristic
57C4×D4D411yescharacteristic
58C4×D4D411yescharacteristic
59C2×C4⋊C4—21no—
60C2×C4⋊C4D411yescharacteristic
61C2×C22⋊C4—21no—
62C2×C42—21no—
63C2×C42C2311yescharacteristic
64#107—21no—
65#107—21no—
66C4⋊C8—21no—
67C4⋊C8C2×C411yescharacteristic
68C4⋊C8C2×C411yescharacteristic
69C4⋊C8D412yes—
70#104—42no—
71#104—42no—
72#102—24no—
73#102—21no—
74#102—21no—
75#102—21no—
76#102—21no—
77C22⋊C8—22no—
78C22⋊C8—22no—
79C22⋊C8—21no—
80#97—21no—
81C4×C8—21no—
82C4×C8—21no—
83C4×C8—21no—
84C4×C8—21no—
85#95—21no—
86C24—21no—
87#41—41no—
88#41—41no—
89#41—41no—
90#41—41no—
91C2×Q8—21no—
92C2×D4—22no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—21no—
97C2×D4—41no—
98C2×D4—41no—
99C2×D4C22⋊C411yescharacteristic
100C2×D4C22⋊C411yescharacteristic
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—41no—
111C22×C4—41no—
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4#4111yescharacteristic
115D8—42no—
116D8—42no—
117M16—21no—
118M16—21no—
119M16—21no—
120M16—44no—
121M16—41no—
122M16—41no—
123M16D812yescomplemented
124C2×C8—21no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—22no—
130C2×C8—44no—
131C2×C8—41no—
132C2×C8—41no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—41no—
138C4⋊C4C22⋊C411yescharacteristic
139C4⋊C4C22⋊C411yescharacteristic
140C22⋊C4—21no—
141C22⋊C4—21no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C22⋊C4—41no—
145C22⋊C4—41no—
146C42—41no—
147C42—41no—
148C42C22×C411yescharacteristic
149C42C2×D411yescharacteristic
150C42C2×D411yescharacteristic
151C42#4111yescharacteristic
152C23—21no—
153C23—21no—
154C23—42no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23—81no—
159C23#12111yescharacteristic
160Q8—41no—
161Q8—41no—
162D4—21no—
163D4—21no—
164D4—21no—
165D4—42no—
166D4—42no—
167D4—41no—
168D4—41no—
169D4—41no—
170D4—41no—
171D4—81no—
172D4—81no—
173D4#10412yescomplemented
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4C2×C22⋊C411yescharacteristic
196C2×C4C4×D411yescharacteristic
197C2×C4C4×D411yescharacteristic
198C2×C4#12011yescharacteristic
199C2×C4#12111yescharacteristic
200C2×C4#12311yescharacteristic
201C2×C4C2×D811yescharacteristic
202C2×C4#13511yescharacteristic
203C2×C4AGL(1,ℤ/8)11yescharacteristic
204C2×C4AGL(1,ℤ/8)11yescharacteristic
205C8—22no—
206C8—42no—
207C8—41no—
208C8—41no—
209C8—42no—
210C8—41no—
211C22—21no—
212C22—21no—
213C22—22no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—41no—
219C22—82no—
220C22—82no—
221C22—81no—
222C22—81no—
223C22—81no—
224C22#38611yescharacteristic
225C22#45911yescharacteristic
226C22#46311yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—41no—
231C4—41no—
232C4—82no—
233C4—81no—
234C4C2×#10411yescharacteristic
235C4#42111yescharacteristic
236C4C4×D811yescharacteristic
237C4#44211yescharacteristic
238C4#44711yescharacteristic
239C4#44911yescharacteristic
240C2—21no—
241C2—21no—
242C2—42no—
243C2—81no—
244C2—161no—
245C2#187111yescharacteristic
246C2#187911yescharacteristic
247C2#188111yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#3100C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×M16C2semidirect
3. #1980C2semidirect
4. C4×#102C2semidirect
5. #1492C2semidirect
6. #1303C2semidirect
7. #1278C2semidirect
12. #474C4semidirect
13. #474C4semidirect
14. #459C4semidirect
15. #459C4semidirect
123. M16D8semidirect
173. D4#104semidirect

Maximal subgroups

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

subgroupindex
#31002
D4×M162
#19802
C4×#1022
#14922
#13032
#12782