Finite Group Databasebeta

← C2×#1615 #18106

Subgroups — C2×#1615 #18106

Order 256.

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

normal subgroups only

This group has 508 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
1C2×#539C211yesmaximal, characteristic
2C2×#484C211yesmaximal, characteristic, complemented
3C2×#481C211yesmaximal, characteristic
4C2×#400C211yesmaximal, characteristic, complemented
5#1615C218yesmaximal, direct product factor
6C2×#328C211yesmaximal, characteristic, complemented
7C2×#327C211yesmaximal, characteristic
8C2×#324C211yesmaximal, characteristic, complemented
9#539—22no—
10#539—22no—
11#539—22no—
12#539C2214yes—
13C2×#126—21no—
14C2×#125—21no—
15C2×#123—21no—
16C2×#123—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4—21no—
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#484—22no—
24#484C2214yescomplemented
25#481—22no—
26#481C2214yes—
27C2×#107—21no—
28C2×#106—21no—
29C2×#103—21no—
30C2×#102—21no—
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33#400—22no—
34#400C2214yescomplemented
35C2×#95—21no—
36C2×#95—21no—
37C2×#95C2211yescharacteristic
38#328C2214yescomplemented
39#327C2214yes—
40#324C2214yescomplemented
41C22×Q8D411yescharacteristic
42C22×D4—21no—
43C22×D4D411yescharacteristic
44C23×C4—21no—
45C23×C4—21no—
46C23×C4—21no—
47C23×C4D411yescharacteristic
48C23×C4D411yescharacteristic
49C23×C4C2311yescharacteristic
50C22×C8—21no—
51C22×C8—21no—
52#126—24no—
53#126—24no—
54#125—24no—
55#125—24no—
56#123—24no—
57#123—24no—
58#123—24no—
59#123—24no—
60C22⋊Q8—24no—
61C22⋊Q8—24no—
62C22⋊Q8—21no—
63C22⋊Q8—21no—
64C22⋊Q8D412yes—
65C22⋊Q8C2312yes—
66#121—21no—
67#121—21no—
68#121D412yes—
69#121C2312yes—
70C4×D4—24no—
71C4×D4—24no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D412yes—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4C2312yes—
89C2×C4⋊C4C2312yes—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—21no—
96#107—24no—
97#106—24no—
98#103—24no—
99#102—24no—
100C22⋊C8—21no—
101C22⋊C8—21no—
102C22⋊C8C2312yes—
103C22⋊C8C2312yes—
104#95—24no—
105#95—24no—
106#95—21no—
107#95C2312yes—
108C24—21no—
109C24C2×D411yescharacteristic
110C2×Q8—21no—
111C2×Q8—42no—
112C2×Q8C2×D412yes—
113C2×D4—22no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4—42no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4C2×D412yes—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—24no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—44no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—42no—
140C22×C4—82no—
141C22×C4C2×D412yes—
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D412yes—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2411yescharacteristic
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—42no—
152C2×C8—42no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4C2×D412yes—
169C4⋊C4C2×D412yes—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—22no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—22no—
177C42—24no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—42no—
182C23—42no—
183C23—42no—
184C23#12011yescharacteristic
185C23#12011yescharacteristic
186C23#13512yes—
187C23#13712yes—
188C23C22×D411yescharacteristic
189Q8—42no—
190Q8—81no—
191D4—42no—
192D4—42no—
193D4—42no—
194D4—44no—
195D4—81no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—44no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—81no—
222C2×C4—84no—
223C2×C4—84no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4#12011yescharacteristic
227C2×C4#12011yescharacteristic
228C2×C4C22×D411yescharacteristic
229C2×C4C22×D411yescharacteristic
230C8—42no—
231C8—42no—
232C22—24no—
233C22—24no—
234C22—21no—
235C22—21no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—84no—
242C22#45212yes—
243C22#45212yes—
244C22#45412yes—
245C22#45812yes—
246C22C2×#12011yescharacteristic
247C22C2×#13511yescharacteristic
248C22C2×#13711yescharacteristic
249C4—21no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—42no—
255C4—42no—
256C4—42no—
257C4—42no—
258C4—82no—
259C4—82no—
260C2—21no—
261C2—21no—
262C2—22no—
263C2—82no—
264C2#161514yesdirect product factor
265C2C2×#45211yescharacteristic
266C2C2×#45411yescharacteristic
267C2C2×#45811yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#484C2semidirect
4. C2×#400C2semidirect
5. #1615C2direct (N × H)
6. C2×#328C2semidirect
8. C2×#324C2semidirect
24. #484C22semidirect
34. #400C22semidirect
38. #328C22semidirect
40. #324C22semidirect
264. C2#1615direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5392
C2×#4842
C2×#4812
C2×#4002
#16152
C2×#3282
C2×#3272
C2×#3242