Finite Group Databasebeta

← C2×#2010 #23413

Subgroups — C2×#2010 #23413

Order 256.

1095 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1095 conjugacy classes of subgroups — too many to draw. Only its 183 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×#575C211yesmaximal, characteristic, complemented
2C2×#525C212yesmaximal, complemented
3C2×#421C212yesmaximal, complemented
4C2×#407C211yesmaximal, characteristic, complemented
5#2010C218yesmaximal, direct product factor
6C2×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
7C22×#41C2212yes—
8C24×C4C2211yescharacteristic
9#575—24no—
10#575—22no—
11#575C2214yescomplemented
12C2×#137—22no—
13C2×AGL(1,ℤ/8)—22no—
14C2×#135—22no—
15C22×M16C2211yescharacteristic
16C22×M16C2211yescharacteristic
17#525—24no—
18#525—24no—
19#525—24no—
20#525C2218yescomplemented
21C2×#123—22no—
22C2×C22⋊Q8—22no—
23C2×#121—22no—
24C2×#120—22no—
25C2×C4×D4—24no—
26C2×#117C2212yes—
27#421—24no—
28#421C2218yescomplemented
29C2×#104—24no—
30#407—22no—
31#407C2214yescomplemented
32C2×C22⋊C8—22no—
33H(3,ℤ/4)C2214yescomplemented
34C25D411yescharacteristic
35C2×#41—22no—
36C2×#41—22no—
37C2×#41—22no—
38C2×#41—22no—
39C2×#41—22no—
40C2×#41—22no—
41C2×#41—22no—
42C2×#41—22no—
43C2×#41—22no—
44C2×#41—22no—
45C2×#41D414yescomplemented
46C2×#41C2314yes—
47C22×Q8D412yescomplemented
48C22×D4—24no—
49C22×D4—22no—
50C22×D4D412yescomplemented
51C23×C4—22no—
52C23×C4—22no—
53C23×C4—22no—
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4—21no—
60C23×C4D411yescharacteristic
61C23×C4D412yes—
62C23×C4C2312yes—
63C23×C4C2311yescharacteristic
64#137—28no—
65#137—28no—
66AGL(1,ℤ/8)—28no—
67AGL(1,ℤ/8)—28no—
68#135—28no—
69#135—28no—
70C2×Q16—22no—
71C2×SD16—24no—
72C2×D8—22no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—21no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16D412yescomplemented
79C2×M16C2312yes—
80C2×M16C2312yes—
81C22×C8—21no—
82C22×C8D412yescomplemented
83#123—28no—
84#123—28no—
85C22⋊Q8—28no—
86C22⋊Q8—28no—
87#121—28no—
88#121—28no—
89#120—28no—
90#120—28no—
91C4×D4—216no—
92C4×D4—216no—
93#117—22no—
94#117—22no—
95#117D414yescomplemented
96#117C2314yes—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—22no—
99C2×C22⋊C4—24no—
100C2×C22⋊C4—24no—
101C2×C22⋊C4—22no—
102C2×C42D414yescomplemented
103#104—216no—
104C22⋊C8—28no—
105C24—22no—
106C24—21no—
107C24—22no—
108C24—22no—
109C24—21no—
110C24—44no—
111C24C2×D412yes—
112C24C2×D411yescharacteristic
113#41—24no—
114#41—24no—
115#41—24no—
116#41—24no—
117#41—24no—
118#41—24no—
119#41—24no—
120#41—24no—
121#41—42no—
122#41—42no—
123#41—42no—
124#41—42no—
125#41—42no—
126#41—42no—
127#41—42no—
128#41—42no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8C2×D414yescomplemented
135C2×D4—216no—
136C2×D4—24no—
137C2×D4—24no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—416no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4C2×D414yescomplemented
149C22×C4—24no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—24no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4—24no—
164C22×C4—22no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4C2×D412yes—
180C22×C4C2×D411yescharacteristic
181C22×C4C2×D412yes—
182C22×C4C2×D412yes—
