Finite Group Databasebeta

← C2×#2031 #23471

Subgroups — C2×#2031 #23471

Order 256.

617 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#533C211yesmaximal, characteristic, complemented
2C2×#421C211yesmaximal, characteristic
3C22×#104C211yesmaximal, characteristic
4C2×#410C211yesmaximal, characteristic
5#2031C218yesmaximal, direct product factor
6C2×#344C211yesmaximal, characteristic, complemented
7C2×#339C211yesmaximal, characteristic, complemented
8C2×H(3,ℤ/4)C211yesmaximal, characteristic, complemented
9C22×#41C2211yescharacteristic
10C22×M16C2211yescharacteristic
11C22×M16C2211yescharacteristic
12#533—22no—
13#533—22no—
14#533—22no—
15#533C2214yescomplemented
16C2×C4⋊Q8—21no—
17C2×#125—21no—
18C2×C22⋊Q8—22no—
19C2×C4×Q8—22no—
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C22×C42C2211yescharacteristic
24#421—22no—
25#421C2214yes—
26C2×#104—21no—
27C2×#104—21no—
28C2×#104—22no—
29C2×#104—22no—
30C2×#104C2214yes—
31#410—22no—
32#410C2214yes—
33C2×#99—22no—
34#344C2214yescomplemented
35#339C2214yescomplemented
36H(3,ℤ/4)C2214yescomplemented
37C2×#41—21no—
38C2×#41—21no—
39C2×#41—21no—
40C2×#41—21no—
41C2×#41—21no—
42C2×#41—21no—
43C2×#41—21no—
44C2×#41—21no—
45C2×#41—21no—
46C2×#41—21no—
47C2×#41Q812yescomplemented
48C2×#41C2312yes—
49C22×Q8—22no—
50C22×Q8Q811yescharacteristic, complemented
51C22×D4—21no—
52C22×D4Q811yescharacteristic, complemented
53C23×C4—21no—
54C23×C4—21no—
55C23×C4C2311yescharacteristic
56C2×M16—21no—
57C2×M16—21no—
58C2×M16—21no—
59C2×M16—21no—
60C2×M16—21no—
61C2×M16—21no—
62C2×M16—21no—
63C2×M16—21no—
64C2×M16—21no—
65C2×M16—21no—
66C2×M16C2312yes—
67C2×M16C2312yes—
68C22×C8—21no—
69C22×C8—21no—
70C4⋊Q8—24no—
71C4⋊Q8—24no—
72#125—24no—
73#125—24no—
74C22⋊Q8—28no—
75C22⋊Q8—28no—
76C4×Q8—28no—
77C4×Q8—28no—
78#117—21no—
79#117—21no—
80#117—21no—
81#117—21no—
82#117—21no—
83#117—21no—
84#117—21no—
85#117D412yes—
86#117D412yes—
87#117C2312yes—
88#117C2312yes—
89#117C2312yes—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4D412yes—
98C2×C42—21no—
99C2×C42—21no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42D412yes—
104C2×C42D411yescharacteristic
105C2×C42D411yescharacteristic
106C2×C42D412yes—
107C2×C42C2312yes—
108#104—24no—
109#104—24no—
110#104—28no—
111#99—28no—
112C24—21no—
113C24C2×D411yescharacteristic
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118#41—22no—
119#41—22no—
120#41—22no—
121#41—22no—
122#41—41no—
123#41—41no—
124#41—41no—
125#41—41no—
126#41—41no—
127#41—41no—
128#41—41no—
129#41—41no—
130C2×Q8—28no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—48no—
137C2×Q8C2×Q812yescomplemented
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—41no—
146C2×D4—41no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4C2×Q812yescomplemented
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—21no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—44no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×Q811yescharacteristic
171C22×C4C2×Q811yescharacteristic
172C22×C4#4111yescharacteristic
173C22×C4#4111yescharacteristic
174C22×C4#4111yescharacteristic
175C22×C4#4111yescharacteristic
176C22×C4#4111yescharacteristic
177C22×C4#4111yescharacteristic
178C22×C4C2411yescharacteristic
179M16—22no—
180M16—22no—
181M16—22no—
182M16—22no—
183M16—41no—
184M16—41no—
185M16—41no—
