Finite Group Databasebeta

← C2×#1637 #18489

Subgroups — C2×#1637 #18489

Order 256.

587 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×D4×Q8C211yesmaximal, characteristic
2C2×#487C211yesmaximal, characteristic, complemented
3C2×#477C211yesmaximal, characteristic, complemented
4C2×#476C211yesmaximal, characteristic
5#1637C218yesmaximal, direct product factor
6C2×#332C211yesmaximal, characteristic, complemented
7C2×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×D4⋊C8C211yesmaximal, characteristic
9C23×Q8—21no—
10D4×Q8—22no—
11D4×Q8—22no—
12D4×Q8—22no—
13D4×Q8C2214yes—
14C2×C4⋊Q8—21no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8—22no—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic
22#487—22no—
23#487C2214yescomplemented
24#477—22no—
25#477C2214yescomplemented
26#476—22no—
27#476C2214yes—
28C2×#107—21no—
29C2×#106—21no—
30C2×C4⋊C8C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×#103—21no—
33C2×#103—22no—
34C2×#102—21no—
35C2×C4×C8C2211yescharacteristic
36#332C2214yescomplemented
37Q8⋊C8C2214yescomplemented
38D4⋊C8C2214yes—
39C22×Q8—22no—
40C22×Q8—24no—
41C22×Q8—24no—
42C22×Q8—21no—
43C22×Q8—21no—
44C22×Q8—44no—
45C22×Q8—44no—
46C22×Q8D411yescharacteristic
47C22×D4D411yescharacteristic
48C23×C4—22no—
49C23×C4—21no—
50C22×C8—21no—
51C22×C8—21no—
52C22×C8—21no—
53C4⋊Q8—24no—
54C4⋊Q8—24no—
55C4⋊Q8—21no—
56C4⋊Q8—21no—
57C4⋊Q8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8D412yes—
61C4⋊Q8C2312yes—
62C4⋊Q8C2312yes—
63C22⋊Q8—24no—
64C22⋊Q8—28no—
65C22⋊Q8—24no—
66C22⋊Q8—28no—
67C4×Q8—21no—
68C4×Q8—21no—
69C4×Q8D412yes—
70C4×Q8C2312yes—
71C4×D4—24no—
72C4×D4—24no—
73C4×D4—21no—
74C4×D4—21no—
75C4×D4D412yes—
76C4×D4C2312yes—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D412yes—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—21no—
87C2×C42—21no—
88C2×C42C2311yescharacteristic
89#107—24no—
90#106—24no—
91C4⋊C8—21no—
92C4⋊C8—21no—
93C4⋊C8C2312yes—
94C4⋊C8C2312yes—
95#103—24no—
96#103—28no—
97#102—24no—
98C4×C8—21no—
99C4×C8C2312yes—
100C24—21no—
101C2×Q8—28no—
102C2×Q8—21no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—48no—
108C2×Q8—44no—
109C2×Q8—48no—
110C2×Q8—416no—
111C2×Q8—48no—
112C2×Q8—41no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8SD1614yes—
116C2×Q8C2×D412yes—
117C2×D4—21no—
118C2×D4—22no—
119C2×D4Q1614yes—
120C2×D4C2×D412yes—
121C22×C4—21no—
122C22×C4—28no—
123C22×C4—21no—
124C22×C4—22no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—48no—
129C22×C4—42no—
130C22×C4—42no—
131C22×C4—41no—
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—42no—
139C2×C8—42no—
140C2×C8—42no—
141C4⋊C4—28no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—22no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4C2×D414yes—
154C4⋊C4C2×D412yes—
155C4⋊C4C2×D412yes—
156C22⋊C4—28no—
157C22⋊C4—22no—
158C22⋊C4—22no—
159C42—22no—
160C42—22no—
161C42C2×D411yescharacteristic
162C42C2×D411yescharacteristic
163C42C2×D411yescharacteristic
164C42C2411yescharacteristic
165C23—22no—
166C23—24no—
167C23—42no—
168C23—42no—
169C23#12011yescharacteristic
170Q8—22no—
171Q8—24no—
172Q8—416no—
173Q8—44no—
174Q8—42no—
175Q8—42no—
176Q8—44no—
177Q8—82no—
178Q8C2×SD1614yes—
179D4—22no—
180D4—24no—
181D4C2×Q1614yes—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—416no—
189C2×C4—44no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—82no—
197C2×C4#12011yescharacteristic
198C2×C4#12011yescharacteristic
199C2×C4#12011yescharacteristic
200C2×C4C2×SD1612yes—
201C2×C4C2×Q1612yes—
202C2×C4AGL(1,ℤ/8)12yes—
203C2×C4#13712yes—
204C2×C4C22×D411yescharacteristic
205C2×C4C22×D411yescharacteristic
206C2×C4C22×D411yescharacteristic
207C8—42no—
208C8—42no—
209C8—42no—
210C22—24no—
211C22—48no—
212C22—42no—
213C22—42no—
214C22#45012yes—
215C22#45112yes—
216C22#45412yes—
217C22C2×#12011yescharacteristic
218C4—24no—
219C4—21no—
220C4—21no—
221C4—44no—
222C4—42no—
223C4—42no—
224C4—82no—
225C4#45012yes—
226C4#45112yes—
227C4C22×SD1611yescharacteristic
228C4C22×Q1611yescharacteristic
229C4C2×AGL(1,ℤ/8)11yescharacteristic
230C4C2×#13711yescharacteristic
231C2—44no—
232C2#163714yesdirect product factor
233C2C2×#45011yescharacteristic
234C2C2×#45111yescharacteristic
235C2C2×#45411yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC42C24
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#487C2semidirect
3. C2×#477C2semidirect
5. #1637C2direct (N × H)
6. C2×#332C2semidirect
7. C2×Q8⋊C8C2semidirect
23. #487C22semidirect
25. #477C22semidirect
36. #332C22semidirect
37. Q8⋊C8C22semidirect
232. C2#1637direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×Q82
C2×#4872
C2×#4772
C2×#4762
#16372
C2×#3322
C2×Q8⋊C82
C2×D4⋊C82