Finite Group Databasebeta

← C2×#1603 #18099

Subgroups — C2×#1603 #18099

Order 256.

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

normal subgroups only

This group has 616 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×#451C211yesmaximal, characteristic, complemented
3C2×#448C211yesmaximal, characteristic
4C2×#395C211yesmaximal, characteristic, complemented
5#1603C218yesmaximal, direct product factor
6C2×#328C211yesmaximal, characteristic, complemented
7C2×#327C211yesmaximal, characteristic
8C2×#324C211yesmaximal, characteristic, complemented
9C23×Q8C2211yescharacteristic
10C22×Q16—21no—
11C22×SD16—21no—
12#539—22no—
13#539—22no—
14#539—22no—
15#539C2214yes—
16C2×#126—21no—
17C2×#125—21no—
18C2×#123—21no—
19C2×#123—21no—
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×D4—21no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2211yescharacteristic
26#451—22no—
27#451C2214yescomplemented
28#448—22no—
29#448C2214yes—
30C2×#103—21no—
31C2×#103—21no—
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34#395—22no—
35#395C2214yescomplemented
36C2×#95—21no—
37C2×#95C2211yescharacteristic
38#328C2214yescomplemented
39#327C2214yes—
40#324C2214yescomplemented
41C22×Q8—21no—
42C22×Q8—21no—
43C22×Q8—21no—
44C22×Q8—21no—
45C22×Q8—21no—
46C22×Q8—44no—
47C22×Q8D411yescharacteristic
48C22×Q8C2312yes—
49C22×D4—21no—
50C22×D4D411yescharacteristic
51C23×C4—21no—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4D411yescharacteristic
55C23×C4D411yescharacteristic
56C23×C4C2311yescharacteristic
57C2×Q16—24no—
58C2×Q16—44no—
59C2×SD16—24no—
60C2×SD16—44no—
61C22×C8—21no—
62C22×C8—21no—
63#126—24no—
64#126—24no—
65#125—24no—
66#125—24no—
67#123—24no—
68#123—24no—
69#123—24no—
70#123—24no—
71C22⋊Q8—24no—
72C22⋊Q8—24no—
73C22⋊Q8—21no—
74C22⋊Q8—21no—
75C22⋊Q8D412yes—
76C22⋊Q8C2312yes—
77#121—21no—
78#121—21no—
79#121D412yes—
80#121C2312yes—
81C4×D4—24no—
82C4×D4—24no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—24no—
92C2×C4⋊C4—44no—
93C2×C4⋊C4D412yes—
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4C2312yes—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C42—21no—
103#103—24no—
104#103—24no—
105C22⋊C8—21no—
106C22⋊C8—21no—
107C22⋊C8C2312yes—
108C22⋊C8C2312yes—
109#95—24no—
110#95—21no—
111#95C2312yes—
112C24—21no—
113C24C2×D411yescharacteristic
114C2×Q8—21no—
115C2×Q8—24no—
116C2×Q8—24no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—44no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—44no—
123C2×Q8—42no—
124C2×Q8—44no—
125C2×Q8—44no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8C2×D412yes—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—42no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4C2×D412yes—
136C22×C4—24no—
137C22×C4—24no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—24no—
149C22×C4—21no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—42no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4—82no—
158C22×C4C2×D412yes—
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D412yes—
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2411yescharacteristic
166Q16—48no—
167SD16—48no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—42no—
171C2×C8—42no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—48no—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D412yes—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—22no—
193C42—24no—
194C23—22no—
195C23—22no—
196C23—22no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23#12011yescharacteristic
201C23#12011yescharacteristic
202C23#13512yes—
203C23#13712yes—
204C23C22×D411yescharacteristic
205Q8—24no—
206Q8—24no—
207Q8—42no—
208Q8—42no—
209Q8—42no—
210Q8—44no—
211Q8—44no—
212Q8—81no—
213D4—42no—
214D4—42no—
215D4—42no—
216D4—44no—
217D4—81no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—24no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—42no—
234C2×C4—48no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—84no—
249C2×C4#12011yescharacteristic
250C2×C4#12011yescharacteristic
251C2×C4C22×D411yescharacteristic
252C2×C4C22×D411yescharacteristic
253C8—42no—
254C8—42no—
255C22—24no—
256C22—24no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—84no—
265C22#45212yes—
266C22#45212yes—
267C22#45612yes—
268C22B(4,𝔽2)12yes—
269C22C2×#12011yescharacteristic
270C22C2×#13511yescharacteristic
271C22C2×#13711yescharacteristic
272C4—21no—
273C4—21no—
274C4—21no—
275C4—21no—
276C4—44no—
277C4—42no—
278C4—42no—
279C4—42no—
280C4—44no—
281C4—42no—
282C4—42no—
283C2—21no—
284C2—21no—
285C2—22no—
286C2—82no—
287C2#160314yesdirect product factor
288C2C2×#45211yescharacteristic
289C2C2×#45611yescharacteristic
290C2C2×B(4,𝔽2)11yescharacteristic
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×#451C2semidirect
4. C2×#395C2semidirect
5. #1603C2direct (N × H)
6. C2×#328C2semidirect
8. C2×#324C2semidirect
27. #451C22semidirect
35. #395C22semidirect
38. #328C22semidirect
40. #324C22semidirect
287. C2#1603direct (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×#4512
C2×#4482
C2×#3952
#16032
C2×#3282
C2×#3272
C2×#3242