Finite Group Databasebeta

← C2×#1597 #18098

Subgroups — C2×#1597 #18098

Order 256.

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

normal subgroups only

This group has 702 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, complemented
2C2×#450C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic
4C2×#394C211yesmaximal, characteristic, complemented
5#1597C218yesmaximal, direct product factor
6C2×#328C211yesmaximal, characteristic, complemented
7C2×#327C211yesmaximal, characteristic, complemented
8C2×#324C211yesmaximal, characteristic, complemented
9C23×D4C2211yescharacteristic
10C22×SD16—21no—
11C22×D8—21no—
12#539—22no—
13#539—22no—
14#539—22no—
15#539C2214yescomplemented
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⋊C4C2211yescharacteristic
25C22×C22⋊C4—21no—
26C22×C22⋊C4—21no—
27#450—22no—
28#450C2214yescomplemented
29#447—22no—
30#447C2214yes—
31C2×#102—21no—
32C2×#102—21no—
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35#394—22no—
36#394C2214yescomplemented
37C2×#95C2211yescharacteristic
38#328C2214yescomplemented
39#327C2214yescomplemented
40#324C2214yescomplemented
41C25—21no—
42C22×Q8D411yescharacteristic
43C22×D4—21no—
44C22×D4—21no—
45C22×D4—21no—
46C22×D4—21no—
47C22×D4—21no—
48C22×D4—21no—
49C22×D4—44no—
50C22×D4D411yescharacteristic
51C22×D4C2312yes—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4D411yescharacteristic
55C23×C4D411yescharacteristic
56C23×C4C2311yescharacteristic
57C2×SD16—24no—
58C2×SD16—44no—
59C2×D8—24no—
60C2×D8—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⋊C4D412yes—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2312yes—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—24no—
102C2×C22⋊C4—44no—
103C2×C22⋊C4—44no—
104C2×C42—21no—
105#102—24no—
106#102—24no—
107C22⋊C8—21no—
108C22⋊C8—21no—
109C22⋊C8C2312yes—
110C22⋊C8C2312yes—
111#95—21no—
112#95C2312yes—
113C24—24no—
114C24—21no—
115C24—22no—
116C24—42no—
117C24—42no—
118C24—42no—
119C24—41no—
120C24—41no—
121C24—42no—
122C24—42no—
123C24C2×D411yescharacteristic
124C2×Q8—21no—
125C2×Q8—42no—
126C2×Q8C2×D412yes—
127C2×D4—24no—
128C2×D4—24no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—21no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—42no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—44no—
138C2×D4—44no—
139C2×D4—44no—
140C2×D4—44no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4C2×D412yes—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—82no—
163C22×C4C2×D412yes—
164C22×C4C2×D411yescharacteristic
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D412yes—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2411yescharacteristic
171SD16—48no—
172D8—48no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—42no—
176C2×C8—42no—
177C4⋊C4—24no—
178C4⋊C4—24no—
179C4⋊C4—24no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—22no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4C2×D412yes—
189C4⋊C4C2×D412yes—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—48no—
198C22⋊C4—48no—
199C42—24no—
200C23—22no—
201C23—22no—
202C23—24no—
203C23—22no—
204C23—42no—
205C23—44no—
206C23—48no—
207C23—48no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—44no—
217C23—84no—
218C23—84no—
219C23—82no—
220C23—82no—
221C23—82no—
222C23—82no—
223C23#12011yescharacteristic
224C23#12011yescharacteristic
225C23#13512yes—
226C23AGL(1,ℤ/8)12yes—
227C23C22×D411yescharacteristic
228Q8—42no—
229Q8—81no—
230D4—24no—
231D4—24no—
232D4—42no—
233D4—42no—
234D4—42no—
235D4—44no—
236D4—44no—
237D4—42no—
238D4—42no—
239D4—44no—
240D4—81no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—84no—
266C2×C4—84no—
267C2×C4#12011yescharacteristic
268C2×C4#12011yescharacteristic
269C2×C4C22×D411yescharacteristic
270C2×C4C22×D411yescharacteristic
271C8—42no—
272C8—42no—
273C22—24no—
274C22—24no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22—44no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—44no—
285C22—82no—
286C22—82no—
287C22—88no—
288C22—84no—
289C22—88no—
290C22—82no—
291C22—82no—
292C22#44912yes—
293C22#44912yes—
294C22#45412yes—
295C22B(4,𝔽2)12yes—
296C22C2×#12011yescharacteristic
297C22C2×#13511yescharacteristic
298C22C2×AGL(1,ℤ/8)11yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—21no—
303C4—42no—
304C4—42no—
305C4—42no—
306C4—42no—
307C4—42no—
308C4—82no—
309C2—21no—
310C2—21no—
311C2—22no—
312C2—82no—
313C2—84no—
314C2#159714yesdirect product factor
315C2C2×#44911yescharacteristic
316C2C2×#45411yescharacteristic
317C2C2×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
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
1. C2×#539C2semidirect
2. C2×#450C2semidirect
4. C2×#394C2semidirect
5. #1597C2direct (N × H)
6. C2×#328C2semidirect
7. C2×#327C2semidirect
8. C2×#324C2semidirect
15. #539C22semidirect
28. #450C22semidirect
36. #394C22semidirect
38. #328C22semidirect
39. #327C22semidirect
40. #324C22semidirect
314. C2#1597direct (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×#4502
C2×#4472
C2×#3942
#15972
C2×#3282
C2×#3272
C2×#3242