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D412yes—
186C22×C4C2×D412yes—
187C22×C4#4111yescharacteristic
188C22×C4#4111yescharacteristic
189C22×C4C2411yescharacteristic
190Q16—28no—
191SD16—216no—
192D8—28no—
193M16—24no—
194M16—24no—
195M16—24no—
196M16—42no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—22no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8C2×D414yescomplemented
203C4⋊C4—216no—
204C4⋊C4—24no—
205C4⋊C4—24no—
206C22⋊C4—216no—
207C22⋊C4—216no—
208C22⋊C4—24no—
209C22⋊C4—24no—
210C42—24no—
211C42C2×D418yescomplemented
212C23—28no—
213C23—24no—
214C23—24no—
215C23—22no—
216C23—24no—
217C23—42no—
218C23—44no—
219C23—44no—
220C23—42no—
221C23—48no—
222C23—44no—
223C23—48no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23—42no—
228C23—42no—
229C23#12012yes—
230C23#12012yes—
231C23#12111yescharacteristic
232C23#12111yescharacteristic
233C23C22×D411yescharacteristic
234Q8—24no—
235Q8—24no—
236Q8—42no—
237Q8—42no—
238D4—24no—
239D4—24no—
240D4—432no—
241D4—44no—
242D4—44no—
243D4—44no—
244D4—44no—
245D4—42no—
246D4—42no—
247C2×C4—28no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—48no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—44no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4#12011yescharacteristic
268C2×C4#12011yescharacteristic
269C2×C4#12012yes—
270C2×C4#12012yes—
271C2×C4#12012yes—
272C2×C4#12112yes—
273C2×C4#12112yes—
274C2×C4#12112yes—
275C2×C4#12112yes—
276C2×C4#12111yescharacteristic
277C2×C4#12111yescharacteristic
278C2×C4#12711yescharacteristic
279C2×C4#12712yes—
280C2×C4#12711yescharacteristic
281C2×C4C22×D412yes—
282C2×C4C22×D411yescharacteristic
283C2×C4C22×D412yes—
284C2×C4C2×#4111yescharacteristic
285C8—24no—
286C8—42no—
287C22—24no—
288C22—22no—
289C22—28no—
290C22—24no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—44no—
295C22—42no—
296C22—48no—
297C22—44no—
298C22—48no—
299C22—42no—
300C22—44no—
301C22—44no—
302C22—84no—
303C22—84no—
304C22#39212yes—
305C22#39211yescharacteristic
306C22#39211yescharacteristic
307C22C2×#12012yes—
308C22C2×#12111yescharacteristic
309C4—48no—
310C4—42no—
311C4—44no—
312C4#39211yescharacteristic
313C4#39212yes—
314C4#39211yescharacteristic
315C4C2×#12011yescharacteristic
316C4C2×#12112yes—
317C4C2×#12711yescharacteristic
318C2—22no—
319C2—21no—
320C2—22no—
321C2—21no—
322C2—44no—
323C2—84no—
324C2#201012yesdirect product factor
325C2C2×#39211yescharacteristic
subgroupG/H
centerC2×C4#127
commutator, Frattini, cosocleC22×C4C24
socleC22#392
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#575C2semidirect
2. C2×#525C2semidirect
3. C2×#421C2semidirect
4. C2×#407C2semidirect
5. #2010C2direct (N × H)
6. C2×H(3,ℤ/4)C2semidirect
11. #575C22semidirect
20. #525C22semidirect
28. #421C22semidirect
31. #407C22semidirect
33. H(3,ℤ/4)C22semidirect
45. C2×#41D4semidirect
47. C22×Q8D4semidirect
50. C22×D4D4semidirect
78. C2×M16D4semidirect
82. C22×C8D4semidirect
95. #117D4semidirect
102. C2×C42D4semidirect
134. C2×Q8C2×D4semidirect
148. C2×D4C2×D4semidirect
202. C2×C8C2×D4semidirect
211. C42C2×D4semidirect
324. C2#2010direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5752
C2×#5252
C2×#4212
C2×#4072
#20102
C2×H(3,ℤ/4)2