186M16—41no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C4⋊C4—216no—
194C4⋊C4—28no—
195C4⋊C4—22no—
196C4⋊C4—22no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—41no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—22no—
204C22⋊C4—22no—
205C22⋊C4—41no—
206C22⋊C4C2×D414yes—
207C42—22no—
208C42—21no—
209C42—21no—
210C42—22no—
211C42—22no—
212C42—22no—
213C42—22no—
214C42—22no—
215C42—41no—
216C42C2×D412yes—
217C42C2×D412yes—
218C42C2×D414yes—
219C23—22no—
220C23—42no—
221C23—41no—
222C23—41no—
223C23—82no—
224C23#12011yescharacteristic
225C23#12011yescharacteristic
226C23C22⋊Q811yescharacteristic
227C23C22⋊Q811yescharacteristic
228C23#12411yescharacteristic
229C23#12411yescharacteristic
230C23C22×D411yescharacteristic
231Q8—22no—
232Q8—22no—
233Q8—416no—
234Q8—41no—
235Q8—41no—
236D4—22no—
237D4—22no—
238D4—42no—
239D4—42no—
240D4—42no—
241D4—42no—
242D4—41no—
243D4—41no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—44no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4#12011yescharacteristic
264C2×C4#12011yescharacteristic
265C2×C4C22⋊Q811yescharacteristic
266C2×C4C22⋊Q811yescharacteristic
267C2×C4C22⋊Q811yescharacteristic
268C2×C4C22⋊Q811yescharacteristic
269C2×C4C22⋊Q811yescharacteristic
270C2×C4C22⋊Q811yescharacteristic
271C2×C4C22⋊Q811yescharacteristic
272C2×C4C22⋊Q811yescharacteristic
273C2×C4C22⋊Q811yescharacteristic
274C2×C4C22⋊Q811yescharacteristic
275C2×C4#12411yescharacteristic
276C2×C4#12411yescharacteristic
277C2×C4#12411yescharacteristic
278C2×C4#12411yescharacteristic
279C2×C4#12411yescharacteristic
280C2×C4#12411yescharacteristic
281C2×C4#12411yescharacteristic
282C2×C4#12411yescharacteristic
283C2×C4#12411yescharacteristic
284C2×C4#12411yescharacteristic
285C2×C4C22×D411yescharacteristic
286C2×C4C22×D411yescharacteristic
287C2×C4C22×Q811yescharacteristic
288C2×C4C2×#4111yescharacteristic
289C2×C4C2×#4111yescharacteristic
290C2×C4C2×#4111yescharacteristic
291C8—42no—
292C8—42no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—42no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22#39311yescharacteristic
305C22#39311yescharacteristic
306C22#39311yescharacteristic
307C22#39311yescharacteristic
308C22C2×#12011yescharacteristic
309C22C2×C22⋊Q811yescharacteristic
310C22C2×#12411yescharacteristic
311C4—24no—
312C4—42no—
313C4—44no—
314C4—44no—
315C4—42no—
316C4—82no—
317C4#39311yescharacteristic
318C4#39311yescharacteristic
319C4#39311yescharacteristic
320C4#39311yescharacteristic
321C4C2×C22⋊Q811yescharacteristic
322C4C2×C22⋊Q811yescharacteristic
323C4C2×#12411yescharacteristic
324C4C2×#12411yescharacteristic
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—82no—
332C2#203112yesdirect product factor
333C2C2×#39311yescharacteristic
subgroupG/H
centerC2×C4#124
commutator, Frattini, cosocleC22×C4C24
socleC22#393
prime radicalC22×#41C22
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×#533C2semidirect
5. #2031C2direct (N × H)
6. C2×#344C2semidirect
7. C2×#339C2semidirect
8. C2×H(3,ℤ/4)C2semidirect
15. #533C22semidirect
34. #344C22semidirect
35. #339C22semidirect
36. H(3,ℤ/4)C22semidirect
47. C2×#41Q8semidirect
50. C22×Q8Q8semidirect
52. C22×D4Q8semidirect
137. C2×Q8C2×Q8semidirect
149. C2×D4C2×Q8semidirect
332. C2#2031direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4212
C22×#1042
C2×#4102
#20312
C2×#3442
C2×#3392
C2×H(3,ℤ/4)2